# Subspace configurations and low degree points on curves

Borys Kadets  Borys Kadets, Department of Mathematics
Boyd Graduate Studies Research Center
University of Georgia
Athens, GA 30602<EMAIL_ADDRESS>http://bkadets.github.io and  Isabel
Vogt  Isabel Vogt, Department of Mathematics
Brown University
Providence, RI 02906<EMAIL_ADDRESS>https://www.math.brown.edu/ivogt

###### Abstract.

This paper is devoted to understanding curves $X$ over a number field $k$ that
possess infinitely many solutions in extensions of $k$ of degree at most $d$;
such solutions are the titular low degree points. For $d=2,3$ it is known
([HS91], [AH91]) that such curves, after a base change to $\overline{k},$
admit a map of degree at most $d$ onto $\mathbb{P}^{1}$ or an elliptic curve.
For $d\geqslant 4$ the analogous statement was shown to be false [DF93]. We
prove that once the genus of $X$ is high enough, the low degree points still
have geometric origin: they can be obtained as pullbacks of low degree points
from a lower genus curve. We introduce a discrete-geometric invariant attached
to such curves: a family of subspace configurations, with many interesting
properties. This structure gives a natural alternative construction of curves
from [DF93]. As an application of our methods, we obtain a classification of
such curves over $k$ for $d=2,3$, and a classification over ${\bar{k}}$ for
$d=4,5$.

##  1\. Introduction

Suppose $k$ is a number field and $X/k$ is a nice curve (smooth, projective,
and geometrically integral). In this paper, we study the infinite collections
of points on $X$ defined over degree $d$ extensions of $k$. A natural first
question is: what is the smallest degree $d$ for which the set of points of
degree $d$ is infinite? This is formalized by the following definition,
introduced in [SV20].

######  Definition 1.1.

The arithmetic degree of irrationality $\operatorname{a.\\!irr}_{k}X$ of a
nice curve $X$ over a field $k$ is the smallest integer $d$ such that $X$ has
infinitely many closed points of degree $d$.

There is also a geometric version $\operatorname{a.\\!irr}_{\bar{k}}X$ of the
arithmetic degree of irrationality of $X$. For $X$ defined over a number field
$k$, it is equal to the minimum of $\operatorname{a.\\!irr}_{L}X_{L}$ as $L$
varies over all finite extensions of $k$. Our motivating problem is:

######  Main Problem 1.2.

Classify the curves $X$ with $\operatorname{a.\\!irr}_{k}X=d$, and the ones
with $\operatorname{a.\\!irr}_{{\bar{k}}}X=d$.

Faltings’ Theorem classifies curves $X$ with
$\operatorname{a.\\!irr}_{{\bar{k}}}X=1$. Main problem 1.2 can therefore be
viewed as a generalization of this fundamental problem.

There are two natural geometric sources of such collections: if $X$ is a
degree $d$ cover of $\mathbb{P}^{1}$ or an elliptic curve $E$ of positive
rank, then pulling back rational points on $\mathbb{P}^{1}$ or $E$ gives an
infinite family of degree $d$ (or less) points on $X$. Previous work on Main
Problem 1.2 has focused on the geometric invariant. Harris–Silverman (for
$d=2$) and Abramovich–Harris (for $d=3$) showed that the above two natural
geometric sources of low degree points characterize when
$\operatorname{a.\\!irr}_{{\bar{k}}}$ takes the value $2$ or $3$. More
precisely, $\operatorname{a.\\!irr}_{\bar{k}}X=2$ or $3$ if and only if
$X_{\bar{k}}$ is a degree $2$ or $3$ cover of $\mathbb{P}^{1}$ or an elliptic
curve. Based on this evidence, Abramovich–Harris conjectured that the same
should hold for all values of $d$. However, Debarre and Fahlaoui [DF93] showed
that more obscure constructions of infinite families of degree $d\geqslant 4$
points exist by cleverly constructing certain curves on the symmetric square
of an elliptic curve. A full classification for any larger values of
$d\geqslant 4$ has remained stubbornly out of reach, and there have been
essentially no classification results for the arithmetic invariant
$\operatorname{a.\\!irr}_{k}$.

In the present paper we refocus on Main Problem 1.2, including the thornier
arithmetic classification. As a result, we obtain the following new
classification.

