# Pluri-cotangent maps of surfaces of general type

Francesco Polizzi, Xavier Roulleau

###### Abstract

Let $X$ be a compact, complex surface of general type whose cotangent bundle
$\Omega_{X}$ is strongly semi-ample. We study the pluri-cotangent maps of $X$,
namely the morphisms
$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
defined by the vector space of global sections $H^{0}(X,\,S^{n}\Omega_{X})$.

2010 Mathematics Subject Classification: 14J29

Keywords: surfaces of general type, pluri-cotangent maps

##  0 Introduction

Let $X$ be a complex surface of general type and assume that its cotangent
bundle is strongly semi-ample, namely, that there exists an integer $n\geq 1$
such that the evaluation map

$H^{0}(X,\,S^{n}\Omega_{X})\otimes\mathcal{O}_{X}\longrightarrow
S^{n}\Omega_{X}$ (1)

is surjective (this condition implies in particular that $X$ is minimal and
$K_{X}$ is ample). Recalling that we have a natural identification between
$H^{0}(X,\,S^{n}\Omega_{X})$ and
$H^{0}(\mathbb{P}(\Omega_{X}),\,\mathcal{O}_{\mathbb{P}(\Omega_{X})}(n))$,
from the surjectivity of (1) we infer that the induced evaluation map

$H^{0}(X,\,S^{n}\Omega_{X})\otimes\mathcal{O}_{\mathbb{P}(\Omega_{X})}\longrightarrow\mathcal{O}_{\mathbb{P}(\Omega_{X})}(n)$
(2)

is also surjective, and so it defines a morphism

$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X})),$
(3)

that we call the $n$_th pluri-cotangent map_ of $X$. The case $n=1$ was
studied by the second author in [Rou09]: it turns out that, as soon as
$\Omega_{X}$ is globally generated and $q(X)\geq 4$, the cotangent map
$\psi_{1}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,\Omega_{X}))\simeq\mathbb{P}^{q(X)-1}$
is a generically finite morphism onto its image.

In the present note we generalize this result to the case $n\geq 2$. Let us
remark that, if $\Omega_{X}$ is strongly semi-ample, then $\psi_{n}$ is a
generically finite morphism onto its image for $n$ sufficiently large if and
only if the second Segre number $c_{1}(X)^{2}-c_{2}(X)$ is strictly positive,
see Proposition 2.2. Our aim is to give effective versions of this statement.
The first result we show is

######  Theorem A (see Theorem 2.19).

Let $n\geq 3$ be such that $S^{n}\Omega_{X}$ is globally generated. If
$h^{0}(X,\,S^{n}\Omega_{X})\neq\frac{1}{2}(n+1)(n+2)$ then $\psi_{n}$ is
generically finite onto its image. In this case, the exceptional locus
$\mathrm{exc}(\psi_{n})$ is a Zariski-closed, possibly empty subset of
$\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ of dimension at most $1$.

The proof of Theorem A is obtained by generalizing the geometrical arguments
used in [Rou09]. We also exploit some results contained in the recent paper
[MU19] by Mistretta and Urbinati, allowing us to prove the finiteness of the
$n$th Gauss map of $X$ (Proposition 2.9), together with the classical
description of surfaces $2$-covered by curves of degree $n$, first obtained by
Bompiani [Bom21] and later rediscovered by Pirio and Russo [PR13], see
Proposition 2.17.

Furthermore, we are also able to provide explicit bounds for the generic
finiteness of the pluri-cotangent maps.

######  Theorem B (see Theorem 3.3).

Let $n\geq 3$ be an integer such that $S^{n}\Omega_{X}$ is globally generated.
If

$c_{1}^{2}>\left(1+\frac{6n-2}{2n^{2}-2n+1}\right)c_{2}+\frac{6n+12}{2n^{2}-2n+1}$
(4)

then the $n$th pluri-cotangent map $\psi_{n}$ is generically finite onto its
image.

The proof of Theorem B uses the explicit computation of
$\chi(X,\,S^{n}\Omega_{X})$, see Lemma 3.1, together with Bogomolov’s
cohomological vanishing (Proposition 1.2), in order to show that (4) implies
the lower bound $h^{0}(X,\,S^{n}\Omega_{X})>\frac{1}{2}(n+1)(n+2)$. Then the
claim follows from Theorem A. As a consequence, we get

######  Corollary C (see Corollary 3.4).

Let $X$ be a minimal surface of general type with $c_{1}^{2}-c_{2}>0$ and such
that $S^{m}\Omega_{X}$ is globally generated. Then the $n$th pluri-cotangent
map $\psi_{n}$ is generically finite onto its image for all multiples $n$ of
$m$ such that

$n>\frac{\beta+\sqrt{\beta^{2}-\alpha\gamma}}{\alpha},$ (5)

where

$\alpha:=2(c_{1}^{2}-c_{2}),\quad\beta:=c_{1}^{2}+2c_{2}+3,\quad\gamma:=c_{1}^{2}+c_{2}-12.$
(6)

Corollary C is proved by rewriting (4) as a quadratic inequality in $n$ with
strictly positive leading coefficient, namely

$2(c_{1}^{2}-c_{2})n^{2}-2(c_{1}^{2}+2c_{2}+3)n+(c_{1}^{2}+c_{2}-12)>0.$ (7)

Section 4 deals with some examples and counterexamples. In Subsections 4.1 and
4.2 we consider the following two constructions giving surfaces $X$ such that
$\Omega_{X}$ is neither ample nor globally generated:

  * (1)

$X$ is a (special) symmetric complete intersection in an abelian fourfold of
the form $A\times E$, where $A$ is an abelian threefold and $E$ is an elliptic
curve;

  * (2)

$X$ is of the form $X=(C\times F)/G$, where $C$ is a smooth hyperelliptic
curve of genus $3$, $F$ is a smooth curve of odd genus and $G=\mathbb{Z}_{2}$
acts with four fixed points on $C$, freely on $F$ and diagonally on the
product.

In both situations, the vector bundle $S^{2}\Omega_{X}$ turns out to be
globally generated, hence $\Omega_{X}$ is strongly semi-ample, and moreover
the pluri-cotangent map $\psi_{n}$ is generically finite onto its image for
all even $n$ (see Propositions 4.1, 4.2, 4.5, 4.6).

In Subsection 4.3 we exhibit some counterexamples to Corollary C when
$c_{1}^{2}-c_{2}=0$, namely, smooth ample divisors in abelian threefolds. In
fact, if $X$ is such a divisor, for all $n\geq 1$ we have

$H^{0}(X,\,S^{n}\Omega_{X})=S^{n}H^{0}(X,\,\Omega_{X})\simeq\mathbb{C}^{\frac{(n+1)(n+2)}{2}},$
(8)

and the image $X_{n}$ of
$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
is projectively equivalent to the $n$th Veronese surface
$\nu_{n}(\mathbb{P}^{2})\subset\mathbb{P}^{\frac{n(n+3)}{2}}$. Thus, no pluri-
cotangent map of $X$ is generically finite onto its image. Under the
additional assumption that the image of the Albanese map is smooth, we also
show that these counterexamples are the only ones up to finite, étale covers
(Proposition 4.10).

By using finite cyclic covers of a surface $X$ as above, we are also able to
construct surfaces of general type all of whose Gauss maps have arbitrarily
large degree, see Remark 4.11.

Finally, in the last section of the paper we stated a couple of open problems.

Acknowledgments. This work started in May 2017, when the first author visited
the Institut de Mathématiques de Marseille. He is grateful to the members of
the équipe Analyse, Géométrie et Topologie for the invitation and the
hospitality, and to GNSAGA-INdAM for the financial support. He also thanks all
the MathOverflow users that generously answered his questions in several
threads, see

MO397682, MO412306, MO412888, MO413988, MO414382, MO414452,

MO417972, MO418607, MO430933, MO431327.

Both authors are grateful to Erwan Rousseau for providing useful references
and to Igor Reider for suggestions and remarks.

Notation and conventions. We work over the field $\mathbb{C}$ of complex
numbers. By _surface_ we mean a smooth, compact complex surface $X$, and for
such a surface $\Omega_{X}$ denotes the holomorphic cotangent bundle, $T_{X}$
the holomorphic tangent bundle, $\omega_{X}=\mathcal{O}_{X}(K_{X})$ the
canonical bundle, $p_{g}(X)=h^{0}(X,\,K_{X})$ is the geometric genus,
$q(X)=h^{1}(X,\,K_{X})$ is the irregularity and
$\chi(\mathcal{O}_{X})=1-q(X)+p_{g}(X)$ is the holomorphic Euler-Poincaré
characteristic. We also set $c_{1}:=c_{1}(T_{X})=-c_{1}(\Omega_{X})$ and
$c_{2}:=c_{2}(T_{X})=c_{2}(\Omega_{X})$. The second Segre number of $X$ is the
integer $c_{1}^{2}-c_{2}$.

For projective spaces and projective bundles we use the same conventions as in
[Laz04, Chapter 6]. More specifically, if $V$ is a vector space,
$\mathbb{P}(V)$ stands by the projective space of $1$-dimensional quotients of
$V$; we denote by $\mathbb{G}(n,\,\mathbb{P}(V))$ the Grassmannian of
$n$-dimensional subspaces of $\mathbb{P}(V)$, and by
$\mathbb{G}(\mathbb{P}(V),\,n)$ the Grassmannian of $n$-dimensional quotients
of $\mathbb{P}(V)$. Moreover, we write
$\nu_{n}\colon\mathbb{P}(V)\longrightarrow\mathbb{P}(S^{n}V)$ for the
$n{\textrm{th}}$ Veronese embedding of $\mathbb{P}(V)$, and we call its image
$\nu_{n}(\mathbb{P}(V))$ the $n{\mathrm{th}}$ Veronese variety of
$\mathbb{P}(V)$.

