# Extensions of curves with high degree
with respect to the genus

Ciro Ciliberto and Thomas Dedieu

###### Abstract

We classify linearly normal surfaces $S\subseteq\mathbf{P}^{r+1}$ of degree
$d$ such that $4g-4\leqslant d\leqslant 4g+4$, where $g>1$ is the sectional
genus (it is a classical result that for larger $d$ there are only cones). We
apply this to the study of the extension theory of pluricanonical curves and
genus $3$ curves, whenever they verify Property $N_{2}$, using and slightly
expanding the theory of integration of ribbons of the authors and E. Sernesi.
We compute the corank of the relevant Gaussian maps, and we show that all
ribbons over such curves are integrable, and thus there exists a universal
extension.

We carry out a similar program for linearly normal hyperelliptic curves of
degree $d\geqslant 2g+3$. We classify surfaces having such a curve $C$ as a
hyperplane section, compute the corank of the relevant Gaussian maps, and
prove that all ribbons over $C$ are integrable if and only if $d=2g+3$. In the
latter case we obtain the existence of a universal extension.

###### Contents

  1. 1 Introduction
  2. 2 General preliminary results
    1. 2.1 Results on projective curves
    2. 2.2 Scrolls as extensions of linearly normal curves
    3. 2.3 The Hartshorne Bound
  3. 3 Classification of extensions with high degree
    1. 3.1 Detailed description of the items in the classification
    2. 3.2 Previously known results
    3. 3.3 Setup of the Proof of Theorem 1.2
    4. 3.4 The irregular case
    5. 3.5 The rational case
  4. 4 Classification of surfaces with a hyperelliptic section
    1. 4.1 Proof of Theorem 1.5 in the general case
    2. 4.2 Proof of Theorem 1.5 in the sporadic cases
    3. 4.3 The Castelnuovo Classification
  5. 5 Gaussian maps and their cokernels
  6. 6 Ribbons and extensions
  7. 7 Extensions of polarized genus three curves
  8. 8 Extensions of polarized hyperelliptic curves
  9. 9 Extensions of pluricanonical curves
    1. 9.1 Corank of the Gaussian map
    2. 9.2 Surface extensions
    3. 9.3 Universal extensions

## Introduction

In this article we study the extensions of certain non-special curves of genus
$g\geqslant 2$. We shall consider smooth, irreducible, linearly normal
projective curves $C$ of genus $g\geqslant 2$ in $\mathbf{P}^{r}$, and of
degree $d$; it will always be the case that $d>2g-2$, hence $r=d-g$. The
_extensions_ we want to study are surfaces $S\subseteq\mathbf{P}^{r+1}$ having
$C$ as a hyperplane section, and more generally $(c+1)$-dimensional varieties
$X\subseteq\mathbf{P}^{r+c}$ having the curve $C$ as a section by a linear
space. An extension of $C$ is _non-trivial_ if it is not a cone. In this
article we will be interested in non-trivial extensions.

A classical theorem by C. Segre says that if a surface extension of $C$ is a
scroll, then it is actually a cone over $C$. We shall give this theorem a
modern proof in the present text, see [39, §2] for the original proof. On the
other hand, a theorem by Hartshorne [25, Thm. (4.1)] says that if $d>4g+4$,
then $S$ is a scroll; together with the previous result by C. Segre, this
implies that it is a cone. The upshot is that for our study, we can assume
without loss of generality that $d\leqslant 4g+4$. Our first result is the
classification of surface extensions of a curve as above in the range
$4g-4\leqslant d\leqslant 4g+4$.

The following notation will be used throughout the text.

######  Notation 1.1.

For all $e\in\mathbf{N}$, we let $\mathbf{F}_{e}$ be the rational ruled
surface
$\mathbf{P}(\mathcal{O}_{\mathbf{P}^{1}}\oplus\mathcal{O}_{\mathbf{P}^{1}}(-e))$,
and we denote by $E$ the section with self-intersection $-e$ (in case $e=0$,
this has to be taken with a grain of salt), by $F$ the class of the fibres,
and set $H=E+eF$.

######  Theorem 1.2.

Let $S\subseteq\mathbf{P}^{r+1}$ be a nondegenerate irreducible, projective
surface of degree $d\geqslant 4g-4$, and whose general hyperplane section $C$
is smooth, of genus $g\geqslant 2$, and linearly normal. If $S$ is not a cone,
then one of the following holds:

  1. (a)

$S$ is the image by the Veronese map $v_{2}$ of a cone over an elliptic normal
curve of degree $g-1$, and the hyperplane sections of $S$ are bielliptic
bicanonical curves, as in Example 3.1;

  2. (b)

$S$ is a rational surface represented by a linear system of smooth plane
$\delta$-ics, $4\leqslant\delta\leqslant 6$, as in Example 3.2;

  3. (c)

$S$ is the image by the Veronese map $v_{2}$ of a Del Pezzo surface, as in
Example 3.3;

  4. (d)

$S$ is a rational surface with hyperelliptic sections, represented by a linear
subsystem of $|2H+(g+1-e)F|$ on $\mathbf{F}_{e}$, as in Example 3.4;

  5. (e)

$S$ is a rational surface with trigonal sections, represented by a linear
subsystem of $|3H+\frac{1}{2}(g-3e+2)F|$ on $\mathbf{F}_{e}$, and $g\leqslant
10$, as in Example 3.5.

In the above statement, linear (sub)systems (in Cases (b), (d), and (e)) are
defined by simple base points, possibly infinitely near but along a
curvilinear scheme. Our proof of Theorem 1.2 is based on a careful analysis of
the adjoint system of $C$ on $S$.

On the other hand, if $C\subseteq\mathbf{P}^{r}$ enjoys Property $N_{2}$ (see
Theorem 2.2), then its extension theory is governed by ribbons. The latter are
nonreduced schemes supported on $C$, that are potential first-infinitesimal
neighbourhoods of $C$ in an extension $S\subseteq\mathbf{P}^{r+1}$. The
salient points of the theory, which we recall from [16] and slightly expand in
Section 6, are the following: (i) isomorphism classes of non-trivial ribbons
are parametrized by the projective space
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$, where
$L=\left.\mathcal{O}_{\mathbf{P}^{r}}(1)\right|_{C}$, and $\gamma_{C,L}$ is a
Gaussian map, the definition of which is recalled in (5.1); (ii) each ribbon
may be the first-infinitesimal neighbourhood of $C$ in at most one extension,
in particular $C$ may have a non-trivial extension only if $\gamma_{C,L}$ is
not surjective; (iii) if all ribbons in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ can be
realized as first-infinitesimal neighbourhoods of $C$ in an extension, then
there exists a _universal extension_ of $C$, i.e., a $(c+1)$-dimensional
variety $X\subseteq\mathbf{P}^{r+c}$, $c=\operatorname{cork}(\gamma_{C,L})$,
having $C$ as a curve section, and such that all surface extensions of $C$ are
realized in a unique way as a section of $X$ by some $(r+1)$-dimensional
linear space containing $C$.

We shall apply our Classification Theorem 1.2 to study the existence of non-
trivial extensions of polarized curves and, in favourable cases, prove the
existence of universal extensions, as follows. First, we have found some
situations in which we can compute the dimension of
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$, which in
general is a very difficult task. Then our idea is to consider the family of
all possible extensions of the curve $C$, using our Classification Theorem,
which gives the dimension of the locus in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ of those
ribbons corresponding to an actual extension of $(C,L)$ by the Unicity
Property (ii) above. When the two dimensions match, we can conclude that there
exists a universal extension by the general Theorem 6.8.

The dimension of the space
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ is
$\operatorname{cork}(\gamma_{C,L})-1$, where
$\operatorname{cork}(\gamma_{C,L})$ denotes the corank of the map
$\gamma_{C,L}$, i.e., the codimension of its image in $H^{0}(C,2K_{C}+L)$, see
(5.1). We compute it in the following situations.

######  Theorem 1.3.

Let $C$ be a smooth projective curve of genus $g\geqslant 2$, and $L$ a line
bundle on $C$ of degree $d$.

  1. (a)

Assume $C$ is hyperelliptic; if either $d\geqslant 2g+3$ or $d\geqslant g+4$
and $L$ is general, then $\operatorname{cork}(\gamma_{C,L})=2g+2$.

  2. (b)

Assume $g=3$ and $C$ is non-hyperelliptic; if either $d\geqslant 2g=6$, or
$d\geqslant g+1=4$ and $L$ is general, then
$\operatorname{cork}(\gamma_{C,L})=h^{0}(C,4K_{C}-L)$.

In fact the above result is an application of Proposition 5.5, which enables
one to compute the corank of $\gamma_{C,L}$ in virtually any situation,
provided $C$ either is hyperelliptic or has genus $3$. For the genus $3$ case,
we essentially give another proof to an earlier result by Knutsen and Lopez
[29, Prop.2.9(a)]. The following on the other hand is essentially a
compilation of previously known results.

######  Theorem 1.4.

Let $C$ be a smooth projective curve of genus $g\geqslant 5$, non-
hyperelliptic, and $L=mK_{C}$ for some integer $m>1$. Then
$\operatorname{cork}(\gamma_{C,mK_{C}})=0$ if $m>2$ or $\mathrm{Cliff}(C)>2$.

If $\mathrm{Cliff}(C)=1$, then either

  1. (a)

$C$ is trigonal, and then
$\operatorname{cork}(\gamma_{C,2K_{C}})=h^{0}(K_{C}-(g-4)\mathfrak{g})$, with
$\mathfrak{g}$ the class of the $g^{1}_{3}$; or

  2. (b)

$C$ is a plane quintic, and then
$\operatorname{cork}(\gamma_{C,2K_{C}})=h^{0}(\mathbf{P}^{2},-2K_{\mathbf{P}^{2}}-C)=3$.

If $\mathrm{Cliff}(C)=2$, then $\operatorname{cork}(\gamma_{C,mK_{C}})=0$
except in the following cases:

  1. (a)

$g=5$, and then $\operatorname{cork}(\gamma_{C,2K_{C}})=3$;

  2. (b)

$C$ is a bi-anticanonical divisor in a Del Pezzo surface $X$, and then
$\operatorname{cork}(\gamma_{C,2K_{C}})=\linebreak h^{0}(X,-2K_{X}-C)=1$;

  3. (c)

$C$ is bielliptic, and then $\operatorname{cork}(\gamma_{C,2K_{C}})=1$.

The case of curves of genus $g\leqslant 4$ is elementary, see (9.4.1).

The next stage of our program is to examine in all three cases above (genus
$3$ curves, hyperelliptic curves, and pluricanonical curves) the families of
all surface extensions of a given polarized curve $(C,L)$. For pluricanonical
curves, Theorem 1.2 above tells us all the surfaces we need to consider. In
general we will assume that $\deg(L)\geqslant 2g+3$, in order for Property
$N_{2}$ for $(C,L)$ (which is needed to apply the theory of ribbons and
extensions) to be granted by the Green Theorem 2.2. For pluricanonical curves,
this condition is automatic except for a few sporadic cases in genus
$g\leqslant 3$. For an arbitrary polarized curve of genus $3$, this condition
is stronger than $\deg(L)\geqslant 4g-4$, so that again Theorem 1.2 tells us
all the surfaces we need to consider.

For a polarized hyperelliptic curve $(C,L)$ of arbitrary genus however, we
need a stronger classification result. We prove the following, which extends
classical results by Castelnuovo [8], and more recent ones by Serrano [41] and
Sommese–Van de Ven [42] (see (4.3) for more comments on these results).

######  Theorem 1.5.

Let $C\subseteq\mathbf{P}^{d-g}$ be a linearly normal hyperelliptic curve of
genus $g\geqslant 2$ and degree $d\geqslant 10$, unless $g=2$ or $3$ in which
case we only make the looser assumption that $d\geqslant 2g+3$. For all
surfaces $S\subseteq\mathbf{P}^{d-g+1}$ having $C$ as a hyperplane section, if
$S$ is not a cone then it is rational, and ruled by conics. In particular, its
general hyperplane section is hyperelliptic.

######  Corollary 1.6.

In the setting of Theorem 1.5, the surface $S$ is represented by a linear
subsystem of $|2H+(g+1-e)F|$ on $\mathbf{F}_{e}$, as in case (d) of Theorem
1.2.

Our main tool in proving Theorem 1.5 is the Reider and Beltrametti–Sommese
Theorem 4.1.

Eventually, we can complete our program, to the effect that we obtain the
following results.

######  Theorem 1.7.

Let $(C,L)$ be a non-hyperelliptic polarized curve of genus $g=3$ and degree
$d\geqslant 2g+3=9$. Then the following hold.

###### (1.7.1)

There exists a non-trivial extension of the polarized curve $(C,L)$ if and
only if there exist points $p_{1},\ldots,p_{16-d}\in C$ such that
$L=4K_{C}-\sum_{i=1}^{16-d}p_{i}$ (in particular, $d\leqslant 16$).

###### (1.7.2)

Every ribbon in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ is the first
infinitesimal neighbourhood of $C$ in some extension of $(C,L)$, hence there
exists a universal extension of $(C,L)$.

We refer to (7.8) for a discussion of the universal extensions of polarized
genus $3$ curves. We also include an analysis of what happens for degrees
below $2g+3$, in which case Property $N_{2}$ is no longer granted by Green’s
Theorem, and many results about ribbons and extensions are no longer
available; in particuliar a given ribbon may a priori be the first
infinitesimal neighbourhood of $C$ in several different extensions. Notably we
give examples of polarized curves having two distinct families of extensions,
one of the expected dimension $\operatorname{cork}(\gamma_{C,L})-1$ and one
superabundant, which would be impossible if Property $N_{2}$ holded.

######  Theorem 1.8.

Let $(C,L)$ be a polarized hyperelliptic curve of genus $g$ and degree
$d\geqslant 2g+3$.

###### (1.8.1)

If $d\leqslant 4g+4$, then there exists a non-trivial extension of $(C,L)$.

###### (1.8.2)

If $d=2g+3$, then every ribbon in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ is the first
infinitesimal neighbourhood of $C$ in some extension of $(C,L)$, hence there
exists a universal extension of $(C,L)$, of degree $2g+3$ and dimension $2g+3$
in $\mathbf{P}^{3g+5}$.

###### (1.8.3)

If $d>2g+3$, for general $(C,L)$ there exist ribbons in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ which may
not be realized as first infinitesimal neighbourhoods of $C$ in some extension
of $(C,L)$.

When $d=4g+4$, $(C,L)$ has in general only finitely many extensions, but more
than one; thus there cannot exist a universal extension of $(C,L)$. Also in
this case we analyze briefly the situation for degrees below $2g+3$, and we
find that in this case the extensions form a superabundant family, i.e., this
family has dimension greater than $\operatorname{cork}(\gamma_{C,L})-1$. This
would be impossible if Property $N_{2}$ holded.

######  Theorem 1.9.

Let $(C,mK_{C})$ be a pluricanonical curve, and assume that either $g\geqslant
4$ and $m\geqslant 2$, or $g=3$ and $m\geqslant 3$. Then there exists a non-
trivial extension of the polarized curve $(C,mK_{C})$ if and only if
$(C,mK_{C})$ falls into one of the cases of Theorem 1.4 or (9.4) in which
$\operatorname{cork}(\gamma_{C,mK_{C}})\neq 0$ (in particular, either
$g\leqslant 10$, or $C$ is bielliptic).

Every ribbon in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,mK_{C}}))$ is the
first infinitesimal neighbourhood of $C$ in some extension of $(C,mK_{C})$,
hence there exists a universal extension of $(C,mK_{C})$.

We refer to Subsection 9.3 for a discussion of the universal extensions of
pluricanonical curves. Except in the trigonal case, we provide an explicit
construction.

The organization of the text is as follows. In Section 2 we recall general
results on projective curves, revisit the C. Segre Theorem and recall the
Hartshorne Bound. Section 3 is devoted to the proof of the Classification
Theorem 1.2, and Section 4 to that of the Classification Theorem 1.5 for
hyperelliptic curves. In Section 5 we recall the definition of the Gaussian
map $\gamma_{C,L}$, and compute its corank in a number of cases, thus proving
Theorem 1.3. In Section 6 we recall the theory of ribbons and extensions, and
provide all the necessary material for its application as described in the
introduction. These applications are to polarized genus three curves (in
Section 7), to polarized hyperelliptic curves (in Section 8), and to
pluricanonical curves (in Section 9). All references in this text are
invisible but clickable hyperlinks.

We work over the field $\mathbf{C}$ of complex numbers throughout. We use the
symbols ’$\equiv$’ and ’$\sim$’ to denote numerical and linear equivalence,
respectively.

Acknowledgements. We thank Andreas Knutsen and Angelo Lopez for useful
comments.

## General preliminary results

### Results on projective curves

We will need the following results. The first one is an improvement on a
theorem by Castelnuovo, see [10] and [13, Thm. (1.11)].

######  Theorem 2.1 (Castelnuovo).

Let $C$ be a smooth curve and $L\to C$ be a globally generated line bundle. If
the image of the map associated to the complete linear series $|L|$ is not a
rational curve, or if $|L|$ is a pencil, then the multiplication map

$H^{0}(L)\otimes H^{0}(K_{C})\to H^{0}(K_{C}+L)$

is surjective.

The other one is due to Green.

######  Theorem 2.2 ([24, Thm. 4.a.1]).

Let $C$ be a smooth curve of any genus $g$, and $L\to C$ a line bundle of
degree $d$. For all $k\geqslant 0$, if

$d\geqslant 2g+1+k,$

then $L$ enjoys Property $N_{k}$, i.e.,

  1. (i)

$L$ defines a projectively normal embedding of $C$, and

  2. (ii)

if $k\geqslant 1$, the ideal of $C$ in this embedding is generated by
quadrics, and all syzygies are generated by linear syzygies up to the $k$-th
step.

### Scrolls as extensions of linearly normal curves

The main object of this subsection is to discuss Theorem 2.1 below. We
distinguish between _ruled surfaces_ , by which me mean “abstract ruled
surfaces”, i.e., surfaces $S$ equipped with a locally trivial morphism $S\to
C$ onto a smooth curve whose fibres are $\mathbf{P}^{1}$, and _scrolls_ , by
which we mean a ruled surface embedded in some projective space in such a way
that the fibres are lines.

######  Theorem 2.1.

Let $C\subseteq\mathbf{P}^{n-1}$ be a smooth linearly normal and non-
degenerate curve of genus $g>0$. Assume there exists a scroll
$\Sigma\subseteq\mathbf{P}^{n}$ such that $C$ is a hyperplane section of
$\Sigma$. Then the scroll $\Sigma$ is necessarily a cone.

This first appeared in [39, §2] by C. Segre, and also later in [40, §14] by
the same author, under the additional assumption that $\mathcal{O}_{C}(1)$ is
non-special. In the particular case when $C$ is a canonical curve, this is
[20, Thm. III.2.1, p. 38].

We shall need the following lemma for the proof.