######  Theorem 1.3.

Suppose $X/k$ is a nice curve. Then the following statements hold:

  1. (1)

If $\operatorname{a.\\!irr}_{k}X=2$, then $X$ is a double cover of
$\mathbb{P}^{1}$ or an elliptic curve of positive rank;

  2. (2)

If $\operatorname{a.\\!irr}_{k}X=3$, then one of the following three cases
holds:

    1. (a)

$X$ is a triple cover of $\mathbb{P}^{1}$ or an elliptic curve of positive
rank;

    2. (b)

$X$ is a smooth plane quartic with no rational points, positive rank Jacobian,
and at least one cubic point;

    3. (c)

$X$ is a genus $4$ Debarre-Fahlaoui curve (see Section 5 for a precise
definition);

  3. (3)

If $\operatorname{a.\\!irr}_{\bar{k}}X=d\leqslant 3$, then $X_{\bar{k}}$ is a
degree $d$ cover of $\mathbb{P}^{1}$ or an elliptic curve;

  4. (4)

If $\operatorname{a.\\!irr}_{\bar{k}}X=d=4,5$, then either $X_{\bar{k}}$ is a
Debarre-Fahlaoui curve, or $X_{{\bar{k}}}$ is a degree $d$ cover of
$\mathbb{P}^{1}$ or an elliptic curve.

Suprisingly, the seemingly clever construction by Debarre–Fahlaoui of
counterexamples to the conjecture of Abramovich–Harris arise perfectly
naturally from our perspective. The next open case is to classify curves of
genus $11$ with $\operatorname{a.\\!irr}_{\bar{k}}X=6$, see Section 7.1. As
can be seen from Case 2b of Theorem 1.3, there are certain arithmetic
subtleties involved in the classification; some open questions concerning
these subtleties are described in Section 7.2.

The classification in Theorem 1.3 is obtained from a systematic study of the
possible infinite collections of degree $d$ points. Our guiding philosophy is
that when $d$ is small compared to the genus of $X$, such infinite collections
_still_ occur for good geometric reasons. The first step in our analysis is
thus to make this precise with the following genus bound, which reduces Main
Problem 1.2 to finitely many genera for each value of $d$.

######  Theorem 1.4.

Suppose $X/k$ is a curve of genus $g$ and $\operatorname{a.\\!irr}_{k}X=d$.
Let $m\colonequals\lceil d/2\rceil-1$ and let $\varepsilon\colonequals
3d-1-6m<6$. Then one of the following holds:

  1. (1)

There exists a nonconstant morphism of curves $\phi\colon X\to Y$ of degree at
least $2$ such that $d=\operatorname{a.\\!irr}_{k}Y\cdot\deg\phi;$

  2. (2)

The genus of $X$ is bounded

$g\leqslant\max\left(\frac{d(d-1)}{2}+1,\ 3m(m-1)+m\varepsilon\right).$

In case (1) there is a clear source of low degree points on $X$: they can be
obtained as pullbacks of low degree point on $Y$ under $\phi$. There is a long
history of genus bounds in problems related to Main Problem 1.2; see [Voj91],
[Voj92], [AH91], [ST01]. Theorem 1.4 is indirectly claimed in [AH91] by
combining [AH91, Lemma 3] with [AH91, Theorem 2]; however, the statement of
[AH91, Lemma 3] has an error, and the proof of [AH91, Theorem 2] contains a
gap. See the discussion in Section 1.1.

Following the ideas introduced by Abramovich and Harris, we study the geometry
of curves with $\operatorname{a.\\!irr}_{k}X=d$ by studying the geometry of
linear systems of the form $|nD|$ for degree $d$ points $D$. The Mordell-Lang
conjecture ensures that these linear systems have positive dimension. An
important step in proving Theorem 1.4 is Theorem 3.5, which states that unless
case (1) holds, the linear systems $|nD|$ are birational for most $D$ and
$n\geqslant 2$.

With birationality proved, we can investigate finer questions concerning the
geometry of the linear systems $|nD|$. We do so by equipping these linear
systems with a discrete-geometric structure: there is an infinite family of
multisecant planes within each of the projective spaces $|nD|$, which form an
combinatorially interesting configuration. The presence of this additional
structure allows us to prove the following finer classification of curves $X$
with $\operatorname{a.\\!irr}_{k}X=d$. To formally state this classification
we require a notion of a sufficiently general degree $d$ point $D$; this is
rigorously defined in Section 2.