If $\mathscr{E}$ is a vector bundle on $X$ and $x\in\mathscr{E}$, we write
$\mathscr{E}(x)$ for the fibre of $\mathscr{E}$ over $x$. Furthermore, we
denote by $\pi\colon\mathbb{P}(\mathscr{E})\longrightarrow X$ the projective
bundle of $1$-dimensional quotients of $\mathscr{E}$, so that
$\pi_{*}\mathcal{O}_{\mathbb{P}(\mathscr{E})}(n)=S^{n}\mathscr{E}$ for all
$n\geq 1$.

##  1 Pluri-cotangent maps and Gauss maps

Let $X$ be a compact, complex surface of general type. We say that its
cotangent bundle $\Omega_{X}$ is _strongly semi-ample_ if the $n$th symmetric
power $S^{n}\Omega_{X}$ is globally generated for some $n\geq 1$, namely, if
the evaluation map

$H^{0}(X,\,S^{n}\Omega_{X})\otimes\mathcal{O}_{X}\longrightarrow
S^{n}\Omega_{X}$ (9)

is surjective. Note that this implies

$h^{0}(X,\,S^{n}\Omega_{X})\geq\operatorname{rank}\,S^{n}\Omega_{X}=n+1$ (10)

and if equality holds then $S^{n}\Omega_{X}\simeq\mathcal{O}_{X}^{n+1}$. In
particular, since we are assuming that $X$ is of general type, the strict
inequality holds in (10). Moreover, the strong semi-ampleness of $\Omega_{X}$
implies that $\mathcal{O}_{\mathbb{P}(\Omega_{X})}(1)$ is semi-ample, see
[MU19, Section 3.1], and so $\Omega_{X}$ is nef.

######  Lemma 1.1.

If $\Omega_{X}$ is strongly semi-ample, then $X$ does not contain any smooth
rational curve. In particular, $X$ is minimal and $K_{X}$ is ample.

###### Proof.

If $C$ is a smooth curve contained in $X$, then $\Omega_{C}$ is a quotient of
the restricted bundle $\Omega_{X}|_{C}$. Since passing to the $n$th symmetric
product preserves epimorphisms, it follows that $S^{n}\Omega_{C}$ is a
quotient of $S^{n}\Omega_{X}|_{C}$. This implies that $\Omega_{C}$ is strongly
semi-ample, hence $g(C)\geq 1$. ∎

Lemma 1.1 allows us to apply to our situation the next result, based on
Bogomolov’s work [Bog78], see [Kob80, Corollary A.1] and [RouRous13, p. 1341].

######  Proposition 1.2.

Let $X$ be a surface of general type with ample canonical class. Then for all
$n\geq 1$ the vector bundle $S^{n}\Omega_{X}$ is semi-stable with respect to
the polarization $K_{X}$, and moreover

$H^{0}(X,\,S^{n}T_{X}\otimes\omega_{X}^{k})=0\quad\text{for}\,\,n-2k>0.$ (11)

Setting $k=1$ in (11) and applying Serre duality, we get Bogomolov’s vanishing

$H^{2}(X,\,S^{n}\Omega_{X})=0\quad\textrm{for}\,\,n\geq 3.$ (12)

######  Corollary 1.3.

Let $X$ be a surface of general type with ample canonical class. Then for all
$n\geq 3$ we have $h^{0}(X,\,S^{n}\Omega_{X})\geq\chi(X,\,S^{n}\Omega_{X})$.

###### Proof.

Immediate consequence of (12). ∎

######  Remark 1.4.

The extremal case $n=2,\,k=1$ in Proposition 1.2 is characterized by the
following result, cf. [Kob80, Theorem B and Corollaries B.1 and B.2]. If $X$
is a minimal surface of general type with ample canonical bundle, then we have

$H^{0}(X,\,S^{2}T_{X}\otimes\omega_{X})=H^{0}(X,\,S^{2}\Omega_{X}\otimes\omega_{X}^{-1})=0$
(13)

if and only if $\Omega_{X}$ is an indecomposable rank $2$ vector bundle. One
direction is clear: if $\Omega_{X}=L_{1}\oplus L_{2}$ is the direct sum of two
line bundles, then a straightforward computation shows that
$S^{2}\Omega_{X}\otimes\omega_{X}^{-1}$ has a direct summand isomorphic to
$\mathcal{O}_{X}$, hence $H^{0}(X,\,S^{2}\Omega_{X}\otimes\omega_{X}^{-1})\neq
0$. Conversely, let us assume that $S^{2}\Omega_{X}\otimes\omega_{X}^{-1}$ has
a non-zero global section and let us show that $\Omega_{X}$ is decomposable;
to this pourpose, we will use the following argument suggested to us by Igor
Reider. Identifying $S^{2}\Omega_{X}\otimes\omega_{X}^{-1}$ with the sheaf
$End_{0}(\Omega_{X})$ of trace-zero endomorphisms of $\Omega_{X}$, a non-zero
global section corresponds to an endomorphism
$f\colon\Omega_{X}\longrightarrow\Omega_{X}$ whose trace is zero at every
point. Now we have two cases:

  * $\boldsymbol{(i)}$

There is a point $x\in X$ such that
$f_{x}\colon\Omega_{X}(x)\longrightarrow\Omega_{X}(x)$ has two non-zero
eigenvalues $\pm\lambda$; then

$\Omega_{X}=\ker(f-\lambda\operatorname{I})\oplus\ker(f+\lambda\operatorname{I})$
(14)

is the desired splitting. 111If we have an endomorphism $g\colon
E\longrightarrow E$ of a vector bundle $E$, then its determinant $\det
g\colon\det E\longrightarrow\det E$ is a scalar multiple of the identity and
so, if it vanishes at one point, it vanishes everywhere. Taking $E=\Omega_{X}$
and $g=f-\lambda I$, $g=f+\lambda I$, we get  $\Omega_{X}(x)=\ker(f-\lambda
I)(x)\oplus\ker(f+\lambda I)(x)$ (15) for all $x\in X$.

  * $\boldsymbol{(ii)}$

The endomorphism $f\colon\Omega_{X}\longrightarrow\Omega_{X}$ is nilpotent
everywhere, hence $f^{2}=0$. We will rule out this case, by exploiting the
ampleness of $K_{X}$. In fact, by the nilpotency condition, the sheaf
$\operatorname{im}(f)$ injects into $\ker(f)$; then, setting

$c_{1}(\ker(f))=L,\quad c_{1}(\operatorname{im}(f))=L^{\prime}$ (16)

the divisor $L-L^{\prime}$ is effective. The semi-stability of $\Omega_{X}$
with respect to the polarization $K_{X}$ now gives

$K_{X}L=\mu(L)\leq\mu(\Omega_{X})=K_{X}^{2}/2,\quad
K_{X}L^{\prime}=\mu(L^{\prime})\geq\mu(\Omega_{X})=K_{X}^{2}/2,$ (17)

and so

$0\leq K_{X}(L-L^{\prime})\leq K_{X}^{2}/2-K_{X}^{2}/2=0,$ (18)

that implies $L=L^{\prime}$. Thus $K_{X}=c_{1}(\Omega_{X})=2L$, hence $L$ is
ample. But this is impossible because, by a result of Bogomolov, $\Omega_{X}$
cannot have ample subsheaves of rank $1$, see [Reid77, Theorem 2].

Recalling that we have a natural identification between
$H^{0}(X,\,S^{n}\Omega_{X})$ and
$H^{0}(\mathbb{P}(\Omega_{X}),\,\mathcal{O}_{\mathbb{P}(\Omega_{X})}(n))$,
from the surjectivity of (9) we infer that the induced evaluation map

$H^{0}(X,\,S^{n}\Omega_{X})\otimes\mathcal{O}_{\mathbb{P}(\Omega_{X})}\longrightarrow\mathcal{O}_{\mathbb{P}(\Omega_{X})}(n)$
(19)

is also surjective, and so defines a morphism

$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X})).$
(20)

######  Definition 1.5.

We call $\psi_{n}$ the ${n}{\mathrm{th}}$ _cotangent map_ of $X$, and we
denote its image by
$X_{n}:=\psi_{n}(\mathbb{P}(\Omega_{X}))\subset\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$.

By [Laz04, Appendix A] there is a relative $n$th Veronese embedding
$\nu_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(S^{n}\Omega_{X})$
such that, for every $x\in X$, the fibre $\mathbb{P}(\Omega_{X}(x))$ of
$\mathbb{P}(\Omega_{X})$ over $x$ is sent to a rational normal curve of degree
$n$ inside the $n$-dimensional projective space
$\mathbb{P}(S^{n}\Omega_{X}(x))$. Moreover, passing to projective bundles in
the evaluation map (9), we get a morphism
$e_{n}\colon\mathbb{P}(S^{n}\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
and a factorization of $\psi_{n}$ of the form

${\mathbb{P}(\Omega_{X})}$${\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))}$${\mathbb{P}(S^{n}\Omega_{X})}$$\scriptstyle{\psi_{n}}$$\scriptstyle{\nu_{n}}$$\scriptstyle{e_{n}}$
(21)

Let us now consider two important examples: the case where $\Omega_{X}$ is
ample and the case where $\Omega_{X}$ is globally generated.

######  Example 1.6.

The case where $\Omega_{X}$ is ample. If $\Omega_{X}$ is ample, then it is
automatically strongly semi-ample, see [Laz04, Theorem 6.1.10], and the Chern
numbers of $X$ satisfy the inequality $c_{1}^{2}-c_{2}>0$, see [Kl69].
Furthermore, by [Laz04, Example 6.1.5 and Theorem 6.1.15], the ampleness of
$\Omega_{X}$ is equivalent to the fact that $e_{n}$ is finite onto its image.
Summing up, we can state what follows:

_Assume that $S^{n}\Omega_{X}$ is globally generate. Then the $n$th cotangent
map $\psi_{n}$ is a finite morphism onto its image $X_{n}$ if and only if
$\Omega_{X}$ is ample_.