######  Lemma 2.2 ([3, Lemme 1]).

Let $C$ be a smooth curve, and let

(2.2.1) $0\to E^{\prime}\to E\to E^{\prime\prime}\to 0$

be an exact sequence of vector bundles on $C$. Assume that

  1. (i)

the boundary map $\partial:H^{0}(E^{\prime\prime})\to H^{1}(E^{\prime})$ is
zero, and

  2. (ii)

the multiplication map $\alpha:H^{0}(E^{\prime\prime})\otimes
H^{0}((E^{\prime})^{\vee}\otimes\omega_{C})\to
H^{0}(E^{\prime\prime}\otimes(E^{\prime})^{\vee}\otimes\omega_{C})$ is
surjective.

Then the exact sequence (2.2.1) is split.

###### Proof of Theorem 2.1.

Let $f:S\to\Sigma$ be the minimal resolution of singularities. The surface $S$
is ruled with $C$ as a section of the ruling, and $\Sigma$ is a scroll, hence
there exists a rank $2$ vector bundle $\mathcal{E}$ on $C$ such that
$S\cong\mathbf{P}(\mathcal{E})$ and the map
$S\to\Sigma\subseteq\mathbf{P}^{n}$ is given by a linear subsystem of
$|H^{0}(\mathcal{O}_{\mathbf{P}({\cal E})}(1))|$. Moreover, there is the
following exact sequence of locally free sheaves on $C$:

(2.2.2)
---
$\textstyle{0\to}$$\textstyle{\mathcal{O}_{C}(1)\otimes
N_{C/S}^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\to{\cal
E}\to\mathcal{O}_{C}(1)\to 0,}$$\textstyle{\mathcal{O}_{C}}$

see [26, Prop. V.2.6]. We shall use Lemma 2.2 to show that this exact sequence
is split. Then $\mathcal{E}=\mathcal{O}_{C}\oplus\mathcal{O}_{C}(1)$ and the
linear system $|H^{0}(\mathcal{O}_{\mathbf{P}({\cal E})}(1))|$ contracts the
section corresponding to the trivial quotient $\mathcal{E}\to\mathcal{O}_{C}$,
and $\Sigma$ is a cone, which is the result we wanted to prove.

It thus only remains to apply Lemma 2.2 to the exact sequence (2.2.2).
Condition (ii) is satisfied by the Castelnuovo Theorem 2.1, i.e., the
multiplication map $H^{0}(L)\otimes H^{0}(K_{C})\to H^{0}(K_{C}+L)$ is
surjective, where $L=\mathcal{O}_{C}(1)$. To see that condition (i) of the
lemma holds as well, we write the long exact sequence associated to (2.2.2):

---
$\textstyle{0\to H^{0}(\mathcal{O}_{C})\to}$$\textstyle{H^{0}({\cal
E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\to
H^{0}(\mathcal{O}_{C}(1))\to}$$\textstyle{H^{1}(\mathcal{O}_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\to}$$\textstyle{H^{1}({\cal
E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\to
H^{1}(\mathcal{O}_{C}(1))\to
0,}$$\textstyle{H^{0}(\mathcal{O}_{S}(1))}$$\textstyle{H^{1}(\mathcal{O}_{S})}$$\textstyle{H^{1}(\mathcal{O}_{S}(1))}$

for the vertical identifications, see, e.g., [26, Lem. V.2.4];
$\mathcal{O}_{S}(1)$ stands for $\mathcal{O}_{\mathbf{P}({\cal E})}(1)$. It
follows from the fact that $C\subseteq\mathbf{P}^{n-1}$ is linearly normal
that $H^{0}(\mathcal{O}_{\Sigma}(1))\to H^{0}(\mathcal{O}_{C}(1))$ is
surjective, hence $H^{0}(\mathcal{O}_{S}(1))\to H^{0}(\mathcal{O}_{C}(1))$ is
surjective, and thus the boundary map $H^{0}(\mathcal{O}_{C}(1))\to
H^{1}(\mathcal{O}_{C})$ is zero, i.e., condition (i) holds. We may thus apply
Lemma 2.2 and, as explained above, this concludes the proof of the theorem. ∎

### The Hartshorne Bound

######  Theorem 2.1 ([25]).

Let $C$ be a smooth curve of genus $g$, sitting in a smooth surface $S$. If
$C^{2}>4g+5$, then there exists a ruled surface $\Sigma$ having $C$ as a
section, and a birational map $S\dashrightarrow\Sigma$ which is an isomorphism
on an open subset containing $C$.

If $C^{2}=4g+5$, the only other possibility is that there is a birational map
$S\dashrightarrow\mathbf{P}^{2}$ which is an isomorphism on an open subset
containing $C$, and identifies $C$ with a cubic curve.

This result is also a particular case of [27, Thm. A].

######  Corollary 2.2.

Let $C\subseteq\mathbf{P}^{n}$ be a smooth curve of genus $g>1$ and degree
$d$, non-degenerate and linearly normal. If $d>4g+4$, then every extension of
$C$ is trivial.

If $g=1$ and $d=4g+5$, the only possibility for $C$ to have a non-trivial
extension is that it is a hyperplane section of the Veronese surface
$v_{3}(\mathbf{P}^{2})\subseteq\mathbf{P}^{9}$.

###### Proof.

Let $S\subseteq\mathbf{P}^{n+1}$ be a surface having $C$ as a hyperplane
section. Then $\deg(S)=\deg(C)=d$, hence $C^{2}=d$ as a divisor in $S$. We
consider a minimal desingularization $\pi:S^{\prime}\to S$ and, abusing
notation, we still denote by $C$ its proper transform on $S^{\prime}$. Note
that $S$ may have at most isolated singularities and, by the minimality of the
resolution, there is no irreducible $(-1)$-curve $\Gamma$ on $S^{\prime}$ such
that $C\cdot\Gamma=0$.

By Hartshorne’s Theorem 2.1 above, there are two cases to be considered.
Assume first that there exists a birational map
$S^{\prime}\dashrightarrow\mathbf{P}(E)$ which is an isomorphism on an open
subset $U\subseteq S^{\prime}$ containing $C$, where $E$ is a rank two vector
bundle over $C$. The pull-back to $S^{\prime}$ of the linear system $|C|$ on
$S$ is the complete linear system $|C|$ on $S^{\prime}$ because
$C\subseteq\mathbf{P}^{n}$ is linearly normal, and it is base-point-free. In
turn, the image on $\mathbf{P}(E)$ of this system is again the complete linear
system $|C|$, because $S^{\prime}\dashrightarrow\mathbf{P}(E)$ is an
isomorphism on $U$. The upshot is that $S$ is the image of $\mathbf{P}(E)$
defined by this linear system, and by Theorem 2.1 this is a cone.

In the remaining case of Hartshorne’s Theorem, which may occur only if $g=1$
and $d=4g+5=9$, similar arguments show that $S$ is the image of
$\mathbf{P}^{2}$ by the complete linear system of plane cubics. ∎

## Classification of extensions with high degree

This section is devoted to Theorem 1.2. We first expand on the description of
the items in the classification, and then give the proof.

### Detailed description of the items in the classification

Let $S\subseteq\mathbf{P}^{N}$ be a degree $d$ surface, of sectional
(geometric) genus $g$. We call _simple internal projection_ of $S$ a surface
$S^{\prime}\subseteq\mathbf{P}^{N^{\prime}}$ obtained by projecting $S$ from a
curvilinear subscheme $Z$ of length $b$ supported on the smooth locus of $S$,
$N^{\prime}=N-\dim(\langle Z\rangle)$, such that the projection map is
birational. We recall that a scheme $Z$ is _curvilinear_ if for all point $p$
in the support of $Z$, the Zariski tangent space of $Z$ at $p$ has dimension
at most one.

For $S^{\prime}$ a simple internal projection of $S$ as above, one has
$\deg(S^{\prime})=d-b$ and $S^{\prime}$ has the same sectional genus $g$ as
$S$. Note that, if $d-b\geqslant 2g+1$ and $S$ is regular and linearly normal,
then any projection from a curvilinear subscheme $Z$ of length $b$ supported
on the smooth locus of $S$ is a simple internal projection, and
$N^{\prime}=N-b$, for in this case the linear system of hyperplane sections of
$S$ passing through $Z$ restricts on its general member to a complete, non-
special, very ample linear system.

######  Example 3.1.

Let $C$ be a bielliptic curve of genus $g\geqslant 4$. Then the canonical
model of $C$ in $\mathbf{P}^{g-1}$ sits on a cone $X$ with vertex a point $p$
over a normal elliptic curve $E$ of degree $g-1$ in a hyperplane $\Pi$ of
$\mathbf{P}^{g-1}$ not containing $p$, and $C$ is the complete intersection of
$X$ with a quadric in $\mathbf{P}^{g-1}$. The bielliptic involution is the
restriction to $C$ of the projection from $p$ to $\Pi$.

Note that the minimal resolution of $X$ is the projective bundle
$\mathbf{P}(\mathcal{O}_{E}\oplus\mathcal{L})$ where $\mathcal{L}$ is the
hyperplane bundle of $E$ in $\Pi\cong\mathbf{P}^{g-2}$. The map
$\mathbf{P}(\mathcal{O}_{E}\oplus\mathcal{L})\to X$ is induced by the
$\mathcal{O}(1)$ bundle on $\mathbf{P}(\mathcal{O}_{E}\oplus\mathcal{L})$.

Consider the $2$–Veronese image $S$ of $X$. Since

$h^{0}(E,\operatorname{\mathrm{Sym}}^{2}(\mathcal{O}_{E}\oplus\mathcal{L}))=h^{0}(E,\mathcal{O}_{E})+h^{0}(E,\mathcal{L})+h^{0}(E,\mathcal{L}^{\otimes
2})=3g-2,$

the surface $S$ is linearly normally embedded in $\mathbf{P}^{3(g-1)}$. The
bicanonical image of $C$ is a hyperplane section of $S$, and it is linearly
normally embedded with degree $d=4(g-1)$. In this case we will say that $S$
presents the _bicanonical bielliptic case_.

######  Example 3.2.

Consider the linear system $|\mathcal{O}_{\mathbf{P}^{2}}(\delta)|$ of plane
curves of degree $\delta$, whose self–intersection is $\delta^{2}$ and whose
genus is

$g=\frac{(\delta-1)(\delta-2)}{2}.$

We assume $\delta\geqslant 4$ so that $g>1$. One has $\delta^{2}\geqslant
4g-4$ if and only if $\delta\leqslant 6$. This means that for
$4\leqslant\delta\leqslant 6$, the degree of the $\delta$–Veronese image of
$\mathbf{P}^{2}$ (and suitable simple internal projections of it) is in the
range $[4g-4,4g+4]$.

More precisely, in case $\delta=4$ one has $\delta^{2}=4g+4$, which is the
maximum possible degree with respect to the sectional genus. We still have
degree in the above range if we make simple internal projections of the
$4$–Veronese of $\mathbf{P}^{2}$ from $b\leqslant 8$ points.

If $\delta=5$, we have $\delta^{2}=4g+1$, and the degree is in the above range
if we make simple internal projections of the $5$–Veronese of $\mathbf{P}^{2}$
from $b\leqslant 5$ points.

Finally, if $\delta=6$ we have $\delta^{2}=4g-4$.

In all these cases we will say that the surfaces present the _planar case_.

######  Example 3.3.

a) Let $X$ be the plane blown up at $h\leqslant 7$ (proper or infinitely near)
points such that there is an irreducible cubic curve passing simply through
these points. Let $E_{1},\ldots,E_{h}$ be the exceptional $(-1)$-divisors over
the blown–up points, set $E=E_{1}+\cdots+E_{h}$ and let $H$ be the pull back
on $X$ of a general line of $\mathbf{P}^{2}$. Note that the anticanonical
system on $X$ is $|3H-E|$. We will say that we are here in a _Del Pezzo
situation_ (even though $X$ is a genuine _Del Pezzo surface_ only if the
anticanonical system is ample).

Consider the linear system $|6H-2E|$. This linear system is base point free
and its general curve is irreducible of genus $g=10-h$ and self–intersection
$4(9-h)=4g-4$. Moreover it is not difficult to see that $\phi_{|6H-2E|}$ is a
birational morphism to the image $S$, that is non-degenerate in
$\mathbf{P}^{27-3h}=\mathbf{P}^{3g-3}$. Note that the hyperplane sections of
these surfaces are bicanonically embedded. For $h=0$ we get again the planar
case for $\delta=6$.

b) Similarly, let $X$ be an irreducible quadric in $\mathbf{P}^{3}$, and
consider the linear system $|-2K_{X}|$, which is the linear system of quadric
sections of $X$. Thus, either $X$ is the image of $\mathbf{F}_{0}$ by the
linear system $|H+F|$ and $|-2K_{X}|=|4H+4F|$ (curves of bidegree $(4,4)$ on
$\mathbf{P}^{1}\times\mathbf{P}^{1}$), or $X$ is the image of $\mathbf{F}_{2}$
by the linear system $|2H|$ and we may identify $X$ with $\mathbf{F}_{2}$ and
$|-2K_{X}|$ with $|4H|$. As in case a), the linear system $|-2K_{X}|$ is base
point free, its general member is irreducible of genus $g=9$ and
self–intersection $4g-4=32$, and the associated map $\phi_{|-2K_{X}|}$ is a
birational morphism to the image $S$, which is a non-degenerate surface in
$\mathbf{P}^{3g-3}=\mathbf{P}^{24}$ with bicanonical hyperplane sections.

We will say that the surfaces in cases a) and b) above present the
_bicanonical Del Pezzo case_.

######  Example 3.4.

Consider, in the Notation 1.1, the linear system

$|2H+kF|=|2E+(k+2e)F|=|H+E+(k+e)F|$

on a rational ruled surface $\mathbf{F}_{e}$, with $k\geqslant\max(0,3-e)$. It
is base point free, and very ample unless $k=0$, in which case the morphism
$\phi_{|2H|}$ maps $\mathbf{F}_{e}$ birationally onto its image, which is the
$2$-Veronese image of the cone in $\mathbf{P}^{e+1}$ over a rational normal
curve in $\mathbf{P}^{e}$. In any event the general curve in $|2H+kF|$ is
smooth and irreducible.

Since

(3.4.1) $K_{\mathbf{F}_{e}}\sim-2E-(e+2)F\sim-2H+(e-2)F,$

the adjoint system to $|2H+kF|$ is $|(k+e-2)F|$, and therefore the curves in
$|2H+kF|$ are hyperelliptic of genus $g=k+e-1$. The assumption that
$k\geqslant 3-e$ grants that $g\geqslant 2$. Moreover,

$(2H+kF)^{2}=4e+4k=4g+4.$

If $C$ is a smooth curve in $|2H+kF|$, from the exact sequence

$0\rightarrow\mathcal{O}_{\mathbf{F}_{e}}\rightarrow\mathcal{O}_{\mathbf{F}_{e}}(2H+kF)\rightarrow\mathcal{O}_{C}(2H+kF)\rightarrow
0$

and from the fact that

$h^{1}(\mathbf{F}_{e},\mathcal{O}_{\mathbf{F}_{e}})=0,\quad
h^{0}(C,\mathcal{O}_{C}(2H+kF))=3g+5$

we deduce that
$h^{0}(\mathbf{F}_{e},\mathcal{O}_{\mathbf{F}_{e}}(2H+kF))=3g+6$. If we set
$S=\phi_{|2H+kF|}(\mathbf{F}_{e})$, then $S$ is non–degenerate of degree
$4g+4$ in $\mathbf{P}^{3g+5}$, and its general curve section is hyperelliptic
of genus $g\geqslant 2$. Any surface which is a simple internal projection of
$S$ from $b\leqslant 8$ points on $S$ still has degree in the range
$[4g-4,4g+4]$, and sectional genus $g$. We will say that surfaces of this type
present the _hyperelliptic case_.

######  Example 3.5.

Let $C$ be a trigonal canonical curve of genus $g\geqslant 4$ in
$\mathbf{P}^{g-1}$. Then $C$ sits on a smooth rational normal scroll $Y$ of
degree $g-2$ in $\mathbf{P}^{g-1}$. We denote by $\mathcal{H}$ the hyperplane
section class of $Y\subseteq\mathbf{P}^{g-1}$, and by $F$ a line of the ruling
of $Y$ (be careful not to mistake $\mathcal{H}$ with the class $H$ in our
Notation 1.1; the minimal resolution of singularities of $Y$ is isomorphic to
some rational ruled surface $\mathbf{F}_{e}$, and one has $\mathcal{H}=H+lF$
for some $l\geqslant 0$).

It is easy to check that $C\in|3\mathcal{H}-(g-4)F|$. Conversely, if $Y$ is a
rational normal scroll of degree $g-2$ in $\mathbf{P}^{g-1}$, and if a smooth
curve $C$ sits in $|3\mathcal{H}-(g-4)F|$, then $C$ is a trigonal canonical
curve. One has $(3\mathcal{H}-(g-4)F)^{2}=3g+6$, hence the linear system
$|\mathcal{O}_{C}(3\mathcal{H}-(g-4)F)|$ is very ample of dimension $2g+6$.
This shows that $\phi_{|3\mathcal{H}-(g-4)F|}$ is a morphism that maps $Y$
birationally to a non-degenerate surface $S\subseteq\mathbf{P}^{2g+6}$. If
$g\leqslant 10$, one has $3g+6\geqslant 4g-4$. We will say that surfaces of
this sort, as well as their simple internal projections of degree at least
$4g-4$, present the _trigonal case_.

To connect with the notation in Theorem 1.2, note that if $\mathcal{H}=H+lF$
on $\mathbf{F}_{e}$, then

$(H+lF)^{2}=e+2l=g-2,$

and

$3\mathcal{H}-(g-4)F=3H+(3l-g+4)F=3H+\frac{1}{2}(g-3e+2)F.$

Note that none of the surfaces in the above Examples is a cone. Indeed, they
have irregularity $q\leqslant 1$, and sectional genus $g\geqslant 2$.

### Previously known results

We quote the following from [32, §7], which we do not use in our proof.

######  Theorem 3.1 ([29, Cor. 2.10], [5, Thm. 2]).

Let $C\subseteq\mathbf{P}^{r}$ be a smooth irreducible nondegenerate linearly
normal curve of genus $g\geqslant 4$ and degree $d$. Then $C$ is not
extendable if

  1. (i)

$C$ is trigonal, $g\geqslant 5$ and $d\geqslant\max\\{4g-6,3g+7\\}$;

  2. (ii)

$C$ is a plane quintic and $d\geqslant 26$;

  3. (iii)

$\mathrm{Cliff}(C)=2$ and $d\geqslant 4g-3$;

  4. (iv)

$\mathrm{Cliff}(C)\geqslant 3$ and $d\geqslant 4g+1-3\mathrm{Cliff}(C)$.

Part (iv) tells us in particular that no curve $C$ with $\mathrm{Cliff}(C)>2$
is extendable in the range of degree under consideration in the present text,
namely $d\geqslant 4g-4$. If $\mathrm{Cliff}(C)\leqslant 2$, our Theorem 1.2
classifies those extensions that indeed exist in the possibilities left open
by the above statement.

If $\mathrm{Cliff}(C)=2$, the only possibility in our range left by the above
theorem is $d=4g-4$. We find that there indeed exist extensions in this
degree, and they are all extensions of bicanonical curves, for some special
curves.

Items (i) and (ii) deal with curves of Clifford index one. For plane quintics
the maximal degree is $25$, and it is indeed realized in our classification,
by rational surfaces representend by a linear system of plane quintics. For
smooth quintics, $g=6$ hence $4g-4=20$, and $4g+4=28$.