######  Theorem 1.5.

Suppose $X$ is a curve with $\operatorname{a.\\!irr}_{k}X=d$. Let
$m\colonequals\lceil d/2\rceil-1$ and let $\varepsilon\colonequals 3d-1-6m<6$.
Then for a sufficiently general degree $d$ point $D$ one of the following
holds:

  1. (1)

$\dim|2D|=1$, and $X$ is a degree $d$ cover of an elliptic curve of positive
rank;

  2. (2)

$\dim|2D|\geqslant 2$, the associated map $X\to\mathbb{P}^{|2D|}$ is not
birational onto its image, and there exists a covering of curves $\phi\colon
X\to Y$ of degree at least $2$ such that
$d=\operatorname{a.\\!irr}_{k}Y\cdot\deg\phi;$

  3. (3)

$\dim|2D|=2$, the associated map $X\to\mathbb{P}^{2}$ is birational onto its
image, and $X$ is one of the Debarre-Fahlaoui curves (see Section 5 for the
precise definition);

  4. (4)

$\dim|2D|>2$, the associated map $X\to\mathbb{P}^{|2D|}$ is birational onto
its image, and the genus $g$ of $X$ satisfies

$g\leqslant\max\left(\frac{(d-1)(d-2)}{2}+2,\ 3m(m-1)+m\varepsilon\right)$

The proof of Theorem 1.5 involves a detailed analysis of the configuration
geometry in $|3D|$; it shows how the geometry of the linear systems $|nD|$
naturally gives rise to the Debarre-Fahlaoui examples.

######  Remark 1.6.

The results of [AH91] imply that the gonality of a curve with
$\operatorname{a.\\!irr}_{k}X=d$ is at most $2d$; this fact was also
independently observed by Frey [Fre94]. One corollary of Theorem 1.5 is that
the geometric gonality can equal $2d$ only for (geometric) degree $d$ covers
of elliptic curves; and can equal $2d-1$ only for (geometrically) Debarre-
Fahlaoui curves. See Remarks 3.4 and 5.9.

The method of examining multisecant configurations can be applied to study the
low degree points on special families of curves. As a demonstration, in
Section 6.1 we prove that projective curves of large genus have finitely many
sufficiently low degree points. The statement of this estimate uses the
Castelnuovo function $\pi(d,r)$; we recall its definition in Section 6.1. The
number $\pi(d,r)$ is an upper bound for the genus of a nondegenrate degree $d$
curve in $\mathbb{P}^{r}$. When $r$ is fixed and $d$ is growing, $\pi(d,r)\sim
d^{2}/(2r-2)$.

######  Theorem 1.7.

Suppose $X\subset\mathbb{P}^{r}$ is an irreducible (possibly singular) curve
of degree $e$ and genus $g$. Suppose $X$ has infinitely many points of degree
$d$ not contained in hyperplanes of $\mathbb{P}^{r}$. Then

$g\leqslant\pi(e+2d,2r+1).$

There are many open questions concerning the geometry of curves with abundant
low degree points, both of arithmetic and purely geometric nature. We survey
these questions in Section 7.

###  1.1. Relation to previous work

The first results on low degree points were obtained by Hindry [Hin87], who
studied quadratic points on modular curves $X_{0}(p)$ and asked if in general
a curve with infinitely many quadratic points is either hyperelliptic or
bielliptic. Later, Faltings [Fal91], and Vojta [Voj91] used diophantine
approximation techniques to describe low degree points on curves of small
gonality. The strongest of these results was obtained by Vojta, who showed
that a degree $s$ cover of $\mathbb{P}^{1}$ with infinitely many degree $d$
points not contracted by the map to $\mathbb{P}^{1}$ has genus at most
$s(d-1)$ [Voj92].

The resulting genus bound is sharp: if $E$ is an elliptic curve of positive
rank, then a $(d,s)$-curve on $E\times\mathbb{P}^{1}$ satisfies the conditions
of the theorem and has genus $g=s(d-1).$ The general question of describing
curves of arithmetic irrationality $d$ was first addressed in [HS91] in the
case $d=2$ and in [AH91] for $d=3$. Based on these results, Abramovich and
Harris proposed the following conjecture, which was soon disproved by Debarre
and Fahlaoui.

######  Conjecture 1.8 ([AH91]; proved for $d=2$ [HS91]; proved for $d=3$
[AH91]; disproved for all $d\geqslant 4$ [DF93]).