For the sake of completeness, let us shortly explain how to construct surfaces
for which $\Omega_{X}$ is strongly semi-ample but not ample. Let $A$ be an
abelian 3-fold containing an elliptic curve $E$, and let $X\subset A$ be a
sufficiently positive, smooth divisor containing $E$. Then $X$ is a surface of
general type whose Albanese morphism $a_{X}\colon
X\longrightarrow\operatorname{Alb}(X)$ coincides with the inclusion
$X\longrightarrow A$. Since $A$ contains no rational curves, the same is true
for $X$, which is therefore a minimal model with ample $K_{X}$. Furthermore,
since $\Omega_{X}$ is a quotient of $\Omega_{A}|_{X}=\mathcal{O}_{X}^{\oplus
3}$, it follows that $\Omega_{X}$ is globally generated, hence strongly semi-
ample. However, $\Omega_{X}$ is not ample: in fact, varieties with ample
cotangent bundle are Kobayashi hyperbolic ([Laz04, Theorem 6.3.26]), in
particular, they do not contain any elliptic curves. For a detailed analysis
of the analogue construction in codimension $2$, see Subsection 4.1.

######  Example 1.7.

The case where $\Omega_{X}$ itself is globally generated. Since the evaluation
map (9) is the co-differential of the Albanese morphism

$a_{X}\colon X\longrightarrow\operatorname{Alb}(X),$ (22)

the cotangent bundle $\Omega_{X}$ is globally generated if and only if $a_{X}$
is a local immersion. In this case, $S^{n}\Omega_{X}$ is globally generated
for all $n\geq 1$, and we have a natural symmetrization homomorphism
$\sigma_{n}\colon S^{n}H^{0}(X,\,\Omega_{X})\longrightarrow
H^{0}(X,\,S^{n}\Omega_{X})$, that fits into a commutative diagram

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57.60608pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
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(23)

Here $\nu_{n}$ stands for $n{\mathrm{th}}$ Veronese embedding and the rational
map $\mathbb{P}(\sigma_{n})$ is an embedding of projective spaces if
$\sigma_{n}$ is surjective, and a linear projection otherwise. 222According to
our understanding, not much is known about the behaviour of $\sigma_{n}$ in
general. A result in this direction would provide a higher-dimensional
generalization of the celebrated Max Noether’s Theorem for curves, see
[ACGH85, p. 117] and the MathOverflow thread [MO273557]. By [Rou09,
Proposition 2.14] it follows that if $q(X)=h^{0}(X,\,\Omega_{X})>3$ then
$\psi_{1}$ is generically finite, hence we can draw the following conclusion.

_Assume that $\Omega_{X}$ is globally generated and $q(X)>3$. Then the $n$th
cotangent map $\psi_{n}$ is generically finite onto its image for all $n\geq
1$_.

Again for the sake of completeness, let us provide examples where $\Omega_{X}$
is strongly semi-ample but not globally generated. If $X$ is a a fake
projective plane (see [PY07]), then $\Omega_{X}$ is ample (this is true for
every smooth compact complex variety uniformized by the ball
$\mathbb{B}^{n}\subset\mathbb{C}^{n}$, see [Laz04, Construction 6.3.36]) and
thus strongly semi-ample. However,
$h^{0}(X,\Omega_{X})=h^{1}(X,\,\mathcal{O}_{X})=0$, namely, $\Omega_{X}$ has
no global sections at all.

##  2 Finiteness of the Gauss map and dimension of the pluri-cotangent image

######  Assumption 2.1.

From now on, $X$ will denote a surface of general type with strongly semi-
ample cotangent bundle $\Omega_{X}$. Note that we are _neither_ assuming that
$\Omega_{X}$ is ample _nor_ that $\Omega_{X}$ is globally generated, having
already analyzed these cases before.

######  Proposition 2.2.

Let $X$ be a surface that satisfies Assumption _2.1_. Then $\psi_{n}$ is
generically finite onto its image $X_{n}$ for $n$ sufficiently large if and
only if $c_{1}^{2}-c_{2}>0$. In this case, we have $\deg
X_{n}=n^{3}(c_{1}^{2}-c_{2})/\deg\psi_{n}$.

###### Proof.

Assume $n\geq 3$. Using the asymptotic form of Riemann-Roch theorem for vector
bundles together with the vanishing (12), we get

$h^{0}(X,\,S^{n}\Omega_{X})\geq\chi(X,\,S^{n}\Omega_{X})=\frac{n^{3}}{6}(c_{1}^{2}-c_{2})+O(n^{2}).$
(24)

Thus, the positivity of the second Segre number $c_{1}^{2}-c_{2}$ implies that
$\Omega_{X}$ is big, hence $\psi_{n}$ is generically finite onto its image for
$n$ sufficiently large. Conversely, suppose that $\psi_{n}$ is generically
finite onto $X_{n}$ for some $n$. Then, if
$\xi\in|\mathcal{O}_{\mathbb{P}(\Omega_{X})}(1)|$, the same argument used in
the proof of [Rou09, Proposition 2.15] shows that

$0<\deg X_{n}\cdot\deg\psi_{n}=(n\xi)^{3}=n^{3}(c_{1}^{2}-c_{2}).$ (25)

∎

######  Remark 2.3.

In Section 3 we will provide a quantitative version of Proposition 2.2, see in
particular Corollary 3.4.

######  Remark 2.4.

The statement of Proposition 2.2 boils down to the fact that, if we assume
that the line bundle $\mathcal{O}_{\mathbb{P}(\Omega_{X})}(1)$ is nef, then it
is big if and only if its top self-intersection is strictly positive.

######  Remark 2.5.

Subsection 4.3 contains a detailed analysis of some examples where
$\Omega_{X}$ is globally generated, $c_{1}^{2}-c_{2}=0$ and $\dim X_{n}=2$ for
all $n\geq 1$. This shows that the assumption about the positivity of the
second Segre number in Proposition 2.2 cannot be removed.

######  Remark 2.6.

There exist examples of surfaces $X$ of general type with big cotangent bundle
and $c_{1}^{2}-c_{2}\leq 0$, see [RouRous13]. They are obtained by taking the
minimal resolution of some singular models with rational double points, hence
they contain smooth rational curves and so $\Omega_{X}$ is not strongly semi-
ample (Lemma 1.1). However, the bigness of $\Omega_{X}$ implies that
$\psi_{n}$ is a birational map onto its image for $n$ large enough. This shows
that the “only if” part in the statement of Proposition 2.2 does not hold if
one drops strongly semi-ampleness in Assumption 2.1.

In this paper we focus on finding explicit lower bounds on $n$ such that
$\psi_{n}$ is generically finite onto its image. Our arguments are geometric
in nature, and generalize the ones used in [Rou09, Section 2]; furthermore, we
use in an essential way some results from [PR13] and [MU19].

Let $\pi\colon\mathbb{P}(\Omega_{X})\longrightarrow X$ be the structure
projection and let us look at the restriction of $\psi_{n}$ to the fibre
$\pi^{-1}(x)$ over a point $x\in X$. Such a fibre is the curve
$\mathbb{P}(\Omega_{X}(x))\simeq\mathbb{P}^{1}$, and the restriction of
$|\mathcal{O}_{\mathbb{P}(\Omega_{X})}(n)|$ to it is the complete linear
system $|\mathcal{O}_{\mathbb{P}^{1}}(n)|$, that embeds $\pi^{-1}(x)$ as a
rational normal curve $C_{x}$ of degree $n$ in
$\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$. There is a unique $n$-dimensional
linear subspace $L_{x}\subset\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
containing $C_{x}$, so we have a morphism

$\mathsf{g}_{n}\colon
X\longrightarrow\mathbb{G}(n,\,\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))),\quad
x\mapsto L_{x}.$

######  Definition 2.7.

We call $\mathsf{g}_{n}$ the ${n}{\mathrm{th}}$ _Gauss map_ of $X$, and we
denote its image by
$Y_{n}:=\mathsf{g}_{n}(X)\subset\mathbb{G}(n,\,\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X})))$.

Let us now provide an alternative description of the Gauss map. Being
$\Omega_{X}$ globally generated, for every point $x\in X$ there is a
surjection

$H^{0}(X,\,S^{n}\Omega_{X})\longrightarrow S^{n}\Omega_{X}(x)\longrightarrow
0.$ (26)

Since the fibre $S^{n}\Omega(x)$ of $S^{n}\Omega$ over $x$ is a vector space
of dimension $n+1$, after passing to projective spaces and dualizing we obtain
a quotient of dimension $n$ of $\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))^{*}$,
hence an element

$s_{x}\in\mathbb{G}(n,\,\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X})))\simeq\mathbb{G}(\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))^{*},\,n).$
(27)

Thus, we get a morphism

$\mathsf{k}_{n}\colon
X\longrightarrow\mathbb{G}(n,\,\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))),\quad
x\mapsto s_{x}.$ (28)

Following [MU19, p. 2230] we call $\mathsf{k}_{n}$ the $n$th _Kodaira map_ of
$X$.

######  Proposition 2.8.

The two morphisms $\mathsf{g}_{n}$ and $\mathsf{k}_{n}$ do actually coincide.

###### Proof.

Diagram (21) shows that the $n$-dimensional linear space $L_{x}$ containing
$C_{x}$ coincides with the image of $\mathbb{P}(S^{n}\Omega_{X}(x))$ via the
map $e_{n}$. By construction, this image is precisely $s_{x}$. ∎

######  Proposition 2.9.

The $n{\mathrm{th}}$ Gauss map $\mathsf{g}_{n}$ is a finite morphism onto its
image $Y_{n}$. In particular, we have $\dim Y_{n}=2$ for all $n\geq 1$.

###### Proof.

By Proposition 2.8, it is equivalent to prove the result for the Kodaira map
$\mathsf{k}_{n}$. Let $\widetilde{Y}_{n}\longrightarrow Y_{n}$ be the
normalization map of $Y_{n}$, and let $\tilde{\mathsf{k}}_{n}\colon
X\longrightarrow\widetilde{Y}_{n}$ be the corresponding lifting of
$\mathsf{k}_{n}$ (which exists since $X$ is smooth, hence normal). If the
result is true for $\tilde{\mathsf{k}}_{n}$ then it is true for
$\mathsf{k}_{n}$ as well, because the normalization is a finite map. Thus, we
may assume that $Y_{n}$ is normal.