For trigonal curves, the above theorem says that extensions may have degree at
most

$\max(4g-7,3g+6)=\begin{cases}4g-7&\text{if $g\geqslant 13$}\\\ 3g+6&\text{if
$g\leqslant 13$}.\end{cases}$

If $g\geqslant 13$, this implies that there is nothing in the range
$[4g-4,4g+4]$. For $g\leqslant 13$ the bound is sharp and, by our
classification, realized exclusively by trisecant scrolls (cf. Example 3.5) as
soon as $3g+6\geqslant 4g-4$, i.e., $g\geqslant 10$.

For curves with Clifford index zero, i.e., hyperelliptic curves, there exist
extensions in all degrees $4g-4,\ldots,4g+4$, and they are all bisecant
scrolls (cf. Example 3.4).

### Setup of the Proof of Theorem 1.2

We now start proving Theorem 1.2. Let $S\subseteq\mathbf{P}^{r+1}$ be a
surface as in the theorem. It may have at most isolated singular points. We
consider its minimal desingularization $\pi:S^{\prime}\to S$ and, abusing
notation, we still denote by $C$ its proper transform on $S^{\prime}$. By the
minimality of the resolution, there is no irreducible $(-1)$-curve $E$ such
that $C\cdot E=0$ on $S^{\prime}$.

Since $K_{S^{\prime}}\cdot C=2g-2-d<0$, the Kodaira dimension of $S^{\prime}$
is $\kappa(S^{\prime})=-\infty$. We let $q$ be the irregularity of
$S^{\prime}$, and consider a minimal model $f:S^{\prime}\to\Sigma$ of
$S^{\prime}$. If $q>0$, then $\Sigma$ is a $\mathbf{P}^{1}$-bundle over a
smooth curve $\Gamma$ of genus $q$, while if $S^{\prime}$ is rational, then
$\Sigma$ is either $\mathbf{P}^{2}$ or a rational ruled surface
$\mathbf{F}_{e}$, with $e\geqslant 0$ and $e\neq 1$.

### The irregular case

In this section we will prove Theorem 1.2 in case $q>0$, which amounts to
proving the following proposition.

######  Proposition 3.1.

Let $S$ be as in Theorem 1.2 with $q>0$. Then $S$ presents the bicanonical
bielliptic case as in Example 3.1, hence we are in case (a) of Theorem 1.2.

In this case the minimal desingularization $S^{\prime}$ of $S$ has a
surjective morphism $\varphi:S^{\prime}\to\Gamma$ to a smooth curve $\Gamma$
of genus $q$ with connected, rational fibres. We denote by $\theta$ the class
in the Néron–Severi group of $S^{\prime}$ of a general fibre of $\varphi$, and
set

$m=C\cdot\theta.$

This is the degree of the images of the fibres of $\varphi$ on $S$. If $m=1$
then, by Theorem 2.1, $S$ is a cone, which is excluded. Thus we have
$m\geqslant 2$. In the case $m=2$, the images of the fibres of $\varphi$ on
$S$ are conics, and we will say we are in the _conic case_.

The proof of Proposition 3.1 consists of a few steps.

####  3.4.1 Reduction to the conic case

The first step consists in the following:

######  Proposition 3.1.

Let $S$ be as in Proposition 3.1. Then $m=2$ and $d=4(g-1)$.

###### Proof.

We apply [25, Thm. (2.3)] to the effect that if $C$ is a smooth, irreducible
curve of genus $g$ on an irregular ruled surface $S^{\prime}$ with $m>1$, then

$C^{2}\leqslant\frac{2m}{m-1}(g-1).$

Hence we have

$4(g-1)\leqslant d=C^{2}\leqslant\frac{2m}{m-1}(g-1),$

and the assertion immediately follows. ∎

####  3.4.2 Passing to a minimal model

Consider the image $\bar{C}$ of $C$ in $\Sigma$ via the map
$f:S^{\prime}\to\Sigma$. This is an irreducible curve and, since $m=2$ by
Proposition 3.1, $\bar{C}$ may have at most double points that can be proper
or infinitely near. Let $h$ be the number of double points of $\bar{C}$. One
has

$\bar{C}^{2}=C^{2}+4h+\nu,\quad p_{a}(\bar{C})=g+h,$

with $\nu$ the number of $(-1)$-curves $E$ contracted by $f$ and such that
$C\cdot E=1$.

By performing a sequence of elementary transformations on $\Sigma$ based at
the double points of $\bar{C}$, we produce a birational map

$\alpha:\Sigma\dasharrow\Sigma^{\prime}$

where $\Sigma^{\prime}$ is still a $\mathbf{P}^{1}$–bundle over $\Gamma$, the
image $C^{\prime}$ of $\bar{C}$ via $\alpha$ is smooth and $C^{\prime
2}=C^{2}+\nu=4(g-1)+\nu$. Applying Proposition 3.1 to
$C^{\prime}\subseteq\Sigma^{\prime}$, one finds that $\nu=0$.

Abusing notation, we will still denote by $\theta$ the class of a fibre of the
structure morphism $\varphi^{\prime}:\Sigma^{\prime}\to\Gamma$.

The pair $(S^{\prime},C)$ is birational to the pair
$(\Sigma^{\prime},C^{\prime})$, i.e., there is a birational map
$S^{\prime}\dasharrow\Sigma^{\prime}$ that maps $C$ to $C^{\prime}$. Hence the
image of $\Sigma^{\prime}$ via the map $\phi_{|C^{\prime}|}$ determined by the
linear system $|C^{\prime}|$ is the original surface $S$. So, rather than
studying the linear system $|C|$ on $S^{\prime}$, we may study the linear
system $|C^{\prime}|$ on $\Sigma^{\prime}$.

####  3.4.3 The adjoint system

Next we consider the _adjoint linear system_
$|K_{\Sigma^{\prime}}+C^{\prime}|$.

Let $\left.\varphi^{\prime}\right|_{C^{\prime}}:C^{\prime}\to\Gamma$ be the
double cover, with branch divisor $B$. By the Riemann–Hurwitz Formula we have

$g=2q-1+b,\quad\text{with}\quad\deg(B)=2b.$

From the cohomology sequence of the exact sequence

$0\rightarrow\mathcal{O}_{\Sigma^{\prime}}(K_{\Sigma^{\prime}})\rightarrow\mathcal{O}_{\Sigma^{\prime}}(K_{\Sigma^{\prime}}+C^{\prime})\rightarrow\mathcal{O}_{C^{\prime}}(K_{C^{\prime}})\rightarrow
0,$

since

$\displaystyle
h^{0}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(K_{\Sigma^{\prime}}))=h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(K_{\Sigma^{\prime}}+C^{\prime}))=0,$
$\displaystyle
h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(K_{\Sigma^{\prime}}))=q,$
$\displaystyle h^{0}(C^{\prime},\mathcal{O}_{C^{\prime}}(K_{C^{\prime}}))=g,$

we get

$\dim(|K_{\Sigma^{\prime}}+C^{\prime}|)=g-1-q=q-2+b.$

Moreover, $q-2+b\geqslant 0$ because we are assuming $q>0$ and $g\geqslant 2$.

Note that $K_{\Sigma^{\prime}}\cdot\theta=-2$, hence
$(K_{\Sigma^{\prime}}+C^{\prime})\cdot\theta=0$, so, since
$K_{\Sigma^{\prime}}+C^{\prime}$ is effective, we have the numerical
equivalence $K_{\Sigma^{\prime}}+C^{\prime}\equiv k\theta$, for some integer
$k$.

######  Lemma 3.1.

In the above setting, one has $q=1$.

###### Proof.

We have

$0=(K_{\Sigma^{\prime}}+C^{\prime})^{2}=K_{\Sigma^{\prime}}^{2}+2(K_{\Sigma^{\prime}}+C^{\prime})\cdot
C^{\prime}-(C^{\prime})^{2}=K_{\Sigma^{\prime}}^{2}+4(g-1)-C^{\prime
2}=K_{\Sigma^{\prime}}^{2}.$

On the other hand, $K_{\Sigma^{\prime}}^{2}=8(1-q)$, and the assertion
follows. ∎

####  3.4.4 The classification

We may now finish the proof of Proposition 3.1. To do so, we first identify
the surface $\Sigma^{\prime}$. We have
$\Sigma^{\prime}=\mathbf{P}(\mathcal{E})$, where $\mathcal{E}$ is a rank two
vector bundle on a curve $\Gamma$ of genus 1. We can suppose that
$\mathcal{E}$ is _normalized_ (see [26, p. 373]), with invariant $e$. We
denote by $E$ the section such that $E^{2}=-e$. We have

$K_{\Sigma^{\prime}}\equiv-2E-e\theta$

by [26, Cor. V.2.11], and

(3.0.1) $C^{\prime}\equiv 2E+a\theta$

for some integer $a$.

One has

$\displaystyle 2g-2=(K_{\Sigma^{\prime}}+C^{\prime})\cdot
C^{\prime}=(a-e)\theta\cdot C^{\prime}=2(a-e),$

hence

(3.0.2) $a=g-1+e.$

######  Lemma 3.1.

The vector bundle $\mathcal{E}$ is decomposable.

###### Proof.

Suppose by contradiction that $\mathcal{E}$ is indecomposable. Then one has
$-1\leqslant e\leqslant 0$ by [26, Thm. V.2.15]. If $e=0$, then by (3.0.2) we
have

(3.1.1) $C^{\prime}-K_{\Sigma^{\prime}}\equiv 4E+(g-1)\theta,$

which is big and nef. Then, by Kawamata–Viehweg Vanishing, one has
$h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(C^{\prime}))=0$. But then
from the cohomology sequence of the exact sequence

$0\rightarrow\mathcal{O}_{\Sigma^{\prime}}\rightarrow\mathcal{O}_{\Sigma^{\prime}}(C^{\prime})\rightarrow\mathcal{O}_{C^{\prime}}(C^{\prime})\rightarrow
0,$

we see that the restriction map

$H^{0}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(C^{\prime}))\rightarrow
H^{0}(\Sigma^{\prime},\mathcal{O}_{C^{\prime}}(C^{\prime}))$

has corank $h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}})=1$, so it is
not surjective, which in turn implies that $C$ is not linearly normal, a
contradiction.

In case $e=-1$, one has
$h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(C^{\prime}))=0$ by [11,
Thm. 1.17], and then one concludes as in the previous case. ∎

We can now finish the:

##### (3.2)Proof of Proposition 3.1

From Lemma 3.1 we have that

$\mathcal{E}=\mathcal{O}_{\Gamma}\oplus\mathcal{L}$

where $\mathcal{L}$ is a line bundle of degree $-e\leqslant 0$, so that
$e\geqslant 0$ (see [26, Thm. V.2.12]). We want to compute
$h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(C^{\prime}))$. To do so,
consider again the structure morphism
$\varphi^{\prime}:\Sigma^{\prime}=\mathbf{P}(\mathcal{E})\to\Gamma$, and note
that

$\displaystyle
h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(C^{\prime}))=h^{1}(\Gamma,\varphi^{\prime}_{*}\mathcal{O}_{\Sigma^{\prime}}(C^{\prime})).$

From (3.0.1) and (3.0.2), we have

(3.2.1)
$\varphi^{\prime}_{*}\mathcal{O}_{\Sigma^{\prime}}(C^{\prime})=\operatorname{\mathrm{Sym}}^{2}(\mathcal{E})\otimes\mathcal{D}$

where $\mathcal{D}$ is a line bundle of degree $g-1+e$ on $\Gamma$. One has

$\operatorname{\mathrm{Sym}}^{2}(\mathcal{E})=\mathcal{O}_{\Gamma}\oplus\mathcal{L}\oplus\mathcal{L}^{\otimes
2},$

hence

$\operatorname{\mathrm{Sym}}^{2}(\mathcal{E})\otimes\mathcal{D}=\mathcal{D}\oplus(\mathcal{D}\otimes\mathcal{L})\oplus(\mathcal{D}\otimes\mathcal{L}^{\otimes
2}),$

with

$\deg(\mathcal{D}\otimes\mathcal{L})=g-1,\quad\deg(\mathcal{D}\otimes\mathcal{L}^{\otimes
2})=g-1-e.$

Therefore,

$h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(C^{\prime}))=h^{1}(\Gamma,\mathcal{D})+h^{1}(\Gamma,\mathcal{D}\otimes\mathcal{L})+h^{1}(\Gamma,\mathcal{D}\otimes\mathcal{L}^{\otimes
2})=h^{1}(\Gamma,\mathcal{D}\otimes\mathcal{L}^{\otimes 2}).$

By the same argument we made in the proof of Lemma 3.1, we must have
$h^{1}(\Sigma^{\prime},\mathcal{O}_{\Sigma^{\prime}}(C^{\prime}))>0$. So we
must have $h^{1}(\Gamma,\mathcal{D}\otimes\mathcal{L}^{\otimes 2})>0$, hence
$g-1-e=\deg(\mathcal{D}\otimes\mathcal{L}^{\otimes 2})\leqslant 0$, and
$e\geqslant g-1$. On the other hand, by (3.0.1) and (3.0.2) we have

$C^{\prime}\cdot E=(2E+(g-1+e)\theta)\cdot E=g-1-e,$

hence $e\leqslant g-1$, thus $e=g-1$, and from
$h^{1}(\Gamma,\mathcal{D}\otimes\mathcal{L}^{\otimes 2})>0$ we deduce that

$\mathcal{D}\otimes\mathcal{L}^{\otimes
2}\cong\mathcal{O}_{\Gamma},\quad\text{i.e.,}\quad\mathcal{D}\cong{(\mathcal{L}^{\vee})}^{\otimes
2}.$

On the other hand, let us compute the linear equivalence class of
$C^{\prime}$. By (3.0.1),

$\mathcal{O}_{\Sigma^{\prime}}(C^{\prime})=\mathcal{O}_{\Sigma^{\prime}}(2E)\otimes\varphi^{\prime*}\mathcal{M}$

for some line bundle $\mathcal{M}$ on $\Gamma$, hence
$\varphi^{\prime}_{*}\mathcal{O}_{\Sigma^{\prime}}(C^{\prime})=\operatorname{\mathrm{Sym}}^{2}\mathcal{E}\otimes\mathcal{M}$.
Then (3.2.1) implies that $\mathcal{M}=\mathcal{D}$. The upshot is that

$\mathcal{O}_{\Sigma^{\prime}}(C^{\prime})=\mathcal{O}_{\Sigma^{\prime}}(2E)\otimes\varphi^{\prime*}({(\mathcal{L}^{\vee})}^{\otimes
2})=\mathcal{A}^{\otimes 2}$

where we set

$\mathcal{A}=\mathcal{O}_{\Sigma^{\prime}}(E)\otimes\varphi^{\prime*}{(\mathcal{L}^{\vee})}.$

The map $\phi_{\mathcal{A}}$ determined by the line bundle $\mathcal{A}$ maps
$\Sigma^{\prime}=\mathbf{P}(\mathcal{O}\oplus\mathcal{L})$ to a cone
$X\subseteq\mathbf{P}^{g-1}$ over the elliptic normal curve of degree $g-1$ in
$\mathbf{P}^{g-2}$ which is the image of $\Gamma$ via the map
$\phi_{\mathcal{L}^{\vee}}$. In this map the curve $C^{\prime}$ is mapped to a
quadratic section of $X$. This implies that we are in the bicanonical
bielliptic case. ∎

### The rational case

In this subsection we finish the proof of Theorem 1.2 by considering the case
in which the surface $S$ is rational. We will thus prove the following:

######  Proposition 3.1.

Let $S$ be as in Theorem 1.2 with $q=0$, i.e., $S$ is rational. Then $S$
presents either the planar case (see Example 3.2), or the bicanonical Del
Pezzo case (see Example 3.3), or the hyperelliptic case (see Example 3.4), or
the trigonal case (see Example 3.5).

The proof will consist of various steps that we will perform in the next
subsections. We consider the adjoint system $|C+K_{S^{\prime}}|$, which we
write

$|C+K_{S^{\prime}}|=F+|M|,$

where $F$ is the fixed part and $|M|$ the movable part, with $\dim(|M|)=g-1$.
Since $|C+K_{S^{\prime}}|$ cuts the complete canonical linear series on $C$,
we have $C\cdot F=0$.

There are two cases to be considered:

###### (3.1.a)

$|M|$ is composed with a pencil $|\Phi|$, including the case $g=2$ in which
$|M|$ itself is a pencil;

###### (3.1.b)

the general curve in $|M|$ is irreducible, and $\dim(|M|)\geqslant 2$, hence
$g\geqslant 3$.

In case (3.1.a) the curves in $|C|$ are hyperelliptic and $C\cdot\Phi=2$. So
the curves in $|\Phi|$ are mapped to conics on $S\subseteq\mathbf{P}^{r+1}$
and therefore are rational.

####  3.5.1 The hyperelliptic case

###### Proof of Proposition 3.1 in case (3.1.a).

There is a birational morphism $\xi:S^{\prime}\to\mathbf{F}_{e}$ such that the
pencil $|\Phi|$ of rational curves is mapped to the system $|F|$ of fibres of
the structure morphism $\mathbf{F}_{e}\to\mathbf{P}^{1}$. Then the linear
system $|C|$ is mapped to a linear system of curves of the type $|2H+kF|$ and,
by acting if necessary with elementary transformations, we may assume that the
general curve in this system is smooth of genus $g$. The adjoint system to
$|2H+kF|$ is $|(k+e-2)F|$, and therefore $g=k+e-1$. Since $g\geqslant 2$ we
must have $k\geqslant 3-e$. Moreover, since $k=(2H+kF)\cdot E\geqslant 0$, we
must also have $k\geqslant 0$. Thus we are in the hyperelliptic case. ∎

####  3.5.2 The non–hyperelliptic case

Next we consider case (3.1.b). In particular, $g\geqslant 3$.

######  Lemma 3.1.

In case (3.1.b) the general curve in $|C|$ is not hyperelliptic.

###### Proof.

Suppose by contradiction that the general curve in $|C|$ is hyperelliptic. Let
$p$ be a general point of $S^{\prime}$. Let us consider a general pencil
$\mathcal{P}$ of curves in $|C|$ having $p$ as base point. If $C$ is a general
curve in $\mathcal{P}$ and $q$ is the point conjugate to $p$ in the
$g^{1}_{2}$ on $C$, then, by the generality of $\mathcal{P}$ we can assume
that $q$ is not a base point of $\mathcal{P}$. Consider the Zariski closure
$D$ in $S^{\prime}$ of the set of points $q\in S^{\prime}$, such that $p+q$ is
a divisor of the $g^{1}_{2}$ on the curves of $\mathcal{P}$, i.e.,

$D=\overline{\bigcup_{C\in\mathcal{P}^{0}}\\{q:p+q\in g^{1}_{2}(C)\\},}$

with $\mathcal{P}^{0}\subseteq\mathcal{P}$ the Zariski dense open subset
parametrizing smooth members of $\mathcal{P}$. Then $D$ is a (rational) curve
on $S^{\prime}$.