Suppose $\operatorname{a.\\!irr}_{k}X=d$. Then $X_{\bar{k}}$ has a degree $d$
map to $\mathbb{P}^{1}$ or an elliptic curve.

The presence of counterexamples makes it hard to analyze arithmetic
irrationality for arbitrary curves; however, the methods used in [AH91] can
still be applied to certain classes of special curves. For example, Debarre
and Klassen [DK94] showed that a smooth plane curve $X$ of degree $d\geqslant
8$ has arithmetic irrationality $d$ or $d-1$ corresponding to the cases
$X(k)=\emptyset$ and $X(k)\neq\emptyset$ respectively. For a generalization to
curves on other surfaces see [SV20].

A word of warning is warranted concerning the work of Abramovich and Harris
[AH91]: as detailed in [DF93], the paper contains several errors (including in
Lemma 3, Lemma 6, Lemma 8, and Corollary 1), which, while not severe enough to
make the main results false, can be misleading. Some corrections are described
in [DF93], however one of the main results – [AH91, Theorem 2] – should be
considered unproved. For this reason, we give full proofs of several simple
lemmas appearing in [AH91] when we need them.

Other related work includes the study of integral points of low degree
[Lev16], generalization of the work of Vojta to covers of curves [ST01],
results on low degree points for curves on product surfaces [Lev07].

###  1.2. Structure of the paper

In Section 2 we describe how the Mordell-Lang conjecture gives geometric
restrictions on the curves with $\operatorname{a.\\!irr}_{k}X=d$; this
observation was also used in [HS91], [AH91].

In Section 3 we prove a key technical result: the birationality of the linear
systems $|nD|$ for $n\geqslant 2$ and for sufficiently general degree $d$
points $D$ on $X$. More precisely, we show that these linear systems are
birational if there does not exist a cover $\phi\colon X\to Y$ of degree at
least $2$ with $d=\operatorname{a.\\!irr}_{k}Y\cdot\deg\phi.$

In Section 4 we enrich the linear systems $|nD|$ with additional discrete-
geometric structure of a configuration of multisecant subspaces to $X$. The
main properties of this structure are summarized in Section 4.2. As an
application, we prove Theorem 1.4.

In Section 5 we use subspace configruations to relate geometry of special
linear systems to the construction of Debarre-Fahlaoui; in particular we prove
most of Theorem 1.5.

In Section 6 we collect geometric corollaries of the results obtained so far.
We prove Theorem 1.7, finish the proof of Theorem 1.5, and summarize what we
know about curves with arithmetic irrationality at most $5$ to prove Theorem
1.3.

In Section 7 we collect some open questions on the geometry of curves with
$\operatorname{a.\\!irr}_{k}X=d$.

###  1.3. Notation

Throughout the paper $k$ will denote a fixed number field and $X/k$ will
denote a nice curve. Let $\bar{k}$ be an algebraic closure of $k$. We write
$\bar{X}=X_{\bar{k}}$ for the base-change of $X$ to $\bar{k}$. By a degree $d$
point on $X$ we mean an algebraic point with residue field a degree $d$
extension of $k$. Write $\operatorname{Sym}^{d}X=X^{d}/\\!\\!/S_{d}$ for the
$d$th symmetric power of the curve $X$, and $\operatorname{Pic}^{d}X$ for the
degree $d$ component of the Picard scheme of $X/k$.

We write $W_{d}X=W^{0}_{d}X$ for the image of the Abel–Jacobi map
$\operatorname{Sym}^{d}X\to\operatorname{Pic}^{d}X$ sending an effective
divisor of degree $d$ to its linear equivalence class. This is a Brill–Noether
locus of $\operatorname{Pic}^{d}X$ (see [ACGH85, Chapter IV $\S$3] for more
details on Brill–Noether loci). The fiber of the Abel–Jacobi map over a line
bundle $L\in\operatorname{Pic}^{d}X$ is isomorphic to the complete linear
system $|L|\simeq\mathbb{P}H^{0}(X,L)$ of $L$. When $\dim|L|=r$, such a linear
system is called a $g^{r}_{d}$. The minimal value of $d$ for which $X$ has a
($k$-rational) $g^{1}_{d}$ is called the gonality of $X$, denoted
$\operatorname{gon}(X)$.