Next, the assumption that $S^{n}\Omega_{X}$ is globally generated implies that
$S^{tn}\Omega_{X}$ is globally generated for all positive integers $t$;
according to [MU19, Lemma 3.3], we have a factorization

${X}$${Y_{n}}$${Y_{tn}}$$\scriptstyle{\mathsf{k}_{n}}$$\scriptstyle{\mathsf{k}_{tn}}$$\scriptstyle{u}$
(29)

where $u\colon Y_{tn}\longrightarrow Y_{n}$ is a finite map. Furthermore, by
[MU19, Theorem 3.4], there exists a diagram

${X}$${Y_{\infty}}$${Y_{\mathbb{G}}}$$\scriptstyle{\mathsf{k}_{\textrm{det}}}$$\scriptstyle{\mathsf{k}_{\mathbb{G}}}$$\scriptstyle{v}$
(30)

where $\mathsf{k}_{\textrm{det}}$ is the Iitaka fibration induced by
$K_{X}=\det\Omega_{X}$ and $v$ is a finite map, such that for $t\gg 0$ we have
$Y_{tn}=Y_{\mathbb{G}}$ and $\mathsf{k}_{tn}=\mathsf{k}_{\mathbb{G}}$. So, for
$t$ sufficiently large, we get

$\mathsf{k}_{n}=u\circ\mathsf{k}_{tn}=u\circ v\circ\mathsf{k}_{\textrm{det}}.$
(31)

Since $K_{X}$ is ample (Lemma 1.1), it follows that
$\mathsf{k}_{\textrm{det}}$ is an isomorphism onto its image; therefore
$\mathsf{k}_{n}$ is a composition of finite maps, and the proof is complete.

∎

######  Lemma 2.10.

For all $p\in\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$, the restriction of
$\pi\colon\mathbb{P}(\Omega_{X})\longrightarrow X$ to $\psi_{n}^{-1}(p)$ is
injective.

###### Proof.

Given $x\in X$, the intersection $\pi^{-1}(x)\cap\psi_{n}^{-1}(p)$ is either
empty or consists of a single point, because the restriction
$\psi_{n}\colon\pi^{-1}(x)\longrightarrow
C_{x}\subset\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ is an embedding. ∎

######  Remark 2.11.

The degree of the Gauss map divides the degree of the pluri-cotangent map.
Denoting by $U_{n}$ the universal vector bundle over the affine Grassmannian
of $(n+1)$-dimensional subspaces of $H^{0}(X,\,S^{n}\Omega_{X})$ and by
$\mathbb{P}(U_{n})$ the corresponding projectivization, we have a commutative
diagram

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.44225pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}(\Omega_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces{\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
98.96025pt\raise 30.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\psi_{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{{\hbox{\kern
181.12303pt\raise 11.63524pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}\ignorespaces{\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}}\ignorespaces{}\ignorespaces{\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-31.30444pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
40.44781pt\raise 7.04167pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.18056pt\hbox{$\scriptstyle{\tilde{\psi}_{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 78.4099pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
78.4099pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}(U_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
94.34125pt\raise-17.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 94.34125pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
135.26134pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 172.24025pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
172.24025pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))}$}}}}}}}{\hbox{\kern-7.53471pt\raise-41.13776pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.03801pt\raise-35.95026pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\mathsf{g}_{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.44225pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.44225pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{G}(n,\,\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))),}$}}}}}}}\ignorespaces}}}}\ignorespaces&\end{split}$

where $\pi_{1}$, $\pi_{2}$ are the natural projections and $\tilde{\psi}_{n}$
is such that $\varphi\circ\tilde{\psi}_{n}=\psi_{n}$. Since the restriction of
$\psi_{n}$ to the fibres of $\pi$ is an embedding, it follows that
$\tilde{\psi}_{n}$ is a finite map onto its image, whose degree is the same as
the degree of the Gauss map $\mathsf{g}_{n}$. So, assuming that the
pluricotangent map $\psi_{n}$ is generically finite onto its image $X_{n}$,
and denoting by $\deg\psi_{n}$ the degree of
$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow X_{n}$, we obtain

$\deg\psi_{n}=\deg\mathsf{g}_{n}\cdot\deg\pi_{1}^{\prime},$ (32)

where $\pi_{1}^{\prime}$ is the restriction of $\pi_{1}$ to the image of
$\tilde{\psi}_{n}$. As a consequence, if $\psi_{n}$ is generically finite onto
its image, then $\deg\mathsf{g}_{n}$ divides $\deg\psi_{n}$.

If $p\in\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$, let us denote by
$D_{p}^{\prime}$ the subvariety $\pi(\psi_{n}^{-1}(p))\subset X$. By Lemma
2.10, we have

$\dim D_{p}^{\prime}=\dim\psi_{n}^{-1}(p).$ (33)

######  Proposition 2.12.

Let $x\in X$. Then $x\in D_{p}^{\prime}$ if and only if $p\in C_{x}$.

###### Proof.

If $x\in D_{p}^{\prime}$ then $x=\pi(z)$, with $z\in\psi_{n}^{-1}(p)$. Thus
$z\in\pi^{-1}(x)$ and so $p=\psi_{n}(z)\in\psi_{n}(\pi^{-1}(x))=C_{x}$.
Conversely, if $p\in C_{x}$ then $p=\psi_{n}(w)$, with $w\in\pi^{-1}(x)$. Then
$x=\pi(w)\in\pi(\psi_{n}^{-1}(p))=D_{p}^{\prime}$. ∎

######  Proposition 2.13.

For all $p\in\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$, the subvariety
$\psi^{-1}_{n}(p)\subset\mathbb{P}(\Omega_{X})$ has dimension at most $1$.
Hence $D_{p}^{\prime}\subset X$ has dimension at most $1$, too.

###### Proof.

Write $Z:=\psi_{n}^{-1}(p)$ and $N:=h^{0}(X,\,S^{n}\Omega_{X})$. Since
$\psi_{n}$ is not a constant map, we have $\dim Z\leq 2$. By contradiction,
assume $\dim Z=2$. Then the restriction $\pi_{Z}\colon Z\longrightarrow X$ is
a bijective morphism by Lemma 2.10 and so, since $X$ is normal, it is an
isomorphism by Zariski’s Main Theorem, see [Mum99, Chapter III]. Thus we get a
regular section $t\colon X\longrightarrow\mathbb{P}(\Omega_{X})$ of the
projective bundle $\pi\colon\mathbb{P}(\Omega_{X})\longrightarrow X$, that in
turn corresponds to a rank $1$ quotient
$\Omega_{X}\longrightarrow\mathcal{L}$, where
$\mathcal{L}=t^{*}\mathcal{O}_{\mathbb{P}(\Omega_{X})}(1)$, see [Ha77, Chapter
II, Proposition 7.12]. Since $\psi_{n}\circ t$ contracts $X$ to the point
$p\in\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))\simeq\mathbb{P}^{N-1}$, it follows
that $(\psi_{n}\circ t)^{*}\mathcal{O}_{\mathbb{P}^{N-1}}(1)$ is the trivial
line bundle on $X$, and so

$\mathcal{L}^{n}=t^{*}\mathcal{O}_{\mathbb{P}(\Omega_{X})}(n)=t^{*}\psi_{n}^{*}\mathcal{O}_{\mathbb{P}^{N-1}}(1)=\mathcal{O}_{X}.$
(34)

So, taking the $n$th symmetric product of
$\Omega_{X}\longrightarrow\mathcal{L}$, we get a quotient
$S^{n}\Omega_{X}\longrightarrow\mathcal{O}_{X}$, contradicting the semi-
stability of $S^{n}\Omega_{X}$ with respect to $K_{X}$, see Proposition 1.2.
Hence the only possibility is $\dim Z=1$, and this proves the first statement.
The second statement follows from (33). ∎

######  Definition 2.14.

A point $p\in\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ is called _exceptional
for_ $\psi_{n}$ if the fibre $\psi_{n}^{-1}(p)$ has dimension $1$. The set of
such exceptional points will be denoted by $\mathrm{exc}(\psi_{n})$.

Let $p$ be an exceptional point for $\psi_{n}$, so that $\dim
D_{p}^{\prime}=1$, and let $D_{p}\subseteq D_{p}^{\prime}$ be an irreducible
component of dimension $1$. If we set $\Sigma:=\psi_{n}(\pi^{-1}(D_{p}))$,
then we have

$\Sigma=\bigcup_{x\in D_{p}}C_{x}.$ (35)

######  Proposition 2.15.

The variety $\Sigma$ is a surface in $\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
containing a $1$-dimensional family of rational normal curves of degree $n$
passing through $p$. More precisely, for every point $q\in\Sigma$, different
from $p$, there exists a rational normal curve of degree $n$ contained in
$\Sigma$ and joining $p$ and $q$.

###### Proof.

The fact that $\Sigma$ has dimension at least $1$ is an immediate consequence
of (35). By contradiction, assume $\dim\Sigma=1$; then there is a rational
normal curve $C$ of degree $n$ such that $C_{x}=C$ for all $x\in D_{p}$. This
in turn implies that the $n$-plane
$L_{x}\subset\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ is constant on $D_{p}$,
and so the $n{\mathrm{th}}$ Gauss map $\mathsf{g}_{n}\colon
X\longrightarrow\mathbb{G}(n,\,\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
contracts $D_{p}$ to a point, against Proposition 2.9. It follows that
$\Sigma\subset\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ is a surface. If
$q\in\Sigma$, then $q\in C_{x}$ for some $x\in D_{p}$; thus we have $p\in
C_{x}$ by Proposition 2.12, hence the rational normal curve $C_{x}$ joins $p$
and $q$. ∎

######  Definition 2.16.

Let $N>3$ be a positive integer. An irreducible variety
$V\subset\mathbb{P}^{N-1}$ is said to be $2$-_covered by curves of degree_ $n$
if a general pair of points of $V$ can be joined by an irreducible curve of
degree $n$.