Any curve of $|M|$ passing through $p$ (which, by the generality of $p$, is a
general curve of $|M|$), contains $D$, and therefore coincides with $D$. On
the other hand, we claim that $D$ does not cut out a canonical divisor on $C$,
hence it cannot be a member of $|M|$, a contradiction.

It thus only remains to prove the claim. Let $m$ be the multiplicity of $p$ in
$D$. Then

$D\cdot C=mp+q+R,$

where $R$ is contained in $\mathrm{Bs}(\mathcal{P})-p$, with
$\mathrm{Bs}(\mathcal{P})$ the base locus of $\mathcal{P}$. If $R=0$, the
claim holds (recall that $g\geqslant 3$). Otherwise, $R$ must contain all
$d-1$ points of $\mathrm{Bs}(\mathcal{P})-p$, since by the generality of
$\mathcal{P}$ there is a monodromy action on $\mathrm{Bs}(\mathcal{P})-p$, and
it acts as the full symmetric group by [1, p.111–113]. Then the claim follows,
as $d\geqslant 4g-4$. ∎

One has $C\cdot M=2g-2$. Hence for an irreducible curve $M$ to be contained in
a curve in $|C|$ is at most $2g-1$ conditions, and equality holds if and only
if $M$ is smooth and rational, and the restriction of $|C|$ to $M$ is a
complete linear series. We thus have

$\dim(|C-M|)\geqslant 3g-3+\varepsilon-(2g-1)=g-2+\varepsilon\geqslant 1,$

where $\varepsilon$ is the nonnegative integer such that
$C^{2}=4g-4+\varepsilon$.

##### (3.2)Proof of Proposition 3.1 in case $\dim(|C-M|)=g-2+\varepsilon$

In this case the general curve in $|M|$ is smooth and rational. Since
$\dim(|M|)=g-1$, we have $M^{2}=g-2$, $|M|$ is base point free, and
$\phi_{|M|}$ is a morphism mapping $S^{\prime}$ to a surface
$Y\subseteq\mathbf{P}^{g-1}$. In this map the curves $C$ are mapped to
canonical curves of degree $2g-2$. Since $Y$ has rational hyperplane sections,
we have only the following possibilities:

  1. (a)

$g=3$ and $Y$ is $\mathbf{P}^{2}$;

  2. (b)

$g=6$ and $Y$ is the 2–Veronese image of $\mathbf{P}^{2}$;

  3. (c)

$Y$ is a rational normal scroll.

In cases (a) and (b) we are in the planar case with $\delta=4$ and $\delta=5$
respectively. In case (c) we are in the trigonal case. ∎

Next we assume $s:=\dim(|C-M|)\geqslant g-1+\varepsilon$. Recall that

$C-M\sim-K_{S^{\prime}}+F.$

######  Lemma 3.3.

In the above setting, $F$ is in the fixed part of $|C-M|$.

###### Proof.

Let $D$ be an irreducible component of $F$. One has $C\cdot D=0$, hence
$D^{2}<0$. Then $K_{S^{\prime}}\cdot D\geqslant 0$, otherwise $D$ would be a
$(-1)$–curve contracted by $|C|$, a contradiction. So we have
$K_{S^{\prime}}\cdot F\geqslant 0$ and therefore $(-K_{S^{\prime}}+F)\cdot
F<0$. Indeed we also have $F^{2}<0$ by the Hodge Index Theorem, because
$C\cdot F=0$. Since $-K_{S^{\prime}}+F\sim C-M$ is effective, there is a
non–zero divisor $G\leqslant F$ that is in the fixed part of
$|-K_{S^{\prime}}+F|$. If $G=F$ we are done. Otherwise set $F_{1}=F-G$ and
consider the linear system $|-K_{S^{\prime}}+F_{1}|$. By the same argument as
above, we have $(-K_{S^{\prime}}+F_{1})\cdot F_{1}<0$, so there is a non–zero
divisor $G_{1}\leqslant F_{1}$ that is in the fixed part of
$|-K_{S^{\prime}}+F_{1}|$. If $G_{1}=F_{1}$ we are done. Otherwise we repeat
this argument till we eliminate all of $F$ from the fixed part of
$|-K_{S^{\prime}}+F|$. ∎

##### (3.4)Conclusion of the proof of Proposition 3.1

By Lemma 3.3 we can assume that $s=\dim(|-K_{S^{\prime}}|)\geqslant
g-1+\varepsilon\geqslant 2$. Consider the map

$\phi_{|-K_{S^{\prime}}|}:S^{\prime}\dasharrow Y\subseteq\mathbf{P}^{s}.$

One has

$(-K_{S^{\prime}})\cdot C=(C-M-F)\cdot
C=4g-4+\varepsilon-(2g-2)=2g-2+\varepsilon.$

Hence the curves in $|C|$ are mapped via $\phi_{|-K_{S^{\prime}}|}$ to curves
of degree $\delta\leqslant 2g-2+\varepsilon$. Let $\gamma$ be the number of
conditions that containing the curve $C$ imposes to the members of
$|-K_{S^{\prime}}|$. One has

$\begin{cases}\gamma=g&\text{if
}\,\,\,\varepsilon=0\,\,\,\text{and}\,\,\,C\,\,\,\text{is mapped
via}\,\,\,\phi_{|-K_{S^{\prime}}|}\,\,\,\text{ to a canonical curve};\\\
\gamma\leqslant g-1+\varepsilon&\text{otherwise}.\end{cases}$

In any event, unless $\varepsilon=0$, $s=g-1$ and $\gamma=g$, one has

$\dim(|-K_{S^{\prime}}-C|)=s-\gamma\geqslant 0$

so that $-K_{S^{\prime}}-C$ is effective. Hence we have $-K_{S^{\prime}}\sim
C+T$, with $T$ effective. Then $0\sim K_{S^{\prime}}+C+T\sim F+M+T$, which is
not possible.

So the only possibility is that $\varepsilon=0$, $s=g-1$, $\gamma=g$, and
$|-K_{S^{\prime}}|$ cuts out the complete canonical series on $C$. Since
$\dim(|-K_{S^{\prime}}|)\leqslant 9$, we have $g\leqslant 10$.

From $\left.-K_{S^{\prime}}\right|_{C}=K_{C}$ we deduce that
$\mathcal{O}_{C}(C)=\mathcal{O}_{C}(2K_{C})=\mathcal{O}_{C}(-2K_{S^{\prime}})$.

Let us set $|-K_{S^{\prime}}|=A+|B|$, where $A$ is the fixed part and $|B|$
the movable part of $|-K_{S^{\prime}}|$. Since $|-K_{S^{\prime}}|$ cuts out
the complete canonical series on $C$, and $C$ is not hyperelliptic by Lemma
3.1, we have that $C\cdot A=0$ and $|B|$ is not composed with a pencil, hence
the general curve in $|B|$ is irreducible.

Suppose first that $A=0$. Then, for any irreducible curve $B\in|B|$, one has
$p_{a}(B)=1$. Moreover, from the exact sequence

$0\rightarrow\mathcal{O}_{S^{\prime}}\rightarrow\mathcal{O}_{S^{\prime}}(B)\rightarrow\mathcal{O}_{B}(B)\rightarrow
0,$

and from $h^{0}(S^{\prime},\mathcal{O}_{S^{\prime}}(B))=g$, we deduce that
$h^{0}(B,\mathcal{O}_{B}(B))=g-1$, which implies that $B^{2}=g-1$. From the
Index Theorem applied to $C$ and $B$, we have that $C\sim 2B$. Moreover, there
is a birational map $\eta:S^{\prime}\dasharrow\mathbf{P}^{2}$, that maps $|B|$
to the linear system of cubics with $10-g$ simple base points. From this we
see we are in the bicanonical Del Pezzo case.

Suppose next that $A$ is non–zero. By Lemma 3.5 below, $A\cdot B=2$ and the
general member of $|B|$ is rational. By the same argument we made above, we
have $B^{2}=g-2$ and $\phi_{|B|}=\phi_{|-K_{S^{\prime}}|}$ is a birational map
of $S^{\prime}$ to its image $Y$ that is a surface of minimal degree in
$\mathbf{P}^{g-1}$.

If $Y=\mathbf{P}^{2}$, then $\phi_{|B|}$ maps $|B|$ to the linear system of
lines and $A$ to a conic $\Gamma$. The curves in $|C|$ are mapped to plane
quartics. Since $C\cdot A=0$, the linear system $|C|$ is mapped to a linear
system of quartics with 8 (proper or infinitely near) base points on the conic
$\Gamma$. This shows that we are in the planar case with $\delta=4$ and $b=8$
(see notation in Example 3.2).

If $Y$ is the 2–Veronese image of $\mathbf{P}^{2}$, then we may identify $Y$
with $\mathbf{P}^{2}$ and $\phi_{|B|}$ maps $|B|$ to the linear system of
conics and $A$ to a line $R$. The curves in $|C|$ are mapped to plane
quintics. Since $C\cdot A=0$, the linear system $|C|$ is mapped to a linear
system of quintics with 5 (proper or infinitely near) base points along the
line $R$. Hence we are in the planar case with $\delta=5$ and $b=5$ (see
Example 3.2 again).

Finally, if $g\geqslant 4$, $Y$ can be a rational normal scroll in
$\mathbf{P}^{g-1}$ and $\phi_{|B|}$ maps $|B|$ to the linear system
$|\mathcal{H}|$, where $\mathcal{H}$ is the hyperplane section class on $Y$.
The curves in $|C|$ are mapped to canonical trigonal curves $C^{\prime}$ on
$Y$. To see that $C^{\prime}$ is trigonal, consider a divisor $D$ cut out on
$C^{\prime}$ by a line of the ruling of $Y$, and let $d$ be its degree; by the
Geometric Riemann–Roch Theorem, the linear series $|D|$ has dimension $d-2$,
and then it follows from Clifford’s Theorem that it is a $g^{1}_{3}$ as
$C^{\prime}$ is non-hyperelliptic. Let $F$ be a line of the ruling of $Y$. As
in Example 3.5, one sees that $C^{\prime}\in|3\mathcal{H}-(g-4)F|$, so that
$Y$ is smooth, unless maybe $g=4$. In the latter case, $Y$ can be a quadric
cone, so its minimal desingularization is $\mathbf{F}_{2}$ and in this case we
will work on $\mathbf{F}_{2}$ rather than on $Y$.

Suppose that $Y=\mathbf{F}_{e}$ and let, as usual, $E$ be the section such
that $E^{2}=-e$. Then $g$ and $e$ have the same parity and

$\mathcal{H}\sim E+\big{(}\frac{e+g}{2}-1\big{)}F$

and accordingly

$C^{\prime}\sim 3\mathcal{H}-(g-4)F\sim 3E+\big{(}\frac{3e+g}{2}+1\big{)}F.$

The morphism $\phi_{|B|}:S^{\prime}\to Y$ consists in blowing down a number of
$(-1)$–exceptional divisors. Let $D$ be the total such exceptional divisor.
Then

$-K_{S^{\prime}}=-\phi_{|B|}^{*}(K_{\mathbf{F}_{e}})-D.$

Since $\phi_{|B|}^{*}(\mathcal{H})=B$, we have

$A\sim\phi_{|B|}^{*}(-K_{\mathbf{F}_{e}}-\mathcal{H})-D.$

By (3.4.1), we have

$-K_{\mathbf{F}_{e}}-\mathcal{H}\sim E+\big{(}\frac{e-g}{2}+3\big{)}F,$

thus

$A\sim\phi_{|B|}^{*}\Big{(}E+\big{(}\frac{e-g}{2}+3\big{)}F\Big{)}-D.$

One has

$C^{\prime}\cdot\Big{(}E+\Big{(}\frac{e-g}{2}+3\Big{)}F\Big{)}=10-g$

and recall that $C\cdot A=0$. Moreover

$-K_{\mathbf{F}_{e}}\cdot\Big{(}E+\Big{(}\frac{e-g}{2}+3\Big{)}F\Big{)}=8-g<10-g.$

In conclusion:

  1. (i)

$S^{\prime}$ is obtained from $\mathbf{F}_{e}$ by blowing up the (curvilinear)
scheme $Z$ of length $10-g$ that is the complete intersection of a smooth
curve $C^{\prime}$ with a curve $N$ of $\big{|}E+(\frac{e-g}{2}+3)F\big{|}$;

  2. (ii)

the linear system $|C|$ on $S^{\prime}$ is the strict transform of the linear
systems of the curves of $|C^{\prime}|$ on $\mathbf{F}_{e}$ containing $Z$ and

$C^{2}=C^{\prime 2}-(10-g)=3g+6-(10-g)=4g-4,$

as we wanted;

  3. (iii)

the strict transform $A$ of $N$ on $S^{\prime}$ splits off the anticanonical
system on $S^{\prime}$.

Thus we are here in the trigonal case. ∎

This concludes the proof of Proposition 3.1, hence also that of Theorem 1.2.
We end this section by an elementary lemma that has been used above.

######  Lemma 3.5.

Let $S$ be a smooth rational surface, and write

$|-K_{S}|=A+|B|$

with $A$ the fixed part and $|B|$ the movable part. We assume that $A$ is
effective and nonzero, and $B$ big and nef. Then $A\cdot B=2$, and the general
member of $|B|$ is rational if it is irreducible.

###### Proof.

First, $h^{0}(-A)=0$. Next, by Kawamata–Viehweg vanishing,
$h^{1}(-A)=h^{1}(B+K_{S})=0$. Eventually,

$h^{2}(-A)=h^{0}(K_{S}+A)=h^{0}(-B)=0.$

Thus $\chi(-A)=0$, hence by Riemann–Roch $A\cdot B=2$. Therefore,
$(B+K_{S})\cdot B=-A\cdot B=-2$, and the result follows. ∎

## Classification of surfaces with a hyperelliptic section

This section is dedicated to Theorem 1.5. We prove it in Subsections 4.1 and
4.2, and formulate it in useful alternative ways in Subsection 4.3.

We consider a minimal resolution of singularities $S^{\prime}\to S$ and work
on $S^{\prime}$. Being a hyperplane section of $S$, $C$ must be contained in
the smooth locus of $S$, so that considered in $S^{\prime}$ it cannot
intersect any curve contracted by $S^{\prime}\to S$. It follows that,
considered in $S^{\prime}$, $C$ is a big and nef divisor such that $C^{2}=d$.

We want to apply the Reider–Beltrametti–Sommese Theorem below to $C$ for
$k=1$, which requires $C^{2}\geqslant 9$. We thus split the proof of Theorem
1.5 in two: the _general case_ in which we assume that $d\geqslant 9$,
equivalently either $g\geqslant 3$ or $g=2$ and $d\neq 7,8$, and the _sporadic
cases_ in which $g=2$ and $d=7$ or $8$.

### Proof of Theorem 1.5 in the general case

######  Theorem 4.1 (Reider [37], Beltrametti–Sommese [4]).

Let $L$ be a nef line bundle on a smooth surface $S$, and $k$ be a positive
integer. Assume that $L^{2}\geqslant 4k+5$, and there exists a $0$-dimensional
subscheme $Z\subseteq S$ of length $k+1$ such that the restriction

$H^{0}(S,K_{S}+L)\to H^{0}(Z,\left.(K_{S}+L)\right|_{Z})$

is not surjective. Then there exists an effective divisor $D$ containing $Z$,
and such that

$L\cdot D-k-1\leqslant D^{2}<k+1.$

##### (4.2)Proof of Theorem 1.5 in the general case

The general case means that we assume $d\geqslant 9$, see above.

For all divisor $x_{1}+x_{2}$ in the $g^{1}_{2}$ of $C$, the adjoint system
$|K_{S^{\prime}}+C|$ does not separate $x_{1}$ and $x_{2}$, so Theorem 4.1
tells us that there exists a divisor $D$ on $S^{\prime}$ such that

$x_{1},x_{2}\in D\quad\text{and}\quad D\cdot C-2\leqslant D^{2}\leqslant 1.$

The inequality $D\cdot C\leqslant 3$ forbids that $D$ contains $C$: indeed if
$D=C+D^{\prime}$ with $D^{\prime}$ effective, then $D\cdot C=C^{2}+C\cdot
D^{\prime}\geqslant C^{2}\geqslant 9$, a contradiction with $D\cdot C\leqslant
3$. This implies that $D$ cannot be fixed as $x_{1}+x_{2}$ moves in the
$g^{1}_{2}$, for otherwise $D$ would necessarily have $C$ as an irreducible
component.

Let $M$ be the part of $D$ that contains $x_{1}$ and $x_{2}$. By the previous
observation, the family of these $M$ has no fixed part. Thus we have

$2\leqslant M\cdot C\leqslant D\cdot C\leqslant 3.$

We first claim that $M$ is irreducible. Indeed, otherwise at least one
component $M^{\prime}$ of $M$ would verify $M^{\prime}\cdot C=1$, hence it
would be mapped to a line by the map $S^{\prime}\to S$, and give a ruling of
$S$ by lines, which implies that $S$ is a cone by Theorem 2.1. So $M$ is
irreducible, and we have the following possibilities 1) and 2) to consider.

1) $M\cdot C=2$. Then $M$ is mapped to a conic by $S^{\prime}\to S$, in
particular it is a rational curve. Since it moves in a family parametrized by
the $g^{1}_{2}$ on $C$, the surface $S$ is then unirational hence rational, so
the conclusion of our theorem holds in this case.

2) $M\cdot C=3$. Then $M$ is mapped to a cubic by $S^{\prime}\to S$, hence it
spans either a) a plane or b) a $3$-space in $\mathbf{P}^{d-g+1}$.

This case may happen only if $d\leqslant 9$, for the following reason. If
$D\cdot C=3$, then $D^{2}=1$, and thus by the Hodge Index Theorem,

$D^{2}\cdot C^{2}\leqslant(D\cdot C)^{2}\Longleftrightarrow C^{2}\leqslant 9.$

The upshot is that $d=9$ and $g=2$ or $3$, hence $C$ is an intersection of
quadrics by Green’s Theorem 2.2.

In case a), the three points of $M\cap C$ are located on the line $\langle
M\rangle\cap\langle C\rangle$: to see that the latter is indeed a line, note
that $\langle C\rangle$ is a hyperplane in $\langle S\rangle$, hence it is
impossible that the plane $\langle M\rangle$ be contained in $\langle
C\rangle$ for general $C$. On the other hand, $C$ is an intersection of
quadrics, so this situation is impossible. In conclusion, case a) cannot
happen.

In case b) the curve $M$ is rational. Hence if $M^{2}=0$, then the curves $M$
are parametrized by a curve $\mathcal{H}_{M}$. The curve $\mathcal{H}_{M}$ is
rational, because there exists a morphism $\mathbf{P}^{1}\to\mathcal{H}_{M}$
mapping the element $x_{1}+x_{2}$ of the $g^{1}_{2}$ to the corresponding
curve $M$. The upshot is that in this case $|M|$ is a base point free pencil.
The restriction of such a pencil to $C$ would be a base point free $g^{1}_{3}$
(remember that $C$ viewed on $S^{\prime}$ does not intersect any curve
contracted by $S^{\prime}\to S$) containing all divisors of the $g^{1}_{2}$:
by Lemma 4.3 below this is impossible. So we must have $M^{2}>0$. But in this
case $|M|$ cuts out a $g^{r}_{3}$ on $C$ with $r\geqslant 2$, in contradiction
with $g\geqslant 2$.