### Acknowledgements

We would like to thank Dan Abramovich, Juliette Bruce, Brendan Hassett, Eric
Larson, Daniel Litt, and Alexander Petrov for helpful conversations and
communications. We would also like to express our admiration for the
foundational papers of Harris–Silverman [HS91], Abramovich–Harris [AH91],
Debarre–Fahlaoui [DF93], and Frey [Fre94] that inspired our work on this
subject. This material is partly based upon work supported by the National
Science Foundation under Grant No. DMS-1928930 while B.K. participated in a
program hosted by the Mathematical Sciences Research Institute in Berkeley,
California, during the Fall 2020 semester. I.V. was partially supported by a
NSF MSPRF under DMS-1902743 and by DMS-2200655.

##  2\. Corollaries of Mordell–Lang

In this section we collect some corollaries of the Mordell-Lang conjecture,
now a theorem of Faltings [Fal91], for the structure of rational points on
subvarieties of the Picard scheme of a curve over $k$.

Suppose that $\operatorname{a.\\!irr}_{k}X=d$ and that the gonality
$\operatorname{gon}_{k}X$ is strictly greater than $d$. Since
$\operatorname{a.\\!irr}_{k}X=d$, the set $\operatorname{Sym}^{d}X(k)$ is
infinite. Since $\operatorname{gon}_{k}X>d$, the Abel-Jacobi map
$\operatorname{Sym}^{d}X(k)\to\operatorname{Pic}^{d}X(k)$ is injective.
Consequently, the rational points of the image $W_{d}X(k)$ are an infinite
set. Since $W_{d}X$ is a subvariety of a torsor under an abelian variety, the
Mordell Lang conjecture implies that there exists a translate $A\subset
W_{d}X$ of a positive-dimensional abelian subvariety
$A^{0}\subset\operatorname{Pic}^{0}X$, such that $A(k)$ is Zariski dense in
$A$.

By the semicontinuity theorem, there is an open dense locus in $A$ consisting
of points for which the corresponding (isomorophism class of) line bundle
$[L]$ has the minimal achieved value of $h^{0}(X,L)$. (By virtue of the fact
that $A\subset W^{0}_{d}X$, this minimal value is positive.) The fibers of
$\operatorname{Sym}^{d}X\to\operatorname{Pic}^{d}X$ are Severi–Brauer
varieties of dimension $h^{0}(X,L)-1$. Since $\operatorname{gon}_{k}X>d$, any
fiber of $\operatorname{Sym}^{d}X\to\operatorname{Pic}^{d}X$ having a rational
point is necessarily a $\mathbb{P}^{0}$. Since
$\operatorname{Sym}^{d}X(k)\neq\emptyset$, the minimal achieved value of
$h^{0}(X,L)$ must be $1$. Hence there is an open dense locus $U\subset A$ over
which $\operatorname{Sym}^{d}X\to\operatorname{Pic}^{d}X$ is an isomorphism.
In the case $\dim A=1$, applying the curve-to-projective extension theorem to
the map $U\to\operatorname{Sym}^{d}X$, we see that there is a canonical
effective divisor associated to every point of $A$ (and in particular
$U(k)=A(k)$). Moreover, even when $\dim A>1$, resolving the indeterminacy of
the inverse $U\to\operatorname{Sym}^{d}X$ (in a similar manner to the proof
the Lang–Nishimura lemma as in [RY00, Proposition A.6]) proves that
$U(k)=A(k)$.

Since we will focus on this setup for the majority of the paper, we codify it
in the following:

######  Setup 2.1.

Let $X$ be a nice curve with $\operatorname{a.\\!irr}_{k}X=d$ and
$\operatorname{gon}_{k}X>d$. Write $A\subset W_{d}X$ for an associated
positive-dimensional abelian translate and $U\subset A$ for the open dense
locus over which $\operatorname{Sym}^{d}X\to\operatorname{Pic}^{d}_{X}$ is an
isomorphism.

Since the rational points of $A$ are Zariski-dense, any nonempty open in $A$
also has Zariski dense rational points. By a general rational point of $A$, we
mean a rational point in an open dense subvariety of $A$. By a general
effective divisor $D$ in $A(k)=U(k)$, we mean a rational point in an open
dense subvariety of $U\subset A$ where there exists a unique effective divisor
representing each isomorphism class of line bundle. A general effective
divisor $D$ in $A(k)$ is a degree $d$ point on $X$.