The case of surfaces $2$-covered by curves was classically considered in
[Bom21]; a modern treatment can be found in [PR13], see also [Ion05, Theorem
2.8] and [Tr10, Théorème 1.1]. It turns out that they are precisely the
Veronese embeddings of $\mathbb{P}^{2}$:

######  Proposition 2.17.

Let $\Sigma\subset\mathbb{P}^{N-1}$ be a non-degenerate surface which is
$2$-covered by curves of degree $n$. Then $N=\frac{1}{2}(n+1)(n+2)$ and
$\Sigma$ is projectively equivalent to the $n{\mathrm{th}}$ Veronese surface
$\nu_{n}(\mathbb{P}^{2})$. In particular, every curve in the $2$-covering
family is a rational normal curve of degree $n$ and there exists a unique such
a curve passing through two distinct points of $\Sigma$.

###### Proof.

See [PR13, Theorem 2.2]. ∎

Let us study now the image of the $n{\mathrm{th}}$ cotangent map and the
geometry the locus $\mathrm{exc}(\psi_{n})$; this generalizes the analysis of
the case $n=1$ that was carried out in [Rou09, Proposition 2.14].

######  Proposition 2.18.

If $n\geq 2$ and the map
$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
is not generically finite, then its image
$X_{n}\subset\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ is a non-degenerate,
linearly normal surface $2$-covered by rational normal curves of degree $n$.
Therefore, by the last result, it is projectively equivalent to
$\nu_{n}(\mathbb{P}^{2})$.

###### Proof.

By Proposition 2.13 we have $2\leq\dim X_{n}\leq 3$. If $\dim X_{n}=2$ then
every fibre of $\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow X_{n}$ has
dimension $1$, i.e. every point $p\in X_{n}$ is an exceptional point for
$\psi_{n}$. Hence $D_{p}$ has dimension $1$, whereas
$\Sigma=\psi_{n}(\pi^{-1}(D_{p}))$ has dimension $2$ and is contained in the
irreducible surface $X_{n}$. Therefore $\Sigma=X_{n}$ and so, by using
Proposition 2.15 and the fact that $p\in X_{n}$ is arbitrary, we infer that
$X_{n}$ is $2$-covered by curves of degree $n$. Finally, $X_{n}$ is non-
degenerate and linearly normal, being the image of the morphism induced by a
complete linear system in $\mathbb{P}(\Omega_{X})$. ∎

######  Theorem 2.19.

Let $n\geq 3$ be an integer such that $S^{n}\Omega_{X}$ is globally generated.
If $h^{0}(X,\,S^{n}\Omega_{X})\neq\frac{1}{2}(n+1)(n+2)$ then $\psi_{n}$ is
generically finite onto its image, namely, $\dim X_{n}=3$, and we have

$\deg\psi_{n}\leq\frac{n^{3}(c_{1}^{2}-c_{2})}{h^{0}(X,\,S^{n}\Omega_{X})-3}.$
(36)

In this case, $\mathrm{exc}(\psi_{n})$ is a Zariski-closed, possibly empty
subset of $\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ of dimension at most $1$.

###### Proof.

The first statement immediately follows from Propositions 2.17 and 2.18,
setting $N=h^{0}(X,\,S^{n}\Omega_{X})$. Since $X_{n}$ is a non-degenerate
threefold in $\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$, we have $\deg X_{n}\geq
h^{0}(X,\,S^{n}\Omega_{X})-3$ and so (36) is a consequence of (25). Regarding
the last statement, if $\psi_{n}$ is generically finite onto its image then we
have $\dim\psi_{n}^{-1}(\mathrm{exc}(\psi_{n}))\leq 2$, hence the exceptional
locus $\mathrm{exc}(\psi_{n})$ has dimension at most $1$. Such a locus is a
(possibly empty) Zariski-closed subset of
$\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$ because $\psi_{n}$ is a proper
morphism, see [EGAIV, Corollaire 13.1.4]. ∎

Note that, as explained in Example 1.6, the exceptional locus
$\mathrm{exc}(\psi_{n})$ is empty if and only if $\Omega_{X}$ is ample.

##  3 An explicit bound for the generic finiteness of the pluri-cotangent maps

We start this section by a straightforward calculation, whose details are
included because we could not find a suitable reference.

######  Lemma 3.1.

Let $X$ be a compact, complex surface. We have

$\chi(X,\,S^{n}\Omega_{X})=\frac{1}{12}(n+1)\left((2n^{2}-2n+1)c_{1}^{2}-(2n^{2}+4n-1)c_{2}\right).$
(37)

###### Proof.

This is a standard application of the splitting principle, as stated in
[Fried98, p. 28]: every universal formula on Chern classes which holds for
direct sum of line bundles holds in general. Let $\mathscr{E}=L_{1}\oplus
L_{2}$ be a decomposable rank $2$ vector bundle on $X$; then

$c_{1}(\mathscr{E})=c_{1}(L_{1})+c_{1}(L_{2}),\quad
c_{2}(\mathscr{E})=c_{1}(L_{1})c_{1}(L_{2}).$ (38)

We can compute $c_{1}(S^{n}\mathscr{E})$ as follows:

$\begin{split}c_{1}(S^{n}\mathscr{E})&=c_{1}\left(S^{n}(L_{1}\oplus
L_{2})\right)=c_{1}\left(\bigoplus_{i=0}^{n}L_{1}^{i}\otimes
L_{2}^{n-i}\right)\\\
&=\sum_{i=0}^{n}\left(ic_{1}(L_{1})+(n-i)c_{1}(L_{2})\right)=\frac{n(n+1)}{2}\left(c_{1}(L_{1})+c_{1}(L_{2})\right)\\\
&=\frac{n(n+1)}{2}c_{1}(\mathscr{E}).\end{split}$ (39)

Let us now compute $c_{2}(S^{n}\mathscr{E})$. We have

$\begin{split}c_{2}(S^{n}\mathscr{E})&=c_{2}\left(S^{n}(L_{1}\oplus
L_{2})\right)=c_{2}\left(\bigoplus_{i=0}^{n}L_{1}^{i}\otimes
L_{2}^{n-i}\right)\\\ &=\sum_{0\leq i<j\leq n}c_{1}(L_{1}^{i}\otimes
L_{2}^{n-i})c_{1}(L_{1}^{j}\otimes L_{2}^{n-j})\\\ &=\sum_{0\leq i<j\leq
n}\left(ic_{1}(L_{1})+(n-i)c_{1}(L_{2})\right)\left(jc_{1}(L_{1})+(n-j)c_{1}(L_{2})\right)\\\
&=Ac_{1}(L_{1})^{2}+Bc_{1}(L_{1})c_{1}(L_{2})+Cc_{1}(L_{2})^{2},\end{split}$
(40)

where

$\begin{split}A&=\sum_{0\leq i<j\leq n}ij,\\\ B&=\sum_{0\leq i<j\leq
n}\left(i(n-j)+(n-i)j\right),\\\ C&=\sum_{0\leq i<j\leq
n}(n-i)(n-j).\end{split}$ (41)

The quantities in (41) can be calculated by means of the standard formulas for
the sum of integers and squares, obtaining

$\begin{split}A=C=&\frac{1}{24}(n-1)n(n+1)(3n+2)\\\
B&=\frac{1}{12}n(n+1)(3n^{2}+n+2).\end{split}$ (42)

Plugging (42) into (40), and taking into account (38), we get

$c_{2}(S^{n}\mathscr{E})=\frac{1}{24}(n-1)n(n+1)(3n+2)c_{1}(\mathscr{E})^{2}+\frac{1}{6}n(n+1)(n+2)c_{2}(\mathscr{E}).$
(43)

Now, the Riemann-Roch theorem for vector bundles on surfaces, see [Fried98, p.
31], implies

$\chi(X,\,S^{n}\mathscr{E})=\frac{c_{1}(S^{n}\mathscr{E})(c_{1}(S^{n}\mathscr{E})-K_{X})}{2}-c_{2}(S^{n}\mathscr{E})+(n+1)\frac{c_{1}^{2}+c_{2}}{12}.$
(44)

Setting $\mathscr{E}=\Omega_{X}$ in (44) and using (39) and (43), by standard
computations we obtain (37). ∎

######  Corollary 3.2.

Let $X$ be surface of general type with ample canonical class. Then for all
$n\geq 3$ we have

$h^{0}(X,\,S^{n}\Omega_{X})\geq\frac{1}{12}(n+1)\left((2n^{2}-2n+1)c_{1}^{2}-(2n^{2}+4n-1)c_{2}\right).$
(45)

###### Proof.

Combine Lemma 3.1 with Corollary 1.3. ∎

######  Theorem 3.3.

Let $X$ be a surface of general type satisfying Assumption 2.1 and let $n\geq
3$ be an integer such that $S^{n}\Omega_{X}$ is globally generated. If

$c_{1}^{2}>\left(1+\frac{6n-2}{2n^{2}-2n+1}\right)c_{2}+\frac{6n+12}{2n^{2}-2n+1}$
(46)

then the $n$th pluri-cotangent map $\psi_{n}$ is generically finite onto its
image $X_{n}$.

###### Proof.

Using (37) one checks that (46) is equivalent to
$\chi(X,\,S^{n}\Omega_{X})>\frac{1}{2}(n+1)(n+2)$. Using Corollary 1.3 we
infer $h^{0}(X,\,S^{n}\Omega_{X})>\frac{1}{2}(n+1)(n+2)$, so the claim follows
from Theorem 2.19. ∎

We can now state the following quantitative version of Proposition 2.2.