The conclusion is that only case 1) may happen, in which the conclusion of our
theorem holds. ∎

######  Lemma 4.3.

Let $C$ be a curve of genus $g$, and assume it has a $g^{1}_{2}$,
$\mathfrak{l}$, and a $g^{1}_{3}$, $\mathfrak{m}$. We consider the following
condition:

(4.3.1) $\forall x_{1}+x_{2}\in\mathfrak{l},\ \exists z\in C:\quad
x_{1}+x_{2}+z\in\mathfrak{m}.$

If (4.3.1) holds, then either $z$ is a base point of $\mathfrak{m}$, or $g=0$.
If (4.3.1) does not hold, then $C$ is birational to a curve of type $(2,3)$ in
$\mathbf{P}^{1}\times\mathbf{P}^{1}$, in particular $g\leqslant 2$.

###### Proof.

First assume that (4.3.1) holds. Consider $z,z^{\prime}\in C$ such that
$x_{1}+x_{2}+z\in\mathfrak{m}$ and
$x^{\prime}_{1}+x^{\prime}_{2}+z^{\prime}\in\mathfrak{m}$ for some members
$x_{1}+x_{2}$ and $x^{\prime}_{1}+x^{\prime}_{2}$ of $\mathfrak{l}$. Then

$x_{1}+x_{2}+z\sim
x^{\prime}_{1}+x^{\prime}_{2}+z^{\prime},\quad\text{hence}\quad z\sim
z^{\prime},$

and either $C$ is rational, or $z=z^{\prime}$. In the latter case $z$ is a
base point of $\mathfrak{m}$. This proves the first part of the lemma.

For the second part, consider the map

$\phi=(\phi_{\mathfrak{l}},\phi_{\mathfrak{m}}):C\to\mathbf{P}^{1}\times\mathbf{P}^{1}$

defined by the two pencils $\mathfrak{l}$ and $\mathfrak{m}$. It is
straightforward to verify that (4.3.1) holds if and only if $\phi$ is not
birational, and this proves the second part of the statement. ∎

### Proof of Theorem 1.5 in the sporadic cases

##### (4.1)Proof of Theorem 1.5 in the sporadic cases

The sporadic cases are those when $g=2$ and $d=7$ or $8$. Let
$S\subseteq\mathbf{P}^{d-1}$, $d=7,8$, be a degree $d$ surface with one
hyperplane section a genus $2$ curve $C$, such that $S$ is not a cone, hence
not a scroll either by Theorem 2.1. Since $d>2g-2$, we have
$\kappa(S)=-\infty$ as in Subsection 3.3.

Assume by contradiction that it is not rational. Then $S$ is abstractly a
ruled surface, necessarily elliptic as it is not a scroll. Letting
$S^{\prime}$ be a minimal resolution of the singularities of $S$, and $\Sigma$
a minimal model of $S^{\prime}$; $\Sigma$ is a ruled surface over an elliptic
curve $B$. By [12, Thm 1.4], since $d\geqslant 7$, if
$S\subseteq\mathbf{P}^{d-1}$ is not $1$-weakly defective then for general
$p_{1},p_{2}\in S$, the general section of $S$ by a hyperplane tangent at both
$p_{1}$ and $p_{2}$ is a curve $C_{0}$ with two nodes at $p_{1}$ and $p_{2}$
and no other singularity.

If $C_{0}$ is irreducible, then it is a rational curve because $S$ has
sectional genus $2$. Moreover, $C_{0}$ dominates the base $B$ since
$C_{0}^{2}>0$, a contradiction to the non-rationality of $S$. Otherwise, there
are the two following possibilities:

  1. (i)

$C_{0}=\bar{C}+F_{1}+F_{2}$, with $F_{i}\cap\bar{C}=p_{i}$ for $i=1,2$ and
$F_{1}\cap F_{2}=\varnothing$, all three $\bar{C},F_{1},F_{2}$ smooth;

  2. (ii)

$C_{0}=\bar{C}+F$, $\bar{C}\cap F=\\{p_{1},p_{2}\\}$, $\bar{C}$ and $F$
smooth.

In case (i), by generality the situation must be symmetric with respect to
$p_{1}$ and $p_{2}$, hence $F_{1}$ and $F_{2}$ are algebraically equivalent.
Let $F$ be their algebraic equivalence class. Necessarily, $F^{2}=0$, and $F$
moves in a $1$-dimensional family. Moreover,

$C\cdot F=\bar{C}\cdot F=1$

hence $F$ is mapped to a line, and $S$ is a scroll, a contradiction.

In case (ii), at least one among $\bar{C}$ and $F$ dominates the base $B$; we
assume it is $\bar{C}$, thus $\bar{C}$ has genus at least one. Since $C$ has
arithmetic genus $2$, the only possibility is that $\bar{C}$ has genus $1$ and
$F$ has genus $0$. Therefore $F$ is in the class of the ruling of $\Sigma$.
This leads to a contradiction, as the two general points $p_{1}$ and $p_{2}$
may not sit on a common ruling.

It thus only remains to examine the finitely many possibilities in which
$S\subseteq\mathbf{P}^{d-1}$ is 1-weakly defective, listed in Proposition 4.2
below.

1) If $S$ were on the cone over the Veronese surface, then its hyperplane
section $C$ would be on the Veronese surface itself. But then necessarily
$S\subseteq\mathbf{P}^{6}$, i.e., $d=7$, and since all curves on the Veronese
surface have even degree this case cannot be realized.

2) In this case $S$ has sectional genus $3$ (the general hyperplane section of
$S$ is birational to a plane quartic everywhere tangent to the branch curve of
the double cover $S\xrightarrow{2:1}\mathbf{P}^{2}$), and this is contrary to
our assumptions.

3) In this case $S$ is contained in a cone
$K(\Lambda,B)\subseteq\mathbf{P}^{d-1}$. Then $B$ is an irreducible curve in
$\mathbf{P}^{d-3}$, hence it has degree $\delta\geqslant d-3$. The cone
$K(\Lambda,B)$ is swept out by a $1$-dimensional family $(\Pi_{b})_{b\in B}$
of planes containing $\Lambda$. Let $m$ be the number of points cut out on $C$
off $\Lambda$ by a general $\Pi_{b}$. If $m=1$, then $S$ is ruled by lines
which is excluded. Otherwise, let us consider the section of $C$ by a general
hyperplane $H$ containing the line $\Lambda$: this is the sum of the points
$\Pi_{b}\cap C$ for $b\in H\cap B$, plus possibly some points on the vertex
line $\Lambda$. The upshot is that

$d=\deg(H\cap S)\geqslant m\delta\geqslant m(d-3),$

which is possible only if $d\leqslant 6$, in contradiction with our
assumptions. Thus this case cannot happen either.

This ends the proof by contradiction that $S$ is rational. Now, let us write

$|K_{S^{\prime}}+C|=F+|M|,$

where as usual $F$ is the fixed part, and $|M|$ the mobile part. Since $S$ is
rational, it cuts out the complete canonical series on $C$, hence $|M|$ has
dimension $1$, and $F\cdot C=0$ and $M\cdot C=2$. Thus $|M|$ is mapped to a
pencil of conics on $S$, and the proof is over. ∎

######  Proposition 4.2 (Chiantini–Ciliberto [12]).

Let $S\subseteq\mathbf{P}^{r}$, $r\geqslant 6$, be a $1$-weakly defective
surface with isolated singularities. Then $S$ falls into one of the following
four cases.

  1. 1)

$S\subseteq\mathbf{P}^{6}$ is contained in the cone over the Veronese surface
and vertex a point, i.e.,

$S\subseteq K\bigl{(}p,v_{2}(\mathbf{P}^{2})\bigr{)}.$

  2. 2)

$S\subseteq\mathbf{P}^{6}$ is a quartic double plane
$\pi:S\xrightarrow{2:1}\mathbf{P}^{2}$ embedded by the complete linear system
$|\pi^{*}2H|$, where $H$ is the line class in $\mathbf{P}^{2}$.

  3. 3)

$S$ is contained in the cone with vertex a line over a curve $B$,

$S\subseteq K\bigl{(}\Lambda,B\bigr{)}\subseteq\mathbf{P}^{r}.$

The proof is a direct application of [12, Thm 1.3] and is left to the reader.

### The Castelnuovo Classification

As a consequence of Theorem 1.5, we can prove the following classification
result, from which Corollary 1.6 will follow.

######  Corollary 4.1.

Let $S$ be as in Theorem 1.5. If $S$ is not a cone, then $S$ is represented by
the complete linear system

$|2E+(g+1+e)F|$

on the rational ruled surface $\mathbf{F}_{e}$, $0\leqslant e\leqslant g+1$,
or possibly by a linear subsystem defined by simple base points along a
curvilinear scheme.

###### Proof.

By Theorem 1.5, we know that $S^{\prime}$ is rational and contains a pencil
$|M|$ of rational curves such that $C\cdot M=2$. Then there is a birational
morphism $S^{\prime}\to\Sigma$, with $\Sigma$ a minimal rational ruled surface
on which $|M|$ is mapped to the ruling. Since $C\cdot M=2$ on $S^{\prime}$,
the linear system $|C|$ is mapped in $\Sigma$ to a linear system with only
simple and double base points. One may get rid of all double base points by
performing elementary transformations. Thus we may assume that
$\Sigma\cong\mathbf{F}_{e}$ and the general member of $|C|$ (in $\Sigma$) is
smooth.

In our usual notation, $C\sim 2E+kF$, and one finds $k=g+1+e$ with the
adjunction formula. Then,

$0\leqslant C\cdot E=g+1-e.$

∎

######  Corollary 4.2.

Let $S$ be as in Theorem 1.5. If $S$ is not a cone, then it is rational and
represented by one of the following linear systems, or one obtained from those
by adding simple base points along a curvilinear subscheme:

  1. a)

$\mathcal{L}_{g+3}(g+1,2)$, the linear system of plane curves of degree $g+3$
with one base point of multiplicity $g+1$ and one double point;

  2. b)

$\mathcal{L}_{2g+2-\mu}([2g-\mu,2^{g-\mu}])$, $\mu=0,\ldots,g$, the linear
system of plane curves of degree $2g+2-\mu$, with one base point $p$ of
multiplicity $2g-\mu$, and $g-\mu$ double base points infinitely near to $p$,
pairwise distinct on the exceptional divisor of the blow-up of $p$.

###### Proof.

Take into account Corollary 4.1. If $e=0$, we are in case a). If $e=1$, we are
in case b) with $\mu=g$. If $e>1$, we perform $e-1$ elementary transformations
at general points, thus ending up on $\mathbf{F}_{1}$, after what we arrive at
$\mathbf{P}^{2}$ by contracting the $(-1)$-curve $E$ on $\mathbf{F}_{1}$, with
a linear system as in case b) with $\mu=g+1-e$. ∎

##### (4.3)

The classification of rational surfaces such that the general hyperplane
section is hyperelliptic had been classically worked out by Castelnuovo [8].
In more recent times, the classification of arbitrary smooth surfaces having
at least one smooth hyperelliptic hyperplane section has been worked out by
Serrano [41] and Sommese–Van de Ven [42]. Our classification takes into
account singular surfaces as well, with the same condition that they have one
hyperelliptic section, albeit with the restriction that the degree $d$ be at
least $10$, or at least $2g+3$ if $g=2$ or $3$. Note that without this
assumption there are other possibilities, including rational surfaces such
that the general hyperplane section is not hyperelliptic, if the pair $(g,d)$
equals $(3,8)$ or $(4,9)$, and an elliptic ruled surface, if $(g,d)=(3,8)$,
cf. [41].

## Gaussian maps and their cokernels

We consider $C\subseteq\mathbf{P}^{r}$ a linearly normal curve, and let
$L=\left.\mathcal{O}_{\mathbf{P}^{r}}(1)\right|_{C}$. In this section we
define the Gaussian map $\gamma_{C,L}$, and compute its corank in a number of
cases, thus proving Theorem 1.3.

##### (5.1)The Gaussian map $\gamma_{C,L}$

Let $R_{C,L}$ be the kernel of the multiplication map

$\mu_{C,L}:H^{0}(K_{C})\otimes H^{0}(L)\to H^{0}(K_{C}+L).$

The Gaussian map

$\textstyle\gamma_{C,L}:R_{C,L}\to H^{0}(2K_{C}+L)$

is the map locally defined as $\sum_{i}s_{i}\otimes
t_{i}\mapsto\sum_{i}(s_{i}\cdot dt_{i}-t_{i}\cdot ds_{i})$.

We are interested in the corank of $\gamma_{C,L}$, i.e., the dimension of its
cokernel in $H^{0}(2K_{C}+L)$. Most of the time, Castelnuovo Theorem 2.1 tells
us that the multiplication map $\mu_{C,L}$ is surjective, which readily gives
the dimension of $R_{C,L}$; then it suffices to compute the dimension of the
kernel of $\gamma_{C,L}$ to find its corank. To do so, we shall use the
canonical identification which we now explain.

##### (5.2)Restriction maps to the diagonal

We shall interpret the maps $\mu_{C,L}$ and $\gamma_{C,L}$ in terms of
operations on the product $C\times C$. We let $\mathrm{pr}_{1}$ and
$\mathrm{pr}_{2}$ be the two projections,

$\textstyle{C\times
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{\mathrm{pr}_{2}}$$\textstyle{C}$$\textstyle{C}$

and $\Delta\subseteq C\times C$ be the diagonal. The multiplication map
$\mu_{C,L}$ identifies with the restriction map $r_{1}$ to the diagonal
$\Delta$, as indicated in the diagram below.

$\textstyle{H^{0}(\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{1}}$$\scriptstyle{=}$$\textstyle{H^{0}(K_{C})\otimes
H^{0}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{C,L}}$$\textstyle{H^{0}\bigl{(}\left.(\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L)\right|_{\Delta}\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{H^{0}(K_{C}+L)}$

From this we deduce the following identification of the kernel of the
multiplication map $\mu_{C,L}$,

$R_{C,L}\cong H^{0}(C\times
C,\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-\Delta).$

In turn, the Gaussian map $\gamma_{C,L}$ identifies with yet again a
restriction map to the diagonal, namely the following map $r_{2}$,

$r_{2}:H^{0}(C\times
C,\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-\Delta)\to
H^{0}\bigl{(}\Delta,\left.(\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-\Delta)\right|_{\Delta}\bigr{)}\cong
H^{0}(C,2K_{C}+L).$

For the last identification, note that $\left.\Delta\right|_{\Delta}\cong
N_{\Delta/C\times C}\cong-K_{C}$. The upshot is the following identification
of the kernel of $\gamma_{C,L}$,

(5.2.1) $\ker(\gamma_{C,L})\cong H^{0}(C\times
C,\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta).$

######  Proposition 5.3.

Let $C$ be a hyperelliptic curve of genus $g$, and $L$ an effective line
bundle. There is an identification

(5.3.1) $\ker(\gamma_{C,L})\cong
H^{0}\bigl{(}C,(g-3)\mathfrak{g}\bigr{)}\otimes
H^{0}\bigl{(}C,L-2\mathfrak{g}\bigr{)},$

where $\mathfrak{g}$ denotes the class of the $g^{1}_{2}$ on $C$.

###### Proof.

We shall prove that the right-hand-sides of (5.3.1) and (5.2.1) are
isomorphic, which suffices to prove the proposition. First assume that there
exists an effective divisor

$E\in|\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta|$

in $C\times C$. Then for all $p\in C$, we have the following equality as
divisors on the fibre $\mathrm{pr}_{2}^{*}p\cong C$,

$\textstyle E\cap\mathrm{pr}_{2}^{*}p=\sum_{i=1}^{2g-4}p_{i},$

with the points $p_{1},\ldots,p_{2g-4}$ subject to the condition that
$2p+\sum_{i=1}^{2g-4}p_{i}\in|K_{C}|.$ In general $2p\not\in\mathfrak{g}$, and
then there must be two of the points $p_{i}$’s, say $p_{1}$ and $p_{2}$, such
that $p+p_{1},p+p_{2}\in\mathfrak{g}$. Then necessarily $p_{1}=p_{2}$, and
thus

$\textstyle
E\cap\mathrm{pr}_{2}^{*}p=2p_{1}+\sum_{i=3}^{2g-4}p_{i}\quad\text{with}\quad\left\\{\begin{aligned}
&p+p_{1}\in\mathfrak{g},\ \text{and}\ \\\
&\textstyle\sum_{i=3}^{2g-4}p_{i}\in|K_{C}-2\mathfrak{g}|=|(g-3)\mathfrak{g}|.\end{aligned}\right.$

This shows that $E$ must contain the graph $I\subseteq C\times C$ of the
hyperelliptic involution with multiplicity $2$; in other words, $2I$ is a
fixed part of the linear system
$|\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta|$, and thus

$H^{0}\bigl{(}C\times
C,\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta\bigr{)}=H^{0}\bigl{(}C\times
C,\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta-2I\bigr{)}.$

The graph $I$ is the unique divisor in
$|\mathrm{pr}_{1}^{*}\mathfrak{g}+\mathrm{pr}_{2}^{*}\mathfrak{g}-\Delta|$,
hence

(5.3.2)
$\displaystyle\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta-2I$
$\displaystyle\sim\mathrm{pr}_{1}^{*}\bigl{(}(g-3)\mathfrak{g}\bigr{)}+\mathrm{pr}_{2}^{*}\bigl{(}L-2\mathfrak{g}\bigr{)},$

and the conclusion follows if
$\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta$ is effective. On the
other hand, the linear equivalence (5.3.2) implies that
$\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta$ is effective if
$\mathrm{pr}_{1}^{*}\bigl{(}(g-3)\mathfrak{g}\bigr{)}+\mathrm{pr}_{2}^{*}\bigl{(}L-2\mathfrak{g}\bigr{)}$
is effective, hence the proposition holds unconditionally. ∎

######  Proposition 5.4.

Let $C$ be a non-hyperelliptic curve of genus $3$, and $L$ an effective line
bundle on $C$. There is an identification

(5.4.1) $\ker(\gamma_{C,L})\cong H^{0}\bigl{(}C,L-3K_{C}).$

###### Proof.

As in proof of Proposition 5.3, we will identify the right-hand-side of
(5.4.1) with that of (5.2.1), and this will end the proof. Assume there exists
an effective divisor

$E\in|\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta|$

on $C\times C$. Then for all $p\in C$, we have the following equality as
divisors on the fibre $\mathrm{pr}_{2}^{*}p\cong C$,

$E\cap\mathrm{pr}_{2}^{*}(p)=p_{1}+p_{2},$

where $p_{1},p_{2}\in C$ are the only two points on $C$ such that
$2p+p_{1}+p_{2}\in K_{C}$: seeing $C$ as a plane quartic, $p_{1},p_{2}$ are
the residual intersection points of $C$ and its tangent line at $p$. Thus $E$
must contain the tangential correspondence

$T=\overline{\left\\{(q,p)\in C\times C:q\neq p\text{ and
}q\in\mathbf{T}_{p}C\right\\}},$

where $\mathbf{T}_{p}C$ denotes the tangent line to $C$ at $p$ in its model as
a plane quartic. In other words, the linear system
$|\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta|$ has $T$ as a fixed
part, and therefore

$H^{0}\bigl{(}C\times
C,\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta\bigr{)}=H^{0}\bigl{(}C\times
C,\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta-T\bigr{)}.$

Let us now compute the class of $T$. For all $(q,p)\in C\times C$, we have the
equalities as divisors on $C$,

$T\cap\mathrm{pr}_{1}^{*}(q)=C\cap\mathrm{D}^{q}C-2q\quad\text{and}\quad
T\cap\mathrm{pr}_{2}^{*}(p)=C\cap\mathbf{T}_{p}C-2p,$

where $\mathrm{D}^{q}C$ is the first polar of $q$ with respect to $C$ (seen as
a plane quartic), see, e.g., [18, Appendix A]. First polars with respect to a
plane quartic are plane cubics, hence $\mathrm{D}^{q}C$ cuts out the divisor
class $3K_{C}$ on $C$, since the latter is canonically embedded. We thus find

$T\cdot\mathrm{pr}_{1}^{*}(q)\sim 3K_{C}-2q\quad\text{and}\quad
T\cdot\mathrm{pr}_{2}^{*}(p)\sim K_{C}-2p,$

hence

$T\sim\mathrm{pr}_{1}^{*}K_{C}+p_{2}^{*}(3K_{C})-2\Delta.$

Eventually, we find

(5.4.2)
$\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta-T\sim\mathrm{pr}_{2}^{*}(L-3K_{C}),$

and this ends the proof if
$\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta$ is effective. On the
other hand it follows from (5.4.2) that
$\mathrm{pr}_{1}^{*}K_{C}+\mathrm{pr}_{2}^{*}L-2\Delta$ is effective if
$L-3K_{C}$ is effective, hence the result holds unconditionally. ∎

######  Proposition 5.5.

Let $C$ be a smooth projective curve of genus $g\geqslant 2$, and $L$ an
effective line bundle on $C$ of degree $d>0$.