Consider the incidence correspondence

$I\colonequals\\{(p,[D])\in X\times U:p\in D\\}.$

Then $I$ necessarily dominates $X$ via the first projection, and is a degree
$d$ cover of $U$ via the second projection. The dimension of $I$ is therefore
equal to the dimension of $A$. The (arithmetic) monodromy of this degree $d$
cover is transitive, since a general effective divisor in $U$ corresponds to a
degree $d$ point on $X$. In particular, we have that

(1) a general pair of effective divisors in $U$ have disjoint support.

The following lemma shows that at the same time, there are many pairs of
divisors in $U$ that share points on $X$.

######  Lemma 2.2.

Suppose we are in Setup 2.1 and that if $\dim A=1$, then $X$ is not a degree
$d$ cover of $A$. Then for any open subset $V\subset U$ and a general point
$P\in X({\bar{k}})$ there exists a pair of distinct divisors $D_{1},D_{2}\in
V$ containing $P$.

###### Proof.

Suppose to the contrary, that for some open $V\subset U$ a general point $P\in
X$ is contained in a unique divisor from $V$. The rational map
$\phi:X\dashrightarrow V$ that sends a point $P\in X$ to the unique divisor in
$V$ that contains $P$ is dominant, and so $\dim A=1$. In this case, the map
$\phi$ extends to a degree $d$ covering $X\to A$. ∎

###  2.1. The abelian translate property

The abelian translate $A$ is a torsor under an abelian variety
$A^{0}\subset\operatorname{Pic}^{0}X$. The group law on $A^{0}$ has the
following consequence, which we term the abelian translate property: for any
three points $L_{1}$, $L_{2}$ and $L_{3}$ of $A$, the line bundle
$L_{1}\otimes L_{2}\otimes L_{3}^{-1}$ is again a point of $A$. Moreover, on
rational points, for effective divisors $D_{1}$, $D_{2}$ and $D_{3}$ in
$U(k)$, the divisor $D_{1}+D_{2}-D_{3}$ is again in $U(k)$.

Fix an effective divisor $D$ in $U(k)$. The abelian translate property implies
that for any $n-1$ divisors $D_{1},\dots,D_{n-1}$ in $U(k)$, there exists a
$n$th divisor $D_{n}$ in $U(k)$ that is linearly equivalent to

$nD-D_{1}-\cdots-D_{n-1}.$

In the next section, we will prove that Setup 2.1 implies that the linear
systems $|\mathcal{O}(nD)|$ for $D$ a general effective divisor in $A(k)$ are
birational unless there is a natural geometric source of degree $d$ points on
$X$. We will then interpret the abelian translate property geometrically in
terms of spans of divisors in $|\mathcal{O}(nD)|$.

##  3\. Birationality

The main result of this section is Theorem 3.5 which shows that in Setup 2.1,
unless an infinite collection of degree $d$ points on $X$ is obtained by
pullback from a lower genus curve, the linear system $|2D|$ is birational for
a general $D\in A(k)$. In particular, this immediately implies that the genera
of such curves are bounded by $(d-1)(2d-1)$. These results are closely related
to [AH91]. Our main Theorem 3.5 is similar to [AH91, Lemma 3]; note, however,
that the statement of [AH91, Lemma 3] has an error (the last formula of the
statement is false), and more importantly the proof does not go through for
curves which are degree $d$ covers of pointless conics – the case that
requires most work in our Theorem 3.5.

We will use the following version of de Franchis theorem due to Kani.

######  Theorem 3.1.

Suppose $k$ is a field of characteristic zero and $X/k$ is a nice curve. Then

  1. (1)

There exist at most finitely many surjective morphisms $X\to Y$ to curves
$Y/k$ of genus at least $2$;

  2. (2)

For any integer $d$ there exists at most finitely many surjective morphisms
$X\to C$ of degree less than $d$ to curves of genus $1$, up to translations on
the target.

###### Proof.

See [Kan86] Theorem 3 and [Kan86] Corollary after Theorem 4. ∎

One way of obtaining infinitely many degree $d$ points on $X$ is via pullback
from an elliptic curve of positive rank. The following lemma describes a
situation in which this is the case.