######  Corollary 3.4.

Let $X$ be a minimal surface of general type with $c_{1}^{2}-c_{2}>0$ and such
that $S^{m}\Omega_{X}$ is globally generated. Then the $n$th pluri-cotangent
map $\psi_{n}$ is generically finite onto its image for all multiples $n$ of
$m$ such that

$n>\frac{\beta+\sqrt{\beta^{2}-\alpha\gamma}}{\alpha},$ (47)

where

$\alpha:=2(c_{1}^{2}-c_{2}),\quad\beta:=c_{1}^{2}+2c_{2}+3,\quad\gamma:=c_{1}^{2}+c_{2}-12.$
(48)

###### Proof.

We can rewrite (46) as a quadratic inequality in $n$ with strictly positive
leading coefficient, namely

$Q(n)=2(c_{1}^{2}-c_{2})n^{2}-2(c_{1}^{2}+2c_{2}+3)n+(c_{1}^{2}+c_{2}-12)>0.$
(49)

Let us define $u=c_{1}^{2}/c_{2}>1$. By the Bogomolov-Miyaoka-Yau inequality,
one has $u\leq 3$. The reduced discriminant $\delta=\beta^{2}-\alpha\gamma$ of
$Q(n)$ is

$\begin{array}[]{c}\delta=(-u^{2}+4u+6)c_{2}^{2}+(30u-12)c_{2}+9.\end{array}$

We see $\delta$ as a quadratic function of $c_{2}$, depending on the parameter
$u\in(1,\,3]$. The reduced discriminant of $\delta$ is the function

$\bar{\delta}(u)=18(u-1)(13u+1),$ (50)

which satisfies $\bar{\delta}(u)>0$ for $u\in(1,\,3]$. Since all the
coefficients of $\delta$ are positive for $u\in(1,\,3]$, it follows that both
zeros of $\delta$ are real and negative for $u$ in the same range. As a
consequence, $\delta$ is positive for all $c_{2}>0$ and $u\in(1,\,3]$. Summing
up, when $n$ is greater than the root $\frac{\beta+\sqrt{\delta}}{\alpha}$ of
$Q(n)$, one has $Q(n)>0$.

∎

##  4 Examples and counterexamples

###  4.1 Example: symmetric complete intersections in abelian fourfolds of
product type

Let us consider an abelian fourfold of the form $A\times E$, where $A$ is an
abelian threefold and $E$ is an elliptic curve. Let $M$ be a polarization on
$A\times E$ and let $x\in A$ be a point which is not $2$-torsion. Up to
replacing the polarization $M$ on $A\times E$ with a suitable positive
multiple, by standard Bertini-type arguments we can find two smooth
hypersurfaces $Y_{1},\,Y_{2}\in|M|$ such that

  * •

$Y_{1}$ and $Y_{2}$ are both smooth and symmetric, i.e. invariant with respect
to the involution $-1\colon A\times E\longrightarrow A\times E$;

  * •

$Y_{1}$ and $Y_{2}$ both contain the elliptic curve $\\{x\\}\times E$;

  * •

$Y_{1}$ and $Y_{2}$ do not contain any $2$-torsion points of $A\times E$;

  * •

the intersection $Y=Y_{1}\cap Y_{2}$ is smooth.

The conditions above imply that $Y$ is a smooth surface on which $-1$ acts
freely; then the quotient $f\colon Y\longrightarrow X$ provide a smooth
surface $X$, containing an elliptic curve $E^{\prime}$ isomorphic to $E$.

######  Proposition 4.1.

The surface $X$ is of general type with

$c_{1}(X)^{2}=2M^{4},\quad c_{2}(X)=\frac{3}{2}M^{4},$ (51)

hence $c_{1}(X)^{2}-c_{2}(X)=\frac{1}{2}M^{4}>0$. Moreover
$H^{0}(X,\,\Omega_{X})=0$, in particular $\Omega_{X}$ is not globally
generated. Finally, $S^{n}\Omega_{X}$ is globally generated for all even $n$,
hence $\Omega_{X}$ is strongly semi-ample; however, $\Omega_{X}$ is not ample.

###### Proof.

Using the short exact sequence of tangent bundles

$0\longrightarrow T_{Y}\longrightarrow T_{A\times E}|_{Y}\longrightarrow
N_{Y}\longrightarrow 0,$ (52)

and recalling that $T_{A\times E}=\mathcal{O}_{A\times E}^{\oplus 4}$ and
$N_{Y}=\mathcal{O}_{Y}(M)^{\oplus 2}$, we get the equality of total Chern
classes

$c(T_{Y})=c(\mathcal{O}_{Y}(M))^{-2}$ (53)

that in turn yields $c_{1}(Y)=\mathcal{O}_{Y}(-2M)$ and $c_{2}(Y)=3M^{4}$.
Since $f\colon Y\longrightarrow X$ is an étale double cover, we deduce (51).
Moreover, we have $\Omega_{Y}=f^{*}\Omega_{X}$, hence the vector space
$H^{0}(X,\,\Omega_{X})$ is isomorphic to the involution-invariant subspace of
$H^{0}(Y,\,\Omega_{Y})$. Now, by Lefschetz theorem for Hodge groups, see
[Laz04, Example 3.1.24], the global holomorphic $1$-forms on $Y$ are precisely
the restrictions of those on $A\times E$, so none of them is invariant and we
get $H^{0}(X,\,\Omega_{X})=0$. On the other hand, if $n$ is even then all
global sections of $S^{n}\Omega_{A\times E}$ are invariant and thus, when
restricted to $Y$, they descend to $X$. These sections generate
$S^{n}\Omega_{A\times E}$, hence they also generate $S^{n}\Omega_{Y}$ and
$S^{n}\Omega_{X}$. Finally, $\Omega_{X}$ is not ample because $X$ contains the
elliptic curve $E^{\prime}$, cf. Example 1.6. ∎

######  Proposition 4.2.

Let $X$ be a surface as above. Then the pluri-cotangent map
$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow
H^{0}(X,\,S^{n}\Omega_{X})$ is generically finite onto its image for all even
$n\geq 2$.

###### Proof.

From $\Omega_{Y}=f^{*}\Omega_{X}$ we infer
$S^{n}\Omega_{Y}=f^{*}(S^{n}\Omega_{X})$ for all $n\geq 1$. This implies that
$H^{0}(X,S^{n}\Omega_{X})$ is isomorphic to the involution-invariant subspace
$H^{0}(Y,\,S^{n}\Omega_{Y})^{+}\subseteq H^{0}(Y,\,S^{n}\Omega_{Y})$. On the
other hand, [Deb05, Proposition 13] implies that the restriction map

$H^{0}(A\times E,\,S^{n}\Omega_{A\times E})\longrightarrow
H^{0}(Y,\,S^{n}\Omega_{Y})$ (54)

is injective for all $n$; moreover, if $n$ is even then every global section
of $S^{n}\Omega_{A\times E}$ is invariant, and so its restriction belongs to
$H^{0}(Y,\,S^{n}\Omega_{Y})^{+}$. Summing up, for all even $n\geq 2$ we have

$\begin{split}h^{0}(X,\,S^{n}\Omega_{X})&=\dim
H^{0}(Y,\,S^{n}\Omega_{Y})^{+}\geq\dim H^{0}(A\times E,\,S^{n}\Omega_{A\times
E})\\\ &=\frac{1}{6}(n+1)(n+2)(n+3)>\frac{1}{2}(n+1)(n+2).\end{split}$ (55)

The claim now follows from Theorem 2.19. ∎

###  4.2 Example: some product-quotient surfaces

We start by considering a hyperelliptic curve $C$ of genus $3$, endowed with
an action of the cyclic group $G=\mathbb{Z}_{2}$ as follows. The curve $C$ has
affine equation of the form

$y^{2}=a_{8}x^{8}+a_{6}x^{6}+a_{4}x^{4}+a_{2}x^{2}+a_{0},$ (56)

where the coefficients $a_{i}$ are such that the zeros of the polynomial at
the right side are distinct. If $g$ is the generator of $G$, we define the
action of $G$ on $C$ as

$g(x,\,y)=(-x,\,y).$ (57)

One checks that $g$ has the four fixed points

$(0,\,\sqrt{a_{0}}),\quad(0,\,-\sqrt{a_{0}}),\quad(\infty,\,\sqrt{a_{8}}),\quad(\infty,\,-\sqrt{a_{8}}),$
(58)

hence the quotient map $C\longrightarrow C/G$ is branched at four points and,
by the Hurwitz formula, the curve $C/G$ has genus $1$.

######  Lemma 4.3.

Let $H^{0}(C,\,\omega_{C})=V^{+}\oplus V^{-}$ be the decomposition of
$H^{0}(C,\,\omega_{C})$ into the $G$-invariant subspace $V^{+}$ and the
$G$-antiinvariant subspace $V^{-}$. Then there exists a basis
$\\{\xi_{1},\,\xi_{2},\,\xi_{3}\\}$ of $H^{0}(C,\,\omega_{C})$ such that

$V^{+}=\langle\xi_{1}\rangle,\quad V^{-}=\langle\xi_{2},\,\xi_{3}\rangle$ (59)

and the three canonical divisors $\operatorname{div}(\xi_{1})$,
$\operatorname{div}(\xi_{2})$, $\operatorname{div}(\xi_{3})$ have pairwise
disjoint supports.

###### Proof.

The vector space $H^{0}(C,\,\omega_{C})$ is generated by the holomorphic
$1$-forms which, in affine coordinates, can be written as

$\omega_{1}:=\frac{dx}{y},\quad\omega_{2}:=x\frac{dx}{y},\quad\omega_{3}:=x^{2}\frac{dx}{y}.$
(60)

Note that $\omega_{2}$ is $G$-invariant, whereas $\omega_{1}$ and $\omega_{3}$
are $G$-antiinvariant. Now set $\xi_{1}:=\omega_{2}$ and take as $\xi_{2}$ and
$\xi_{3}$ two general elements in the $G$-antiinvariant subspace
$\langle\omega_{1},\,\omega_{3}\rangle$. ∎

Let us consider now another curve $F$ with a $G$-action.

######  Lemma 4.4.

Let $g=2k+1\geq 5$ be an odd integer. Then there exists a curve $F$ of genus
$g$, endowed with a free $G$-action having the following property. Denoting by
$H^{0}(F,\,\omega_{F})=W^{+}\oplus W^{-}$ the decomposition of
$H^{0}(F,\,\omega_{F})$ into $G$-invariant and $G$-antiinvariant subspace, we
can find a basis $\\{\tau_{1},\ldots,\tau_{g}\\}$ of $H^{0}(F,\,\omega_{F})$
such that

$W^{+}=\langle\tau_{1},\ldots,\tau_{k+1}\rangle,\quad
W^{-}=\langle\tau_{k+2},\ldots,\tau_{g}\rangle$ (61)

and the $g$ canonical divisors
$\operatorname{div}(\tau_{1}),\ldots,\operatorname{div}(\tau_{g})$ have
pairwise disjoint supports.