  1. (a)

If $C$ is hyperelliptic, then

$\operatorname{cork}(\gamma_{C,L})=2g+2-g\cdot h^{1}(L)+(g-2)\cdot
h^{1}(L-2\mathfrak{g})-\operatorname{cork}(\mu_{C,L}),$

where $\mathfrak{g}$ is the $g^{1}_{2}$ on $C$.

  2. (b)

If $C$ is non-hyperelliptic of genus $3$, then

$\operatorname{cork}(\gamma_{C,L})=h^{0}(4K_{C}-L)-3\cdot
h^{1}(L)-\operatorname{cork}(\mu_{C,L}).$

Note that if $L$ is not effective of positive degree, the multiplication map
$\mu_{C,L}$ has kernel $R_{C,L}=0$, so that

$\operatorname{cork}(\gamma_{C,L})=h^{0}(2K_{C}+L).$

Moreover we emphasize that the corank of the multiplication map $\mu_{C,L}$
may be computed in virtually any situation, using for instance Castelnuovo’s
Theorem 2.1 or the Base-Point-Free Pencil Trick; see also [13, §1], [24, Thm.
(4.e.1)], and [35]. Part (b) of the above proposition had already appeared as
[29, Prop. 2.9(a)].

###### Proof.

One has

$\displaystyle\operatorname{cork}(\gamma_{C,L})$
$\displaystyle=h^{0}(2K_{C}+L)-\dim(R_{C,L})+\dim(\ker\gamma_{C,L})$
$\displaystyle=h^{0}(2K_{C}+L)-h^{0}(K_{C})h^{0}(L)+h^{0}(K_{C}+L)-\operatorname{cork}(\mu_{C,L})+\dim(\ker\gamma_{C,L})$
$\displaystyle=(3g-3+d)-g\bigl{(}1-g+d+h^{1}(L)\bigr{)}+(g-1+d)-\operatorname{cork}(\mu_{C,L})+\dim(\ker\gamma_{C,L})$
$\displaystyle=(g+4)(g-1)+d(2-g)-gh^{1}(L)-\operatorname{cork}(\mu_{C,L})+\dim(\ker\gamma_{C,L})$

by Riemann–Roch and the fact that $L$ is effective of positive degree. For
hyperelliptic $C$, one has

$h^{0}\bigl{(}(g-3)\mathfrak{g}\bigr{)}=g-2\quad\text{and}\quad
h^{0}(L-2\mathfrak{g})=d-g-3+h^{1}(L-2\mathfrak{g}),$

and thus Proposition 5.3 gives the result. For $C$ a genus $3$ curve,
Proposition 5.4 gives the result, noting that

$14-d+h^{0}(C,L-3K_{C})=h^{0}(C,4K_{C}-L)$

by Riemann–Roch and Serre duality. ∎

##### (5.6)Proof of Theorem 1.3

Let us first consider the case when $C$ is hyperelliptic. If $d\geqslant
2g+3$, then $L$ is very ample and non-special, hence $\mu_{C,L}$ is surjective
by Castelnuovo’s Theorem 2.1; moreover $h^{1}(L-2\mathfrak{g})=0$ for degree
reasons as well, so that the result follows from Proposition 5.5.

If $d\geqslant g+4$ and $L$ is general, then

$L\sim p_{1}+\cdots+p_{g}+2\mathfrak{g}+D_{0}$

for some general points $p_{1},\ldots,p_{g}\in C$ and some effective divisor
$D_{0}$. In particular we may assume that $p_{1},\ldots,p_{g}$ impose
independent conditions to the canonical series $|K_{C}|$, hence
$h^{1}(L)=h^{1}(L-2\mathfrak{g})=0$ by Serre duality, and moreover
$h^{1}(L-q)=h^{1}(L-q-q^{\prime})=0$ for all $q,q^{\prime}\in C$. It follows
that $L$ is very ample, hence $\mu_{C,L}$ is surjective by Castelnuovo’s
Theorem 2.1, and the result follows from Proposition 5.5.

We now consider the case when $C$ is non-hyperelliptic of genus $3$. If
$d\geqslant 2g+1=7$, then $L$ is very ample and non-special, hence $\mu_{C,L}$
is surjective by Castelnuovo’s Theorem 2.1, and the result follows from
Proposition 5.5.

If $d=2g=6$, then $L$ is base-point-free and non-special, and the map induced
by $|L|$ may identify at most two points $p$ and $q$, which happens if and
only if $L=K_{C}+p+q$ (all this can be seen by standard considerations
involving the Riemann–Roch Theorem). It follows that $\mu_{C,L}$ is surjective
in this case as well, and then Proposition 5.5 gives the result.

If $g+1=4\leqslant d\leqslant 5$ and $L$ is general, then $|L|$ is non-
special, and the following may happen: if $d=5$, $|L|$ is base-point-free and
maps $C$ to a plane quintic; if $d=4$, $|L|$ is a base-point-free $g^{1}_{4}$.
In all cases, it follows from Theorem 2.1 that $\mu_{C,L}$ is surjective, and
the result follows from Proposition 5.5 as in the previous cases. ∎

## Ribbons and extensions

In this section we interpret the extensions of a smooth polarized curve
$(C,L)$ in terms of the integration of ribbons over $(C,L)$, under the
assumption that it satisfies Property $N_{2}$. This leads to a necessary
condition for $(C,L)$ to be extendable. We also define universal extensions,
and give a criterion for their existence. Most results in this section are
essentially an adaptation of some in [16], and we will thus be brief; we also
propose various enhancements with respect to [16].

######  Definition 6.1.

Let $(X,L)$ be a polarized variety such that $L$ is very ample, and consider
$X\subseteq\mathbf{P}^{N}$ the projective embedding defined by $|L|$. For all
$k\in\mathbf{N}^{*}$, the polarized variety $(X,L)$ is _$k$ -extendable_ if
there exists $Y\subseteq\mathbf{P}^{N+k}$, not a cone, and an $N$-dimensional
linear subspace $\Lambda\subseteq\mathbf{P}^{N+k}$ such that $X=Y\cap\Lambda$.

In the above situation, we say that $Y$ is a _non-trivial $k$-extension_, or
simply a _non-trivial extension_ , of $(X,L)$. We say that $(X,L)$ is
_extendable_ if it is $k$-extendable for some $k>0$. The trivial extension of
$(X,L)$ denotes the cone with vertex a point over $X$, in its embedding
defined by $|L|$.

######  Definition 6.2.

Let $(X,L)$ be a polarized variety. A _ribbon_ over $(X,L)$, a.k.a. a ribbon
over $X$ with normal bundle $L$, is a scheme $\tilde{X}$ such that
$\tilde{X}_{\mathrm{red}}=X$, and the ideal $\mathcal{I}_{X/\tilde{X}}$
defining $X$ in $\tilde{X}$ verifies the two conditions
$\mathcal{I}_{X/\tilde{X}}^{2}=0$ and $\mathcal{I}_{X/\tilde{X}}=L^{-1}$.

If $X\subseteq\mathbf{P}^{N}$ is smooth and $Y\subseteq\mathbf{P}^{N+1}$ is a
$1$-extension of $(X,\mathcal{O}_{X}(1))$, then the first infinitesimal
neighbourhood of $X$ in $Y$ is a ribbon over $(X,\mathcal{O}_{X}(1))$ which we
denote by $2X_{Y}$. A ribbon $\tilde{X}$ over $(X,L)$ is _integrable_ if there
exists an extension $Y$ such that $\tilde{X}=2X_{Y}$; in this situation we say
that the variety $Y$ is an _integral_ of the ribbon $\tilde{X}$.

A ribbon over $(X,L)$ is uniquely determined by its extension class
$e_{\tilde{X}}\in\mathrm{Ext}^{1}(\Omega^{1}_{X},L^{-1})$, and two ribbons are
isomorphic if and only if their extension classes are proportional. We will
say a ribbon is _trivial_ if its extension class is zero.

Let $(X,L)$ be a smooth polarized manifold with $L$ very ample, consider the
corresponding embedding $X\subseteq\mathbf{P}^{N}$, and identify this
$\mathbf{P}^{N}$ with a hyperplane $H\subseteq\mathbf{P}^{N+1}$. If
$\tilde{X}$ is a ribbon over $(X,L)$ contained in the first infinitesimal
neighbourhood $2H_{\mathbf{P}^{N+1}}$, then its extension class lies in the
kernel of the map

$\eta:\mathrm{Ext}^{1}(\Omega^{1}_{X},L^{-1})\to\mathrm{Ext}^{1}(\left.\Omega^{1}_{\mathbf{P}^{N}}\right|_{X},L^{-1})$

induced by the restriction map
$\left.\Omega^{1}_{\mathbf{P}^{N}}\right|_{X}\to\Omega^{1}_{X}$, as has been
first observed in [44]. When $X$ is a curve, the map $\eta$ identifies with
$\vphantom{\gamma}{}^{\mathsf{T}\\!}\gamma_{C,L}$, the transpose of the
Gaussian map defined in (5.1).

######  Theorem 6.3.

Let $(C,L)$ be a smooth polarized curve of genus $g\geqslant 2$ and degree
$d\geqslant 2g+3$. Then for all
$v\in\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})$, the ribbon
$\tilde{C}_{v}$ with extension class $v$ is the first infinitesimal
neighbourhood of $C$ in at most one surface, up to automorphism (see below).
In particular, if $(C,L)$ is extendable then $\gamma_{C,L}$ is not surjective.

The precise meaning of the unicity statement above is the following. Consider
$C\subseteq\mathbf{P}^{d-g}$ in its embedding defined by $|L|$, and identify
this $\mathbf{P}^{d-g}$ with a hyperplane $H\subseteq\mathbf{P}^{d-g+1}$. Let
$S,S^{\prime}\subseteq\mathbf{P}^{d-g+1}$ be two surfaces, such that $S\cap
H=S^{\prime}\cap H=C$. If $2C_{S}\cong 2C_{S^{\prime}}$, then there is exists
a projectivity of $\mathbf{P}^{d-g+1}$ acting as the identity on $H$ and
mapping $S$ to $S^{\prime}$.

One gets the necessary condition for the integrability of $(C,L)$ by applying
the unicity statement to the zero vector
$0\in\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})$. Indeed, the trivial
ribbon $\tilde{C}_{0}$ is the first infinitesimal neighbourhood of $C$ in the
cone over $C$, so the unicity statement tells us that if $S$ is a non-trivial
extension of $C$ (thus, $S$ is not a cone), then the two ribbons $2C_{S}$ and
$\tilde{C}_{0}$ are distinct, hence $2C_{S}$ comes from a non-zero vector in
$\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})$.

##### (6.4)

We now outline the proof of Theorem 6.3, as this will be needed later on. It
follows a construction given in [45]. By Theorem 2.2, the curve $(C,L)$
satisfies Property $N_{2}$. Thus the homogeneous ideal of $C$ in its embedding
in $\mathbf{P}^{d-g}$ defined by $|L|$ has a minimal resolution as follows,

(6.4.1) $\cdots\longrightarrow\mathcal{O}_{\mathbf{P}^{d-g}}(-3)^{\oplus
m_{1}}\stackrel{{\scriptstyle\mathbf{r}}}{{\longrightarrow}}\mathcal{O}_{\mathbf{P}^{d-g}}(-2)^{\oplus
m}\stackrel{{\scriptstyle\mathbf{f}}}{{\longrightarrow}}\mathcal{I}_{C/\mathbf{P}^{d-g}}\longrightarrow
0.$

We view $\mathbf{f}$ as a vector of quadratic equations defining scheme
theoretically $C$, in the homogeneous coordinates
$\mathbf{x}=(x_{0}:\ldots:x_{d-g})$ on $\mathbf{P}^{d-g}$.

On the other hand, there is an exact sequence of vector spaces

(6.4.2) $0\longrightarrow H^{0}(C,L)^{\vee}\longrightarrow
H^{0}(C,N_{C/\mathbf{P}^{d-g}}(-1))\longrightarrow\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})\longrightarrow
0,$

cf. [16, Lem. 3.2] and the references given there. Let
$v\in\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})$, and choose a lift
in $H^{0}(C,N_{C/\mathbf{P}^{d-g}}(-1))$. By (6.4.1) the latter space is a
subspace of $H^{0}(C,\mathcal{O}_{C}(1))^{\oplus m}$, so we can represent the
lift of $v$ as a length $m$ vector $\mathbf{f}_{v}$ of linear forms in the
variable $\mathbf{x}$.

Then the ribbon $\tilde{C}_{v}$ with extension class $v$ is the subscheme of
$\mathbf{P}^{d-g+1}$ defined by the equations

(6.4.3) $\mathbf{f}(\mathbf{x})+t\mathbf{f}_{v}(\mathbf{x})=\mathbf{0},\quad
t^{2}=0$

in the homogeneous coordinates $(\mathbf{x}:t)$. In turn, any surface
$S\subseteq\mathbf{P}^{d-g+1}$ containing $\tilde{C}_{v}$ is defined by the
equations

(6.4.4)
$\mathbf{f}(\mathbf{x})+t\mathbf{f}_{v}(\mathbf{x})+t^{2}\mathbf{h}=\mathbf{0},$

where $\mathbf{h}$ is a length $m$ vector of constants, subject to conditions
that we will not discuss here (see paragraph (4.9) in [16]). The upshot of
these conditions however is that two vectors $\mathbf{h}$ and
$\mathbf{h}^{\prime}$ defining two surfaces $S$ and $S^{\prime}$ containing
the ribbon $\tilde{C}_{v}$ differ by an element of
$H^{0}(C,N_{C/\mathbf{P}^{d-g}}(-2))$. Then the unicity statement follows from
the vanishing of this space when $(C,L)$ satisfies Property $N_{2}$, cf. [16,
Lem. 3.6] and the references given there. ∎

One should keep in mind the following conclusions from the above
considerations. The projective space
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ parametrizes
isomorphism classes of non-trivial ribbons over $(C,L)$ likely to be
integrated to a non-trivial extension $S$ of $(C,L)$. Each such ribbon may be
integrated to at most one extension, and each $1$-extension conversely
corresponds to a point in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$.

In analogy with the terminology from deformation theory111in fact, if one
looks at the construction in [45], one sees that this is more than a mere
analogy., we will say that the extension theory of $(C,L)$ is _unobstructed_
if every ribbon corresponding to a point of
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ is
integrable; otherwise we say it is _obstructed_. When the extension theory is
unobstructed, we will see that we can construct a universal extension, in the
following sense.

######  Definition 6.5.

Let $(C,L)$ be a smooth polarized curve of genus $g\geqslant 2$ and degree
$d\geqslant 2g+3$. Let $r=\operatorname{cork}(\gamma_{C,L})$. An $r$-extension
$Y\subseteq\mathbf{P}^{d-g+r}$ of $(C,L)$ is _universal_ if the following
condition holds: for all
$[v]\in\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$, there
exists a unique $(d-g+1)$-plane $\Lambda\subseteq\mathbf{P}^{d-g+r}$
containing $C$, such that the surface $Y\cap\Lambda$ is an integral of the
ribbon over $(C,L)$ defined by the extension class $v$.

##### (6.6)

Note that, under the above assumptions, if $Y\subseteq\mathbf{P}^{d-g+k}$ is a
$k$-extension of $(C,L)$, then it is defined by equations

$\mathbf{f}(\mathbf{x})+\mathbf{F}(\mathbf{x})\cdot\vphantom{\mathbf{t}}^{\mathsf{T}\\!}\mathbf{t}+\mathbf{H}(\mathbf{t})=0,$

in homogeneous coordinates $(\mathbf{x}:\mathbf{t})$,
$\mathbf{x}=(x_{0}:\ldots:x_{d-g})$ and $\mathbf{t}=(t_{1}:\ldots:t_{k})$,
such that the span $\langle C\rangle$ is defined by $\mathbf{t}=0$; see [36,
Thm. 20.3] for instance. Here, $\mathbf{F}$ is an $m\times k$ matrix of linear
forms in $\mathbf{x}$, and $\mathbf{H}$ is a length $m$ vector, constant in
$\mathbf{x}$ and quadratic in $\mathbf{t}$. One thus sees that the map

$\Lambda\in\mathbf{P}^{d-g+k}/\langle
C\rangle\longmapsto[2C_{Y\cap\Lambda}]\in\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$

is linear, given by the matrix $\mathbf{F}$. (Here $\mathbf{P}^{d-g+k}/\langle
C\rangle$ denotes the $(k-1)$-dimensional projective space of $(d-g+1)$-planes
$\Lambda$ containing $C$, and the map associates a $(d-g+1)$-plane
$\Lambda\subseteq\mathbf{P}^{d-g+r}$ containing $C$ with the isomorphism class
of the ribbon $2C_{Y\cap\Lambda}$).

######  Lemma 6.7.

Let $(C,L)$ be a smooth polarized curve of genus $g\geqslant 2$ and degree
$d\geqslant 2g+3$. Let $Y\subseteq\mathbf{P}^{N}$ be an extension of $(C,L)$,
of dimension $1+\operatorname{cork}(\gamma_{C,L})$. Assume that for general
$[e]\in\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$, there
is a linear subspace $\Lambda\subseteq\mathbf{P}^{N}$ containing $C$ and
cutting out a surface on $Y$ such that the ribbon $2C_{Y\cap\Lambda}$ has
extension class $e$ (in other words, $2C_{Y\cap\Lambda}\cong\tilde{C}_{e}$).
Then $Y$ is a universal extension of $C$.