######  Lemma 3.2.

Assume that we are in Setup 2.1 and $\dim A=1$. Let $D\in A$ be a general
divisor. For every effective divisor $E\in A$, the divisor
$E^{\prime}\colonequals 2D-E$ belongs to $A$ by the abelian translate
property. Suppose that there exists a map

$\pi\colon X\to\mathbb{P}^{1}$

of degree $2d$ such that all $E\in A$ are contracted to a point $\psi(E)$ by
$\pi$ and further $\psi(E)=\psi(E^{\prime})$. Then there exists a degree $d$
map $\pi^{\prime}\colon X\to A$ factoring the map $\pi=\psi\circ\pi^{\prime}$.

###### Proof.

The morphism $\psi\colon A\to\mathbb{P}^{1}$ that sends a divisor $E\in A$ to
the point $\pi(E)$ evidently factors through the quotient by the involution
sending $E$ to $2D-E$. By computing the ramification of $\psi$, we will show
that $\psi$ has degree $2$ and is hence equal to this quotient map. As a
result, the original map $\pi$ factors
$X\xrightarrow{d:1}A\xrightarrow{\psi}\mathbb{P}^{1}$, and $X$ is a degree $d$
cover of the elliptic curve $A$.

Since $\dim A=1$, we can extend an inverse of the Abel-Jacobi map to a regular
map $A\to\operatorname{Sym}^{d}X$. Since the effective divisor corresponding
to a general point of $A$ is reduced, the union of the supports of all
nonreduced divisors from $A$ is finite. In particular, we may assume that $D$,
and all of its translates $D^{\prime}$ by the finitely many $2$-torsion points
of $A^{0}$, are disjoint from this finite set. The points of $D^{\prime}$ are
ramification points of $\pi$, and in particular we have the equality of sets
$\pi^{-1}(\psi(D^{\prime}))=\operatorname{supp}D^{\prime}$. Since no
nonreduced divisor intersects
$\operatorname{supp}D^{\prime}=\pi^{-1}(\psi(D^{\prime}))$, any divisor from
$A$ supported on the fiber $\pi^{-1}(\psi(D^{\prime}))$ is equal to
$D^{\prime}$. Therefore the map $\psi\colon A\to\mathbb{P}^{1}$ is totally
ramified over the $4$ points of the form $\psi(D^{\prime})$ satisfying
$2D^{\prime}=2D$. Since $A$ has genus $1$ and $\psi$ is totally ramified over
at least $4$ points, the Riemann-Hurwitz formula gives

$0=-2\deg\psi+\sum_{P}(e_{P}-1)\geqslant-2\deg\psi+4(\deg\psi-1)=2\deg\psi-4.$

Therefore $\deg\psi=2$. This means that there are exactly two divisors from
$A$ supported on a general fiber of $\pi$. Therefore a fiber of $\pi$ above a
general point of $\mathbb{P}^{1}$ is a disjoint union of two divisors from
$A$. The association $P\mapsto(\text{unique }D\in A\text{ with }P\in D)$
defined on an open dense subset of $X$ is the desired morphism
$\pi^{\prime}:X\to A$. ∎

######  Proposition 3.3.

Suppose we are in Setup 2.1 and additionally that $X$ is not a degree $d$
cover of an elliptic curve. Then for any divisor class $D\in A(k)$ the linear
system $|2D|$ is basepoint free and $\dim|2D|\geqslant\max(2,\dim A)$.

###### Proof.

By [AH91, Lemma 1], $|2D|$ is basepoint free and $\dim|2D|\geqslant\dim A$.
(Indeed, by the abelian translate property, for all $E\in A$, the class $2D-E$
is effective. The association $E\mapsto E+|2D-E|$ defines a $(\geqslant\dim
A)$-dimensional family of effective divisors in class $2D$. Since the divisors
$E\in U$ don’t have a shared point, the family of divisors of the form
$E\cup[2D-E]$ do not have any common points either, and so $|2D|$is base point
free.) This completes the proof when $\dim A>1$, so we assume for the
remainder of the proof that $\dim A=1$.

Since $\dim|2D|$ is upper-semicontinuous, it suffices to prove the result for
a general $D\in A$. Suppose that $\dim|2D|=1$, and let
$\phi:X\to\mathbb{P}^{1}$ be the associated map. For every effective divisor
$E\in A$, the divisor $2D-E$ belongs to $A$ by the abelian translate property,
and therefore is effective. Hence every divisor $E\in A$ is supported on a
fiber of $\phi$ and both $E$ and $2D-E$ are supported on the same fiber.
Therefore the assumptions of Lemma 3.2 are satisfied, which is a contradiction
since we assumed that $X$ is not a degree $d$ cover of an elliptic curve. ∎

######  Remark 3.4.

Abramovich and Harris, and, independently, Frey [Fre94], observed that the
gonality of a curve with $\operatorname{a.\\!irr}_{k}X=d$ is at most $2d$. An
immediate corollary of Proposition 3.3 is that the geometric gonality of a
curve with $\operatorname{a.\\!irr}_{k}X=d$ that is not a degree $d$ cover of
an elliptic curve is at most $2d-1$.