###### Proof.

Let $D$ be a curve of genus $k+1$ and let $\mathscr{L}$ be a non-trivial line
bundle on $D$ such that $\mathscr{L}^{2}=\mathcal{O}_{D}$. Then there exists
an étale double cover $f\colon F\longrightarrow D$, with $F$ of genus $2k+1$
and $f_{*}\mathcal{O}_{F}=\mathcal{O}_{D}\oplus\mathscr{L}^{-1}$; the curve
$F$ comes with a free $G$-action, corresponding to the automorphism exchanging
the two sheets of the cover. Furthermore, since
$f_{*}\omega_{F}=\omega_{D}\oplus(\omega_{D}\otimes\mathscr{L}),$ we deduce

$W^{+}=f^{*}H^{0}(D,\,\omega_{D}),\quad
W^{-}=f^{*}H^{0}(D,\,\omega_{D}\otimes\mathscr{L}).$ (62)

Therefore the desired result follows if both $\omega_{D}$ and
$\omega_{D}\otimes\mathscr{L}$ are globally generated. In fact, for a base-
point free line bundle, a general section avoids any given finite set of
points, so we can choose recursively a basis where each section avoids the
base loci of the previous ones; moreover, this property is preserved by étale
pullbacks. It is well known that $\omega_{D}$ is base-point free, see for
instance [Ha77, Lemma 5.1 p. 341]. Regarding $\omega_{D}\otimes\mathscr{L}$, a
point $p$ is in its base locus if and only if

$H^{0}(D,\,\omega_{D}\otimes\mathscr{L}(-p))=H^{0}(D,\,\omega_{D}\otimes\mathscr{L}),$
(63)

namely, if and only if $h^{1}(D,\,\omega_{D}\otimes\mathscr{L}(-p))=1$. By
Serre duality, this is equivalent to the fact that there exists $q\in D$ such
that the divisor class of $\mathscr{L}$ is of the form $q-p$. But then
$\mathcal{O}_{D}(2q-2p)=\mathcal{O}_{D}$, hence the linear system spanned by
$2p$ and $2q$ is a $g^{1}_{2}$ on $D$ and so $D$ is hyperelliptic. Summing up,
$\omega_{D}\otimes\mathscr{L}$ is globally generated if and only if one of
these conditions (both implying $g(D)\geq 3$, and so $g=g(F)\geq 5$) hold:

  * •

$D$ is non-hyperelliptic;

  * •

$D$ is hyperelliptic and the divisor class of $\mathscr{L}$ is not the
difference of two Weierstrass points.

This concludes the proof. ∎

Taking $C$ and $F$ as above, for all $k\geq 2$ we can now define a product-
quotient surface $X=(C\times F)/G$, where $G$ acts diagonally on the product.
Such action is free (because the action of $G$ on $F$ is so), hence $X$ is a
smooth surface of general type, whose invariants are

$p_{g}(X)=3k+1,\quad q(X)=k+2,\quad K_{X}^{2}=16k.$ (64)

Thus $h^{0}(X,\,\Omega_{X})=k+2$ and $c_{1}(X)^{2}-c_{2}(X)=8k>0$. The natural
projections of $C\times F$ induce two isotrivial fibrations $X\longrightarrow
F/G$ and $X\longrightarrow C/G$, whose general fibres are isomorphic to $C$
and $F$, respectively. In particular, the cotangent bundle $\Omega_{X}$ is not
ample, see [Rou09, Corollaire 3.8].

Let us now show that $\Omega_{X}$ is not globally generated, either, but that
it is nevertheless strongly semi-ample.

######  Proposition 4.5.

Let $C$ and $F$ be curves with a $G$-action as above, and $X=(C\times F)/G$.
Then $\Omega_{X}$ is not globally generated, whereas its second symmetric
power $S^{2}\Omega_{X}$ is globally generated $($and so $S^{n}\Omega_{X}$ is
globally generated for all even $n)$.

###### Proof.

Denoting by $\pi_{C}\colon C\times F\longrightarrow C$, $\pi_{F}\colon C\times
F\longrightarrow F$ the two natural projections, we have

$\Omega_{C\times F}=L\oplus M,$ (65)

where $L=\pi_{C}^{*}\omega_{C}$, $M=\pi_{F}^{*}\omega_{F}$. Moreover, the
covering map $C\times F\longrightarrow X$ being étale, for all $n\geq 1$ we
have $H^{0}(X,\,S^{n}\Omega_{X})=H^{0}(C\times F,\,S^{n}\Omega_{C\times
F})^{G}$. Thus, since the action of $G$ on $C\times F$ does not exchange the
two factors, in order to show that $S^{n}\Omega_{X}$ is globally generated it
suffices to show that it is possible to generate every summand $L^{k}\otimes
M^{n-k}$ of $S^{n}\Omega_{C\times F}$ by using $G$-invariant global sections.

In the case $n=1$, the space of invariant global sections of $\Omega_{C\times
F}$ is

$V^{+}\oplus
W^{+}=\langle\xi_{1}\rangle\oplus\langle\tau_{1},\ldots,\tau_{k+1}\rangle.$
(66)

This shows that $\Omega_{X}$ is not globally generated, since $V^{+}$ is
$1$-dimensional and so it is not possible to generate the summand $L$ by means
of $G$-invariant global sections.

In the case $n=2$, we have

$S^{2}\Omega_{C\times F}=L^{2}\oplus(L\otimes M)\oplus M^{2}.$ (67)

We recall that $\operatorname{div}(\xi_{i})$ are pairwise disjoint divisors,
and the same is true for $\operatorname{div}(\tau_{j})$, see Lemmas 4.3 and
4.4; thus, using the notation $\alpha\boxtimes\beta$ for
$\pi_{C}^{*}\alpha\otimes\pi_{F}^{*}\beta$, we can say that

  * •

the $G$-invariant sections $(\xi_{1})^{2},\,\,(\xi_{2})^{2}$ generate $L^{2}$;

  * •

the $G$-invariant sections
$\xi_{1}\boxtimes\tau_{1},\,\,\xi_{2}\boxtimes\tau_{k+2},\,\,\xi_{3}\boxtimes\tau_{k+3}$
generate $L\otimes M$;

  * •

the $G$-invariant sections $(\tau_{1})^{2},\,\,(\tau_{2})^{2}$ generate
$M^{2}$.

This shows that $S^{2}\Omega_{X}$ is globally generated. ∎

######  Proposition 4.6.

Let $k\geq 2$ and $X$ be the surface constructed above. Then the pluri-
cotangent map $\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow
H^{0}(X,\,S^{n}\Omega_{X})$ is generically finite onto its image for all even
$n$.

###### Proof.

We first analyze the case $n=2$. Using the $G$-invariant global sections of
$S^{2}\Omega_{X}$ produced of at the end of the proof of Proposition 4.5, we
get

$\begin{split}h^{0}(X,\,S^{2}\Omega_{X})&=\dim\left[H^{0}(X,\,L^{2})^{G}\oplus
H^{0}(X,\,L\otimes M)^{G}\oplus H^{0}(X,\,M^{2})^{G}\right]\\\ &\geq
2+3+2=7.\end{split}$ (68)

For all $k\geq 2$ we obtain $h^{0}(X,\,S^{2}\Omega_{X})>6$, hence $\psi_{2}$
is generically finite onto its image by Theorem 2.19. Now, let us consider the
general case. We have $c_{1}(X)^{2}=16k^{2}$ and $c_{2}(X)=8k$ and so, by
Corollary 3.4, the $n$th pluricotangent map $\psi_{n}$ is generically finite
onto its image for all even $n$ such that

$n>\frac{32k+3+\sqrt{640k^{2}+384k+9}}{16k}.$ (69)

Straightforward calculations show that function
$h(k)=\frac{32k+3+\sqrt{640k^{2}+384k+9}}{16k}$ is strictly decreasing in the
interval $[2,\,+\infty)$ and that $h(2)\simeq 3.9$. Thus, for $k$ in the same
interval, we have $n>h(k)$ as soon as $n\geq 4$, and this completes the proof.
∎

###  4.3 Counterexamples to the generic finiteness of $\psi_{n}$: ample
divisors in abelian threefolds

We will now show that the requirement
$h^{0}(X,\,S^{n}\Omega_{X})\neq\frac{1}{2}(n+1)(n+2)$ in Theorem 2.19 is an
essential one. In fact, we will provide examples of surfaces $X$ of general
type, with $\Omega_{X}$ globally generated, such that
$h^{0}(X,\,S^{n}\Omega_{X})=\frac{1}{2}(n+1)(n+2)$ and $X_{n}$ is the
$n{\mathrm{th}}$ Veronese surface for all $n\geq 1$. Thus, no pluri-cotangent
map of $X$ is generically finite onto its image. All these surfaces satisfy
$c_{1}^{2}-c_{2}=0$.

Let $(A,\,M)$ be a polarized abelian threefold, with polarization $M$ of type
$(d_{1},\,d_{2},\,d_{3})$, such that there exists a smooth element $X\in|M|$.
By the Riemann-Roch Theorem and the ampleness of $M$, we have

$h^{0}(A,\,M)=\chi(A,\,M)=\frac{1}{6}M^{3}=d_{1}d_{2}d_{3},$ (70)

see [BL04, Chapter 3]. Moreover, by adjunction we get
$\omega_{X}=\mathcal{O}_{X}(X)$ and so

$K_{X}^{2}=M^{3}=6h^{0}(A,\,M).$ (71)

From the short exact sequence

$0\longrightarrow\mathcal{O}_{A}\longrightarrow\mathcal{O}_{A}(X)\longrightarrow\omega_{X}\longrightarrow
0$ (72)

we infer

$p_{g}(X)=h^{0}(A,\,M)+2,\,\quad q(X)=3,$ (73)

hence $\chi(\mathcal{O}_{X})=h^{0}(A,\,M)$. Summing up, $X$ is a minimal
surface of general type with $K_{X}^{2}=6\chi(\mathcal{O}_{X})$, namely
$c_{1}(X)^{2}-c_{2}(X)=0$; moreover the canonical bundle $\Omega_{X}$ is
globally generated (cf. Example 1.7) and so $K_{X}$ is ample (Lemma 1.1).