###### Proof.

We consider the map

$\Lambda\in\mathbf{P}^{d-g+\operatorname{cork}(\gamma_{C,L})}/\langle
C\rangle\longmapsto[2C_{Y\cap\Lambda}]\in\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$

as in (6.6) above. The assumption made on $Y$ means that this map is dominant.
It is moreover linear, as has been observed in (6.6). Since its target and its
source are projective spaces of the same dimension, namely
$\operatorname{cork}(\gamma_{C,L})-1$, it is an isomorphism, which means that
$Y$ is a universal extension of $(C,L)$. ∎

######  Theorem 6.8.

Let $(C,L)$ be a smooth polarized curve of genus $g\geqslant 2$ and degree
$d\geqslant 2g+3$. If the general ribbon in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ is
integrable, then all such ribbons are integrable, and there exists a universal
extension of $(C,L)$.

###### Proof.

Let us first prove, to fix ideas, that if all ribbons in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ are
integrable, then there exists a universal extension of $(C,L)$. The proof is
identical to that of [16, §5], so we will be brief. The idea is that we can
package together all ribbons in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ and their
integrals in a projective bundle
$\mathbf{P}(\mathcal{O}_{\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))}^{\oplus
d-g}\oplus\mathcal{O}_{\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))}(1))$,
and then the universal extension is the image of the family of all surface
integrals by the map defined by the relative $\mathcal{O}(1)$ of this
projective bundle. This works as follows.

One first chooses a section

$v\in\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})\longmapsto\mathbf{f}_{v}\in
H^{0}(C,N_{C/\mathbf{P}^{d-g}}(-1))$

of (6.4.2). For all $v\in\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})$,
we let $\mathbf{h}_{v}$ be the unique vector of constants such that the
integral of the ribbon $\tilde{C}_{v}$ is defined by the equations (6.4.4)
with $\mathbf{h}=\mathbf{h}_{v}$. For all $\lambda\in\mathbf{C}$, one has

(6.8.1) $\mathbf{f}_{\lambda v}=\lambda\mathbf{f}_{v},$

so the ribbon $\tilde{C}_{\lambda v}$ (isomorphic to $\tilde{C}_{v}$) is
defined by the equations

$\mathbf{f}(\mathbf{x})+t\mathbf{f}_{\lambda
v}(\mathbf{x})=\mathbf{f}(\mathbf{x})+\lambda
t\mathbf{f}_{v}(\mathbf{x})=\mathbf{0},\quad t^{2}=0.$

One may thus deduce the equations of the integral of $\tilde{C}_{\lambda v}$
from those of $\tilde{C}_{v}$, namely they are:

$\mathbf{f}(\mathbf{x})+\lambda
t\mathbf{f}_{v}(\mathbf{x})+\lambda^{2}t^{2}\mathbf{h}_{v}=\mathbf{0}.$

By the unicity of the vector of constants $\mathbf{h}$ attached to $\lambda
v$, we conclude that

(6.8.2) $\mathbf{h}_{\lambda v}=\lambda^{2}\mathbf{h}_{v}.$

Next, let us construct the family $\mathcal{S}$ of all surface extensions of
$(C,L)$ in
$\mathbf{P}(\mathcal{O}_{\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))}^{\oplus
d-g}\oplus\mathcal{O}_{\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))}(1))$
by glueing affine pieces. We choose a basis $v_{1},\ldots,v_{r}$ of
$\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L})$,
$r=\operatorname{cork}(\gamma_{C,L})$, and for $i=1,\ldots,r$ we consider the
subscheme $\mathcal{S}_{i}$ of $\mathbf{P}^{d-g+1}\times\mathbf{A}^{r-1}$
defined by the equations

$\mathbf{f}(\mathbf{x})+t\mathbf{f}(\mathbf{x})_{a_{1}v_{1}+\cdots+v_{i}+\cdots+a_{r}v_{r}}+t^{2}\mathbf{h}_{a_{1}v_{1}+\cdots+v_{i}+\cdots+a_{r}v_{r}}=\mathbf{0},$

in the homogeneous coordinates $(\mathbf{x}:t)$ on $\mathbf{P}^{d-g+1}$ and
affine coordinates $(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n})$ on
$\mathbf{A}^{r-1}$ (with the convention that the term under the hat should be
omitted); it is flat over $\mathbf{A}^{r-1}$. The homogeneity properties
(6.8.1) and (6.8.2) ensure that any two pieces $\mathcal{S}_{i}$ and
$\mathcal{S}_{j}$ glue along their open subsets defined by $(a_{j}\neq 0)$ and
$(a_{i}\neq 0)$, via the isomorphism

$([\mathbf{x}:t],a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n})\mapsto([\mathbf{x}:a_{i}t],\frac{a_{1}}{a_{j}},\ldots,\widehat{a_{j}},\ldots,\frac{a_{n}}{a_{j}}).$

The glueing of all $\mathcal{S}_{1},\ldots,\mathcal{S}_{r}$ gives the family
$\mathcal{S}\subseteq\mathbf{P}(\mathcal{O}_{\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))}^{\oplus
d-g}\oplus\mathcal{O}_{\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))}(1))$
as we wanted. Eventually, the construction of the universal extension as the
image of $\mathcal{S}$ by the relative $\mathcal{O}(1)$ of the projective
bundle is exactly the same as in [16, Cor. 5.5].

It remains to prove that the integrability of the general ribbon implies that
of all ribbons in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$. Let
$[v_{0}]\in\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$.
If the general ribbon is integrable, we can find an arc
$\mathbf{D}\subseteq\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$
centered at $[v_{0}]$, such that for all
$[v]\in\mathbf{D}^{\circ}=\mathbf{D}-[v_{0}]$, the corresponding ribbon
$\tilde{C}_{v}$ is integrable. Then we have the two following families,
defined by equations as above:

  1. (i)

the flat family
$\tilde{\mathcal{C}}\subseteq\mathbf{P}^{d-g+1}\times\mathbf{D}$ of all
ribbons $\tilde{C}_{v}$, $[v]\in\mathbf{D}$, and

  2. (ii)

the flat family
$\mathcal{S}_{\mathbf{D}}^{\circ}\subseteq\mathbf{P}^{d-g+1}\times\mathbf{D}^{\circ}$
of the surface integrals of the ribbons $\tilde{C}_{v}$, $[v]\neq[v_{0}]$.

Taking the closure $\mathcal{S}_{\mathbf{D}}$ of
$\mathcal{S}_{\mathbf{D}}^{\circ}$ in $\mathbf{P}^{d-g+1}\times\mathbf{D}$, we
obtain a flat family of surfaces over $\mathbf{D}$. Since
$\mathcal{S}_{\mathbf{D}}^{\circ}$ contains
$\left.\tilde{\mathcal{C}}\right|_{\mathbf{D}^{\circ}}$,
$\mathcal{S}_{\mathbf{D}}$ will contain ${\tilde{\mathcal{C}}}$, hence the
central fibre of $\mathcal{S}_{\mathbf{D}}$ is an integral of the ribbon
$\tilde{C}_{v_{0}}$. Therefore all ribbons in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ are
integrable, and the theorem is proved. ∎

##### (6.9)

In the following sections we shall apply the above Theorem 6.8 to various
specific situations, in which we know the dimension of
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$. Our
strategy to verify that the general ribbon over $(C,L)$ is integrable is to
produce a family of extensions of $(C,L)$ of the same dimension as
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$. Then, by
the Unicity Theorem 6.3, there is an injective map from the parameter space of
this family of extensions to
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$, which is
dominant for dimension reasons.

Eventually, let us note that all the above considerations may be adapted to
polarized manifolds $(X,L)$ of arbitrary dimension. The only difference is
that the exact sequence (6.4.2) should be slightly modified, see [16, Lem.
3.5].

## Extensions of polarized genus three curves

In this section, we study closely the extensions of polarized curves of genus
$3$ and degree $d\geqslant 2g+3$, in order to determine whether their ribbons
are obstructed or not. Our main output in this direction is Theorem 1.7.

##### (7.1)Classification of surfaces with sectional genus $3$

The classification of rational surfaces with hyperplane sections non-
hyperelliptic curves of genus $3$ had been classically worked out by
Castelnuovo [9]. He proved that all such surfaces are represented by a linear
system of plane quartics. More recently Lanteri and Livorni [31] have
classified all pairs $(S,C)$ where $S$ is a smooth surface, $C\subseteq S$ is
a smooth genus $3$ curve, and the linear system $|C|$ is globally generated
and ample. For $d\geqslant 8$, Theorem 1.2 provides more generally the
classification of surfaces, possibly singular, with one hyperplane section a
linearly normal non-hyperelliptic curve of genus $3$.

######  Corollary 7.2.

Let $(C,L)$ be a non-hyperelliptic polarized curve of genus $g=3$ and degree
$d\geqslant 4g-4=8$. Then all surface extensions of $(C,L)$ are rational. If
$d\geqslant 9$, they are all realized by a linear system of plane quartics; if
$d=8$, they are realized either by a linear system of plane quartics, or by a
complete linear system of plane sextics with seven base points of multiplicity
two as in Example 3.3.

###### Proof.

This is a direct application of Theorem 1.2. ∎

######  Lemma 7.3.

Let $C$ be a non-hyperelliptic curve of genus $3$, and $L$ be a line bundle of
degree $d\geqslant 0$ on it. Then $h^{0}(C,4K_{C}-L)$ takes the following
values:

  * —

if $d>16$, then $h^{0}(4K_{C}-L)=0$;

  * —

if $d=16$, then $h^{0}(4K_{C}-L)=1$ if $L=4K_{C}$ and $0$ otherwise;

  * —

if $d=15$, then $h^{0}(4K_{C}-L)=1$ if $L=4K_{C}-p$ for some $p\in C$, and $0$
otherwise;

  * —

if $d=14$, then $h^{0}(4K_{C}-L)=1$ if $L=4K_{C}-p-q$ for some $p,q\in C$, and
$0$ otherwise;

  * —

if $d=13$, then $L$ may always be written as $4K_{C}-p-q-r$ for some $p,q,r\in
C$, and $h^{0}(4K_{C}-L)=2$ if these three points are aligned on the canonical
model of $C$, and $1$ otherwise;

  * —

if $d=12$, then $L$ may always be written as $4K_{C}-p-q-r-s$ for some
$p,q,r,s\in C$, and $h^{0}(4K_{C}-L)=3$ if these four points are aligned on
the canonical model of $C$, and $2$ otherwise;

  * —

if $d<12$, then $h^{0}(4K_{C}-L)=14-d$.

###### Proof.

If $d\geqslant 16$, then $\deg(4K_{C}-L)\leqslant 0$ and the result is clear.
If $d=15$, we can always write $L=4K_{C}-p_{0}+N$ for some arbitrarily chosen
point $p_{0}\in C$ and some degree $0$ line bundle $N$. If
$h^{0}(4K_{C}-L)>0$, then $p_{0}-N\sim p$ for some $p\in C$, hence
$L=4K_{C}-p$. In this case, $h^{0}(4K_{C}-L)=h^{0}(p)=1$. If $d=14$, it
follows as in the previous case that $L=4K_{C}-p-q$ if $h^{0}(4K_{C}-L)>0$. In
this case, $h^{0}(4K_{C}-L)=h^{0}(p+q)=1$ since $C$ is non-hyperelliptic. If
$d\leqslant 13$, by Jacobi’s inversion theorem (see [1, p. 19]) we can always
write $L=4K_{C}-\sum_{i=1}^{16-d}p_{i}$ for some points
$p_{1},\ldots,p_{16-d}$, and then the result follows by Riemann–Roch and Serre
duality. ∎

##### (7.4)Proof of Theorem 1.7

The only if part of (1.7.1) is a direct consequence of Theorem 6.3, taking
into account Theorem 1.3 and Lemma 7.3. On the other hand, Example 3.2
provides an extension for all $(C,L)$ with $L=4K_{C}-\sum_{i=1}^{16-d}p_{i}$,
thus proving the if part of (1.7.1): explicitly, this goes as follows.
Consider $C$ in its canonical embedding as a smooth plane quartic, and let
$\varepsilon:S\to\mathbf{P}^{2}$ be the blow-up of the plane at the points
$p_{1},\ldots,p_{16-d}\in C\subseteq\mathbf{P}^{2}$, with exceptional divisors
$E_{1},\ldots,E_{16-d}$ (when some $p_{i}$’s coincide, this means that we
blow-up infinitely near points). Let $H$ denote the pull-back to $S$ of the
line class on $\mathbf{P}^{2}$. The linear system
$|4H-\sum_{i=1}^{16-d}E_{i}|$ on $S$ cuts out the complete linear series
$|4K_{C}-\sum_{i=1}^{16-d}p_{i}|=|L|$ on the proper transform of $C$, which is
very ample; it is thus base point free, and defines a birational morphism from
$S$ to an extension of $(C,L)$.

Let us now prove (1.7.2). If $d\geqslant 14$, there is nothing to add to
(1.7.1), since in these cases $\operatorname{cork}(\gamma_{C,L})$ is either
$0$ or $1$ by Lemma 7.3, so we will suppose $d\leqslant 13$. Given
$[Z]\in(\mathbf{P}^{2})^{[16-d]}$ a length $16-d$, $0$-dimensional subscheme
$Z\subseteq\mathbf{P}^{2}$, we let $S_{Z}$ be the blow-up of $\mathbf{P}^{2}$
along $Z$, with total exceptional divisor $E_{Z}$, and call $H$ the pull-back
of the line class on $\mathbf{P}^{2}$. The locus
$\mathcal{S}\subseteq(\mathbf{P}^{2})^{[16-d]}$ parametrizing those $Z$ such
that the linear system $|4H-E_{Z}|$ on $S_{Z}$ contains a smooth curve is
dense. Moreover, for all $[Z]\in\mathcal{S}$, since $d\geqslant 9$, this
linear system has dimension $d-2$, is base point free, and defines a
birational morphism. We consider the universal family

$\mathcal{L}\to\mathcal{S}$

of these linear systems, and the dense open subset
$\mathcal{L}^{\circ}\subseteq\mathcal{L}$ consisting of those pairs $(Z,C)$
such that $C$ is a smooth member of $|4H-E_{Z}|$ on $S_{Z}$. After dividing
out by the automorphism group of $\mathbf{P}^{2}$, we get the moduli space
$\mathcal{SC}$ of such pairs, which has dimension

$\dim\bigl{(}(\mathbf{P}^{2})^{[16-d]}\bigr{)}+(d-2)-8=22-d.$

Next we consider the universal Jacobian $\mathcal{J}_{3}^{d}$ parametrizing
degree $d$ line bundle on genus $3$ curves, which has dimension $4g-3=9$, and
its dense subset $\mathcal{J}^{\circ}$ corresponding to non-hyperelliptic
curves. We shall examine the map

$c:(Z,C)\in\mathcal{SC}\mapsto\bigl{[}C,\left.\mathcal{O}_{S_{Z}}(4H-E_{Z})\right|_{C}\bigr{]}\in\mathcal{J}^{\circ},$

the fiber of which over a point $(C,L)$ consists of distinct isomorphism
classes of extensions of $(C,L)$. By our proof of the if part of (1.7.1), the
image of $c$ is the locus of those $(C,L)$ such that $L$ may be written as
$4K_{C}-\sum_{i=1}^{16-d}p_{i}$, which is the whole $\mathcal{J}^{\circ}$
since we are assuming $d\leqslant 13$. Therefore all fibres of $c$ have
dimension at least

$\dim(\mathcal{SC})-\dim(\mathcal{J}^{\circ})=13-d.$

If $d<12$, this proves that for all $(C,L)$, the general ribbon in
$\mathbf{P}(\ker(\vphantom{\gamma}^{\mathsf{T}\\!}\gamma_{C,L}))$ is
integrable, by Lemma 7.3 and the argument given in (6.9). We conclude by
Theorem 6.8 that all ribbons are integrable, and there exists a universal
extension.

If $d=12$ or $13$, the same argument proves (1.7.2) for all $(C,L)$ such that
$L=4K_{C}-\sum_{i=1}^{16-d}p_{i}$ with $p_{1},\ldots,p_{16-d}$ not all on a
line in $\mathbf{P}^{2}$.

We now treat the two remaining cases separately, in the same spirit as the
others. First assume $d=13$, and let $S_{1}$ be the blow-up of
$\mathbf{P}^{2}$ along three pairwise distinct points lying on a line, with
total exceptional divisor $E_{1}$. Consider the locus of smooth quartics in
the linear system $|4H-E_{1}|$, which has dimension $11$, and take its
quotient by the automorphism group of $S_{1}$, which is the subgroup of the
projectivities of $\mathbf{P}^{2}$ acting as the identity on a line, hence has
dimension $3$ (it is the group of homotheties and translations of the affine
plane). We thus get the moduli space $\mathcal{SC}_{1}$ of pairs $(S_{1},C)$,
of dimension $8$. The image of $\mathcal{SC}_{1}$ by the map
$c:\mathcal{L}^{\circ}\to\mathcal{J}^{\circ}$ is the locus $\mathcal{J}^{1}$
of pairs $(C,L)$ such that $L=4K_{C}-D$ with $|D|$ a $g^{1}_{3}$, which has
dimension $7$ (note that if $p_{1},p_{2},p_{3}\in C\subseteq\mathbf{P}^{2}$
are aligned, then they move in a base-point-free $g^{1}_{3}$, and thus up to
linear equivalence we may always assume that they are pairwise distinct). The
upshot is that the fibers of the map

$\left.c\right|_{\mathcal{SC}_{1}}:\mathcal{SC}_{1}\to\mathcal{J}^{1}$

all have dimension at least $1$, and we conclude as in the previous cases.

In the case $d=12$, we fix three pairwise distinct points $q_{1},q_{2},q_{3}$
on a line in $\mathbf{P}^{2}$, and let $\mathcal{S}_{2}$ be the complement of
$\\{q_{1},q_{2},q_{3}\\}$ in this line. For all $q\in\mathcal{S}_{2}$, we let
$S_{q}$ be the blow-up of $\mathbf{P}^{2}$ at $q_{1},q_{2},q_{3}$ and $q$,
with total exceptional divisor $E_{q}$. The $S_{q}$’s form a $1$-dimensional
family of pairwise non-isomorphic surfaces, all with automorphism group the
subgroup of the projectivities of $\mathbf{P}^{2}$ acting as the identity on a
line, of dimension $3$. Quotienting the universal family of the linear systems
$|4H-E_{q}|$, each of dimension $10$, by this automorphism group, we get the
moduli space $\mathcal{SC}_{2}$ of pairs $(S_{q},C)$, of dimension $10+1-3=8$.
The image of $\mathcal{SC}_{2}$ by the map
$c:\mathcal{L}^{\circ}\to\mathcal{J}^{\circ}$ is the locus $\mathcal{J}^{2}$
of pairs $(C,3K_{C})$, which has dimension $6$. Therefore the fibers of the
map

$\left.c\right|_{\mathcal{SC}_{2}}:\mathcal{SC}_{2}\to\mathcal{J}^{2}$

all have dimension at least $2$, and we conclude as in the previous cases. ∎

######  Remark 7.5.

In the case when $L=4K_{C}-D$ with $|D|$ a $g^{1}_{3}$, there exist extensions
of $(C,L)$ supported on a surface different from $S_{1}$. Indeed, choosing a
member of $|D|$ of the form $2p_{1}+p_{2}$, we see that the blow-up
$S_{1}^{\prime}$ of $\mathbf{P}^{2}$ along three aligned points, two of which
are infinitely near, provides an extension of $(C,L)$. This surface is rigid
as $S_{1}$ is, but its automorphism group is larger: it is the subgroup of
those projectivities fixing the two points $p_{1}$ and $p_{2}$ (hence also
leaving the line $\langle p_{1},p_{2}\rangle$ stable), which has dimension
$4$. We thus get a moduli space $\mathcal{SC}_{1}^{\prime}$ of dimension
$11-4=7$ which surjects onto $\mathcal{J}^{1}$. Thus for all
$(C,L)\in\mathcal{J}^{1}$, there is at least one extension supported on the
surface $S_{1}^{\prime}$.