######  Proposition 4.7.

For all $n\geq 1$, we have

$H^{0}(X,\,S^{n}\Omega_{X})=S^{n}H^{0}(X,\,\Omega_{X})\simeq\mathbb{C}^{\frac{(n+1)(n+2)}{2}}.$
(74)

Furthermore, the image $X_{n}$ of
$\psi_{n}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))$
is projectively equivalent to the $n{\emph{th}}$ Veronese surface
$\nu_{n}(\mathbb{P}^{2})\subset\mathbb{P}^{\frac{n(n+3)}{2}}$.

###### Proof.

Since $\Omega_{X}$ is globally generated and $h^{0}(X,\,\Omega_{X})=3$, we
have a short exact sequence

$0\longrightarrow\mathcal{O}_{X}(-K_{X})\longrightarrow
H^{0}(X,\,\Omega_{X})\otimes\mathcal{O}_{X}\longrightarrow\Omega_{X}\longrightarrow
0.$ (75)

From [Eis94, p. 577] it follows that (75) gives rise to a short exact sequence

$0\longrightarrow
S^{n-1}H^{0}(X,\,\Omega_{X})\otimes\mathcal{O}_{X}(-K_{X})\longrightarrow
S^{n}H^{0}(X,\,\Omega_{X})\otimes\mathcal{O}_{X}\longrightarrow
S^{n}\Omega_{X}\longrightarrow 0,$ (76)

where the exactness on the left follows by comparing ranks. The canonical
divisor $K_{X}$ is effective and ample, so
$h^{0}(X,\,-K_{X})=h^{1}(X,\,-K_{X})=0$; taking cohomology in (76), we obtain
(74). As a consequence, the map $\mathbb{P}(\sigma_{n})$ in diagram (23) is an
isomorphism for all $n$. Since the $1$-cotangent map
$\psi_{1}\colon\mathbb{P}(\Omega_{X})\longrightarrow\mathbb{P}(H^{0}(X,\,\Omega_{X}))\simeq\mathbb{P}^{2}$
is surjective, the image of $\psi_{n}$ must coincide, up to a projective
transformation, with the image of $\nu_{n}$. ∎

######  Remark 4.8.

Another way to state (74) is saying that the restriction map

$H^{0}(A,\,S^{n}\Omega_{A})\longrightarrow H^{0}(X,\,S^{n}\Omega_{X})$ (77)

is an isomorphism for all $n\geq 1$, cf. [Deb05, Proposition 13].

######  Remark 4.9.

When $M$ is a principal polarization, namely
$(d_{1},\,d_{2},\,d_{3})=(1,\,1,\,1)$, the surface $X$ is a smooth theta
divisor of $A$ and we get $p_{g}(X)=q(X)=3$, $K_{X}^{2}=6$. In this case, $X$
is isomorphic to the second symmetric product $\operatorname{Sym}^{2}C$, where
$C$ is a smooth, non-hyperelliptic curve of genus $3$, see [CaCiML98, p. 304].

The next result shows that, if one assumes that the Albanese image is smooth,
then the previous counterexamples are the only ones up to finite étale covers.
Recall that an abelian cover is a Galois cover with abelian Galois group.

######  Proposition 4.10.

If $Y$ is a smooth surface of general type with smooth Albanese image, then
the following are equivalent.

  * $\boldsymbol{(1)}$

$\Omega_{Y}$ is globally generated and the pluri-cotangent image $Y_{n}$ has
dimension $2$ for all $n\geq 1$.

  * $\boldsymbol{(2)}$

$\Omega_{Y}$ is globally generated and the $1$-cotangent image $Y_{1}$ has
dimension $2$.

  * $\boldsymbol{(3)}$

$Y$ is a finite, étale cover of a smooth, ample divisor in an abelian
threefold.

  * $\boldsymbol{(4)}$

$Y$ is a finite, étale abelian cover of a smooth, ample divisor in an abelian
threefold.

###### Proof.

$\boldsymbol{(1)}\Longrightarrow\boldsymbol{(2)}$ Obvious.

$\boldsymbol{(2)}\Longrightarrow\boldsymbol{(3)}$ If $\boldsymbol{(2)}$ holds,
then by [Rou09, Proposition 2.14] we have $q(Y)=3$, hence
$A:=\operatorname{Alb}(Y)$ is an abelian threefold and the Albanese map
$a_{Y}\colon Y\longrightarrow A$ is a local immersion onto its smooth image
$X\subset A$. This implies that $Y\longrightarrow X$ is a local bihomomorphism
between compact complex manifolds, hence an unramified analytic cover ([Lee11,
Problem 11-9 p. 303]), which is actually an algebraic cover by GAGA. By
adjunction, the surface $X$ satisfies $0<K_{X}^{2}=X^{3}$, so the divisor $X$
is ample in $A$ by [BL04, Proposition 4.5.2].

$\boldsymbol{(3)}\Longrightarrow\boldsymbol{(4)}$ Let $X\subset A$ be a
smooth, ample divisor in an abelian threefold and let $f\colon
Y\longrightarrow X$ be a finite, étale cover. By Lefschetz hyperplane theorem
[Laz04, Theorem 3.1.21] it follows $\pi_{1}(X)=\pi_{1}(A)=\mathbb{Z}^{6}$;
thus, since the fundamental group of $X$ is abelian, the cover $f\colon
Y\longrightarrow X$ is Galois, with abelian Galois group.

$\boldsymbol{(4)}\Longrightarrow\boldsymbol{(1)}$ Let $X\subset A$ be a smooth
ample divisor in an abelian threefold and let $f\colon Y\longrightarrow X$ be
a finite, étale abelian cover. By [Par91, p. 200], there exist non-trivial
torsion divisors $L_{1},\ldots,L_{s}\in\operatorname{Pic}^{0}(X)$ such that

$f_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}\oplus L_{1}\oplus\cdots\oplus L_{s}.$
(78)

Since the cover is étale, we have $S^{n}\Omega_{Y}=f^{*}(S^{n}\Omega_{X})$ for
all $n\geq 1$. Thus $S^{n}\Omega_{Y}$ is globally generated (because
$S^{n}\Omega_{X}$ is) and, by projection formula, we get

$f_{*}S^{n}\Omega_{Y}=S^{n}\Omega_{X}\oplus(S^{n}\Omega_{X}\otimes
L_{1})\oplus\cdots\oplus(S^{n}\Omega_{X}\otimes L_{s}).$ (79)

Tensoring (76) with $L_{j}$, and using the fact that $K_{X}+L_{j}$ is
effective and ample, we deduce $H^{0}(X,\,S^{n}\Omega_{X}\otimes L_{j})=0$ for
all $j$. Hence (79) yields
$H^{0}(Y,\,S^{n}\Omega_{Y})=f^{*}H^{0}(X,\,S^{n}\Omega_{X})$, which in turn
implies $Y_{n}=X_{n}$. By Proposition 4.7 it follows that $Y_{n}$ has
dimension $2$ for all $n\geq 1$. ∎

######  Remark 4.11.

The argument in the last part of the proof of Proposition 4.10 can be also
used in order to construct examples of surfaces of general type having Gauss
maps of arbitrarily large degree. For the sake of simplicity, let us just
consider finite, étale cyclic covers. Let $X\subset A$ be a smooth ample
divisor in an abelian threefold and let $L$ be a non-trivial a non-trivial
element of order $k$ in $\mathrm{Pic}^{0}(X)$. These data define an étale
$\mathbb{Z}_{k}$-cover $f\colon Y\longrightarrow X$, where

$f_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}\oplus L\oplus\cdots\oplus L^{k-1}.$ (80)

As before, we get
$H^{0}(Y,\,S^{n}\Omega_{Y})=f^{*}H^{0}(X,\,S^{n}\Omega_{X})$, which in turn
implies

$\mathbb{G}(n,\,\mathbb{P}(H^{0}(Y,\,S^{n}\Omega_{Y})))=\mathbb{G}(n,\,\mathbb{P}(H^{0}(X,\,S^{n}\Omega_{X}))).$

This shows that, for all $n\geq 1$, the $n$th Gauss map of $Y$ factors through
the degree $k$ cover $f\colon Y\longrightarrow X$, and so its degree is a
multiple of $k$. Since $k$ is arbitrary, we can construct in this way smooth,
minimal surfaces of general type $Y$, all of whose Gauss maps have arbitrarily
large degree.

##  5 Open problems

We end the paper with a couple of open problems.

######  Open Problem 1.

In Proposition $\operatorname{\ref{prop:image-n-cotangent}}$, does the case
$h^{0}(X,\,S^{n}\Omega_{X})<\frac{1}{2}(n+1)(n+2)$ actually occur?

This question what asked by the first Author in the MathOverflow thread
MO430570, without any answer so far. It is motivated by the fact that, in all
the examples that we are able to compute, if $S^{n}\Omega_{X}$ is globally
generated then $h^{0}(X,\,S^{n}\Omega_{X})\geq\frac{1}{2}(n+1)(n+2)$. The
equality is attained, for instance, by finite étale covers of ample divisors
in abelian threefolds, see Subsection 4.3.

######  Open Problem 2.

How should one modify Proposition $\operatorname{\ref{prop:all-pluricotangent-
dim-2}}$ if one removes the smoothness assumption for the Albanese image of
$Y$? Are there new examples with non-generically finite pluricotangent maps
that appear?

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Francesco Polizzi
Dipartimento di Matematica e Informatica, Università della Calabria
Cubo 30B, 87036 Arcavacata di Rende (Cosenza), Italy.
_E-mail address:_<EMAIL_ADDRESS>

Xavier Roulleau
Laboratoire angevin de recherche en mathématiques, LAREMA, UMR 6093 du CNRS,
UNIV. Angers, SFR MathStic, 2 Bd Lavoisier, 49045 Angers Cedex 01, France
_E-mail address:_<EMAIL_ADDRESS>