For all $C$, there are finitely many $g^{1}_{3}$’s having a member of the form
$3p_{1}$, and the corresponding pairs $(C,L)$ form a $6$-dimensional locus in
$\mathcal{J}^{\circ}$. These pairs have an extension supported on
$S_{1}^{\prime\prime}$, the blow-up of $\mathbf{P}^{2}$ at three infinitely
near points lying on a line. The surface $S_{1}^{\prime\prime}$ is rigid, and
has an automorphism group of dimension $5$.

Similar considerations may be made about the extensions of the polarized
curves $(C,3K_{C})$.

######  Remark 7.6.

One may want to prove Corollary 7.2 for $d\geqslant 9$ directly with the above
considerations, without resorting to Theorem 1.2. Our proof of (1.7.2) shows
that for all $(C,L)$, the general extension of $(C,L)$ is rational and
realized by a linear system of plane quartics. However it is not clear to us
why these two properties should be preserved when one specializes to an
arbitrary extension of $(C,L)$.

##### (7.7)Remarks on the extensions in degree $d<2g+3$

In this case Green’s Theorem 2.2 no longer applies to guarantee that $(C,L)$
verifies Property $N_{2}$, so that in particular one ribbon may a priori have
several different integrals.

We will mostly concentrate on the case $d=2g+2=8$. If $L\neq 2K_{C}$ then all
extensions of $(C,L)$ are realized by a linear sytem of plane quartics by
Corollary 7.2, and the analysis carried out in the proof of Theorem 1.7
applies mutatis mutandis. We find that if $(C,L)$ is general, then the
extensions of $(C,L)$ form a family of the expected dimension

$5=\operatorname{cork}(\gamma_{C,L})-1=h^{0}(4K_{C}-2L)-1.$

We cannot say much more, however because of the possible failure of Property
$N_{2}$.

The case $L=2K_{C}$ is more interesting. In this case we still have, for
general $C$, a $5$-dimensional family of extensions of $(C,2K_{C})$ given by
linear systems of plane quartics: these extensions are obtained by projecting
$v_{4}(\mathbf{P}^{2})$ from eight points that, in $\mathbf{P}^{2}$, lie on a
conic, so that they sum to a bicanonical divisor of any quartic containing
them. There is however another family of extensions, given by complete linear
systems of plane sextics with seven base points of multiplicity two. One
finds, using similar arguments as in the proof of Theorem 1.7 above, that for
general $C$ these form a $6$-dimensional family. We thus find two independent
families of extensions of $(C,2K_{C})$, one of the expected dimension and one
superabundant.222We note in addition that those surfaces obtained by a linear
system of quartics with eight base points lying on a conic have a singularity
of type $\frac{1}{4}(1,1)$, i.e., it is locally the cone over a rational
normal quartic curve. We emphasize also that sextics with seven double points
are Cremona-minimal hence not Cremona-equivalent to smooth quartics, see [7]
and [33].

In degree $d<8$ the situation is similar. In fact, linear systems of sextics
with seven double base points of degree $d<8$ necessarily have further base
points, so that the funny situation for $2K_{C}$ analyzed above now happens
for more line bundles (e.g., in degree $7$ it happens for those line bundles
that may be written as $2K_{C}-p$ for some $p\in C$, which form a
$1$-dimensional family in the Jacobian of $C$). Note however that for $d<8$ we
are out of the range of application of Theorem 1.2 and Corollary 7.2 so that
there may be even more families of extensions.

##### (7.8)Invariants of the universal extensions

We now list the degrees and dimensions of the universal extensions of genus
$3$ curves gotten above, limiting ourselves to the case when the dimension is
at least $3$:

  * —

if $d=13$, and $L=4K_{C}-p-q-r$ for some $p,q,r\in C$ on a line in the
canonical model of $C$, we find a threefold $X$ of degree $d=13$ in
$\mathbf{P}^{12}$;

  * —

if $d=12$ and $L=4K_{C}-p-q-r-s$ for some $p,q,r,s\in C$ not on a line in the
canonical model of $C$, we find a threefold $X$ of degree $d=12$ in
$\mathbf{P}^{11}$;

  * —

if $d=12$ and $L=3K_{C}$, we find a fourfold $X$ of degree $d=12$ in
$\mathbf{P}^{12}$;

  * —

if $9\leqslant d\leqslant 11$, we find a $(15-d)$-dimensional variety of
degree $d$ in $\mathbf{P}^{11}$ (note that $4\leqslant 15-d\leqslant 6$).

Smooth projective varieties of degree $d=9,10,11$ have been classified in
[21], [22], and [6] respectively. Running through the correspondings lists, we
notice that there is no smooth $6$-fold of degree $9$ in $\mathbf{P}^{11}$,
hence the universal extension in degree $9$ is certainly singular. By
contrast, in degree $10$ and $11$, there are smooth $5$-fold and $4$-fold of
sectional genus $3$ in $\mathbf{P}^{11}$, namely
$\mathbf{P}^{1}\times\mathbf{Q}^{4}$ in the Segre embedding, and a scroll over
$\mathbf{P}^{2}$, respectively. It is possible that these coincide with the
universal extensions above, but we will not investigate this now.

## Extensions of polarized hyperelliptic curves

In this section, we study closely the extensions of polarized hyperelliptic
curves of genus $g$ and degree $d\geqslant 2g+3$, in order to determine
whether their extension theory is obstructed or not. Our main output in this
direction is Theorem 1.8. Our general strategy is similar to that employed for
genus $3$ curves in Section 7, but the situation for hyperelliptic curves is
slightly more complicated and requires more care.

##### (8.1)

For all $\mu=g+1,g,\ldots,0$, we let $\mathcal{H}_{\mu}$ be the complete
linear system

$|2H+\mu F|=|2E+(2g+2-\mu)F|\quad\text{on }\mathbf{F}_{g+1-\mu}.$

It defines a projective surface of degree $4g+4$ in $\mathbf{P}^{3g+5}$, which
is the maximum possible degree for a non-trivial extension of a linearly
normal curve of genus $g\geqslant 2$, see Corollary 2.2. Since we focus on
polarized curves of degree $d\geqslant 2g+3$, in order for Property $N_{2}$ to
hold, the maximal number of points from which we may project these surfaces is

(8.1.1) $b_{\max}=2g+1.$

######  Proposition 8.2.

Let $C$ be a hyperelliptic curve of genus $g$, and $\mathfrak{e}$ an effective
divisor of degree $\mu$ on $C$, $0\leqslant\mu\leqslant g+1$. There exists an
embedding of $C$ as a member of the linear system $|2H+\mu F|$ on
$\mathbf{F}_{g+1-\mu}$ such that $\left.E\right|_{C}=\mathfrak{e}$ if and only
if no two points of $\mathfrak{e}$ are conjugate with respect to the
hyperelliptic involution on $C$.

###### Proof of the only if part of Proposition 8.2.

Suppose $C$ is a smooth member of the linear system $|2H+\mu F|$ on
$\mathbf{F}_{g+1-\mu}$, and let

$\left.E\right|_{C}=e_{1}+\cdots+e_{\mu}.$

Consider two points $e_{i}$ and $e_{j}$ of $\left.E\right|_{C}$, distinct in
the sense that $i\neq j$. Since $E\cdot F=1$ and the $g^{1}_{2}$ on $C$ is cut
out by $F$, the only possibility for $e_{i}$ and $e_{j}$ to be conjugate is
that $e_{i}=e_{j}$ and it is a ramification point of the $g^{1}_{2}$. However,
the condition that $e_{i}=e_{j}$ means that $C$ is tangent to $E$ at this
point, and the condition that it is a ramification point of the $g^{1}_{2}$
means that $C$ is tangent to the fiber at this point, so these two conditions
may not be realized simultaneously as $C$ is smooth. The conclusion is thus
that for all $i\neq j$, $e_{i}$ and $e_{j}$ are not conjugate. ∎

In case $\mu\leqslant g$, this can also be proved by cohomological
considerations, as the condition we want to prove is then equivalent to
$h^{0}(C,\left.E\right|_{C})=1$, which in turn is equivalent to
$h^{1}(\mathbf{F}_{g+1-\mu},E-C)=0$, as can be seen by considering the
restriction exact sequence. The latter vanishing holds because $E-C$ has
vanishing $h^{0}$ and $h^{2}$ (the latter by Serre duality), and $\chi(E-F)=0$
by Riemann–Roch.

###### Proof of the if part of Proposition 8.2.

Let $C$ be a hyperelliptic curve of genus $g$, and $\mathfrak{e}$ an effective
divisor of degree $\mu$ on $C$, no two points of which are conjugate. We let
$\mathfrak{f}$ be the class of the $g^{1}_{2}$, and
$\mathfrak{h}=\mathfrak{e}+(g+1-\mu)\mathfrak{f}$. One has

$K_{C}-\mathfrak{h}=(g-1)\mathfrak{f}-\mathfrak{e}-(g+1-\mu)\mathfrak{f}=(\mu-2)\mathfrak{f}-\mathfrak{e},$

so $\mathfrak{h}$ is non-special since our assumption on $\mathfrak{e}$
implies that it imposes independent conditions to any multiple of
$\mathfrak{f}$. Therefore, $h^{0}(C,\mathfrak{h})=g-\mu+3$ by Riemann–Roch.

We first consider the case $\mu\leqslant g$. Let $v$ be a basis of
$H^{0}(C,\mathfrak{e})$, and $s,t$ be a basis of $H^{0}(C,\mathfrak{f})$. Then
$vs^{g+1-\mu},\ldots,vt^{g+1-\mu}$ are linearly independent, and span a
hyperplane in $H^{0}(C,\mathfrak{h})$. Eventually, we choose $u$ such that
$u,vs^{g+1-\mu},\ldots,vt^{g+1-\mu}$ form a basis of $H^{0}(C,\mathfrak{h})$.
Arguing in the same way as above, we find that
$H^{0}(C,\mathfrak{h}+\mathfrak{f})$ has dimension $g-\mu+5$, and
$us,ut,vs^{g-\mu+2},\ldots,vt^{g-\mu+2}$ is a basis of this space. Thus the
linear system $|\mathfrak{h}+\mathfrak{f}|$ maps $C$ in $\mathbf{P}^{g-\mu+4}$
in such a way that it sits in a rational normal scroll built on a line and a
rational normal curve of degree $g-\mu+2$ spanning complementary subspaces,
that is, $\mathbf{F}_{g+1-\mu}$ in its embedding given by $|H+F|$, and
$\left.F\right|_{C}=\mathfrak{f}$ and $\left.H\right|_{C}=\mathfrak{h}$. This
ends the proof in this case, as $H=E+(g+1-\mu)F$. The argument is similar when
$\mu=g+1$, and we leave it to the reader. ∎

##### (8.3)Proof of (1.8.1)

Let $(C,L)$ be a polarized hyperelliptic curve of genus $g$ and degree $d$. We
first prove the result in case $d=4g+4$. We shall see that there are two ways
of proceeding in order to write $L$ as $2\mathfrak{h}+\mu\mathfrak{f}$, so as
to be able to apply Proposition 8.2. We call these the even and odd way,
respectively (note that one way is sufficient to prove (1.8.1)). We keep the
notation $\mathfrak{f}$ for the class of the $g^{1}_{2}$ on $C$, and write
$g=2\gamma+\varepsilon$, $\varepsilon\in\\{0,1\\}$ and $\gamma\in\mathbf{N}$.

In the even way, one first chooses a line bundle $M^{+}$ of (even) degree
$2g+2$ such that $L=2M^{+}$. Then, there is a unique integer
$k\in\\{0,\ldots,\gamma+\varepsilon\\}$ such that $M^{+}$ may be written as

$M^{+}=\mathfrak{e}+(\gamma+1+k)\mathfrak{f},$

with the condition that $\mathfrak{e}$ is the sum of $g+\varepsilon-2k$ points
pairwise not conjugate with respect to the $g^{1}_{2}$ ($k$ is thus the
largest integer such that $M^{+}-(\gamma+1+k)\mathfrak{f}$ is effective). We
set

$\mu=g+\varepsilon-2k\
\text{(even)}\quad\text{and}\quad\mathfrak{h}=\mathfrak{e}+(g+1-\mu)\mathfrak{f}.$

By Proposition 8.2 there exists an embedding of $C$ as a member of the linear
system $|2H+\mu F|$ on $\mathbf{F}_{g+1-\mu}$ such that
$\left.E\right|_{C}=\mathfrak{e}$ and $\left.F\right|_{C}=\mathfrak{f}$. The
normal bundle of $C$ in this embedding is

$\displaystyle N_{C/\mathbf{F}_{g+1-\mu}}$
$\displaystyle=2\mathfrak{h}+\mu\mathfrak{f}$
$\displaystyle=2\mathfrak{e}+(2g+2-\mu)\mathfrak{f}$
$\displaystyle=2\mathfrak{e}+2(\gamma+1+k)\mathfrak{f}=2M^{+}=L,$

hence the embedding of $\mathbf{F}_{g+1-\mu}$ defined by the complete linear
system $|2H+\mu F|$ is an extension of $(C,L)$, as we wanted.

In the odd way, one chooses instead a line bundle $M^{-}$ of (odd) degree
$2g+1$ such that $L=2M^{-}+\mathfrak{f}$. Then, there is a unique integer
$k\in\\{0,\ldots,\gamma\\}$ such that $M^{-}$ may be written as

$M^{-}=\mathfrak{e}+(\gamma+\varepsilon+k)\mathfrak{f},$

with $\mathfrak{e}$ is the sum of $g+1-\varepsilon-2k$ points pairwise not
conjugate with respect to the $g^{1}_{2}$. We set

$\mu=g+1-\varepsilon-2k\
\text{(odd)}\quad\text{and}\quad\mathfrak{h}=\mathfrak{e}+(g+1-\mu)\mathfrak{f}.$

By Proposition 8.2 there exists an embedding of $C$ as a member of the linear
system $|2H+\mu F|$ on $\mathbf{F}_{g+1-\mu}$ such that
$\left.E\right|_{C}=\mathfrak{e}$ and $\left.F\right|_{C}=\mathfrak{f}$. The
normal bundle of $C$ in this embedding is

$\displaystyle N_{C/\mathbf{F}_{g+1-\mu}}$
$\displaystyle=2\mathfrak{h}+\mu\mathfrak{f}$
$\displaystyle=2\mathfrak{e}+(2g+2-\mu)\mathfrak{f}$
$\displaystyle=2\mathfrak{e}+[2(\gamma+\varepsilon+k)+1]\mathfrak{f}=2M^{-}+\mathfrak{f}=L,$

which proves that $(C,L)$ is extendable in the same fashion as in the even
way.

It remains to prove (1.8.1) when $d<4g+4$. In this case, we choose $b=4g+4-d$
distinct points $p_{1},\ldots,p_{b}$ on $C$, to the effect that

$L^{\sharp}=L(p_{1}+\cdots+p_{b})$

is a line bundle of degree $4g+4$ on $C$. Thus there exists a non-trivial
extension $S^{\sharp}$ of $(C,L^{\sharp})$ by the case $d=4g+4$, and we get an
extension of $(C,L)$ by simple internal projection of $S^{\sharp}$ from the
points $p_{1},\ldots,p_{b}$, as in Example 3.4. ∎

To prove the remaining parts of Theorem 1.8, we will apply Proposition 8.2 in
the following way.

######  Corollary 8.4.

Let $C$ be a hyperelliptic curve of genus $g$, $\mu\in\\{g+1,g,\ldots,0\\}$,
and $b$ a non-negative integer. The locus in the Jacobian $J^{4g+4-b}(C)$ of
degree $4g+4-b$ line bundles $L$ on $C$ such that $L$ is the normal bundle of
$C$ in a simple internal projection from $b$ points of the surface
$(\mathbf{F}_{g+1-\mu},2H+\mu F)$ has dimension $\min(g,\mu+b)$.

More precisely, the locus in the corollary parametrizes those $L$ such that
the following holds: there exists an embedding of $C$ as a smooth member of
the linear system $|2H+\mu F|$ on $\mathbf{F}_{g+1-\mu}$ passing through $b$
points $p_{1},\ldots,p_{b}\in\mathbf{F}_{g+1-\mu}$, such that $L$ is the
normal bundle of the proper transform of $C$ in the blow-up of
$\mathbf{F}_{g+1-\mu}$ at $p_{1},\ldots,p_{b}$.

###### Proof.

It follows from Proposition 8.2 that the locus $\mathcal{Z}\subseteq
J^{2g+2-\mu}(C)$ of degree $2g+2-\mu$ line bundles $\mathfrak{h}$ such that
there exists an embedding of $C$ as a member of $|2H+\mu F|$ on
$\mathbf{F}_{g+1-\mu}$ such that $\left.H\right|_{C}=\mathfrak{h}$ has
dimension $\min(g,\mu)$. In turn, the locus we are interested in is the image
of $\mathcal{Z}\times C^{[b]}$, where $C^{[b]}$ denotes the $b$-th symmetric
power of $C$, by the map

(8.4.1) $j_{\mu,b}:(\mathfrak{h},D)\in J^{2g+2-\mu}(C)\times
C^{[b]}\longmapsto 2\mathfrak{h}+\mu\mathfrak{f}-D\in J^{4g+4-b}(C),$

with $\mathfrak{f}$ the class of the $g^{1}_{2}$, hence its dimension is
$\min(g,\mu+b)$. ∎

##### (8.5)Proof of (1.8.2) and (1.8.3)

For all $\mu=g+1,g,\ldots,0$, we consider as in the proof of Theorem 1.7 the
universal family $\mathcal{L}_{\mu,b}\to\mathcal{S}_{\mu,b}$ of the linear
systems $|2H+\mu F-E_{D}|$ on the blow-up of $\mathbf{F}_{g+1-\mu}$ along
$D\in\mathcal{S}_{\mu,b}=(\mathbf{F}_{g+1-\mu})^{[b]}$, with total exceptional
divisor $E_{D}$. The total space $\mathcal{L}_{\mu,b}$ has dimension $3g+5+b$.
Then we divide out by the automorphism group of $\mathbf{F}_{g+1-\mu}$, which
has dimension $5+\max(1,g+1-\mu)$, and end up with a moduli space
$\mathcal{SC}_{\mu,b}$ of pairs $(S,C)$ which has dimension

$2g+b-1+\min(g,\mu).$