We now prove the main result of this section.

######  Theorem 3.5.

Suppose that we are in Setup 2.1 and $D\in U(k)$ is a general divisor. Then
one of the following holds:

  1. (1)

there exists a covering of curves $\phi\colon X\to Y$ of degree at least $2$
with $\operatorname{a.\\!irr}_{k}Y=d/\deg\phi$;

  2. (2)

the associated map $X\to\mathbb{P}^{\dim|2D|}$ is birational. (The basepoint
free line bundle $|2D|$ is birationally very ample).

###### Proof.

By Theorem 3.1, $X$ has only finitely many nonconstant maps $f_{1},...,f_{N}$
of degree at most $d$ to curves of genus $\geqslant 1$ up to automorphisms of
the base. Since $D$ is general, we can assume that $D$ does not intersect the
preimage of the branch locus of any of the $f_{i}$.

Suppose that case (2) does not hold, i.e., that the morphism
$X\to\mathbb{P}^{|2D|}$ factors as
$X\xrightarrow{f}Y\hookrightarrow\mathbb{P}^{|2D|}$ with $m\colonequals\deg
f\geqslant 2$. Then we will show that case (1) holds. Note that it suffices to
show that $\operatorname{a.\\!irr}_{k}Y\leqslant d/\deg\phi$, since $X$ has
finitely many points of degree less than $d$. Write
$\widetilde{f}:X\to\widetilde{Y}$ for the map to the normalization of $Y$.
Since the nondegenerate curve $Y\subset\mathbb{P}^{|2D|}$ has degree at least
$\dim|2D|\geqslant 2$, the degree of $\widetilde{f}$ is at most $d$.

First suppose that the geometric genus of $Y$ is at least $1$. By assumption,
the divisor $D$ has trivial intersection with the preimage of the branch locus
of the map $\widetilde{f}$. Observe that for any curve $C$ and any effective
divisor $\Delta$ the following property holds: if for some positive $k$ the
linear system $k\Delta$ is base-point free and $\Phi:C\to\mathbb{P}^{N}$ is
the associated morphism, then we have the equality of sets
$\Phi^{-1}(\Phi(\Delta))=\Delta$. Applying this to $\Delta=D$ and $k=2$ gives
$\widetilde{f}^{-1}\left(\widetilde{f}(D)\right)=D$. Since $D$ does not
intersect the preimage of the branch locus, we have

$d=\\#D=\\#\widetilde{f}^{-1}\left(\widetilde{f}(D)\right)=m\\#\widetilde{f}(D).$

Hence the image of $D$ in $\widetilde{Y}$ is a point of degree equal to $d/m$.
Since there are infinitely many choices of $D$ and only finitely many choices
for the morphism $\widetilde{f}$, by Theorem 3.1, there exists a map
$\widetilde{f}:X\to Y$ of degree $m$ such that for infinitely many
$D\in\operatorname{Sym}^{d}(k)$ the image $\widetilde{f}(D)$ has degree $d/m$,
in which case (1) holds.

Now consider the case that $\widetilde{Y}$ has genus $0$. Since a genus $0$
curve has infinitely many quadratic points either (1) holds, or
$m=\deg\widetilde{f}>d/2$, and so $\deg Y=2d/m<4$. If $\deg Y=3$ and the genus
of $\widetilde{Y}$ is zero, then $Y$ has odd degree points, and thus
$Y=\mathbb{P}^{1}$; this implies that case (1) holds.

It therefore remains to consider the case $\deg Y=2$, $\dim|2D|=2$, $Y$ is a
smooth pointless conic and $f$ is a covering of degree $d$. In this way, a
general divisor $D\in U$ defines a map to a rational curve $f_{D}:X\to Y_{D}$;
let $U^{\prime}\subset U$ be the open subset of divisors that define such
maps. For every pair $(D_{1},D_{2})\in U^{\prime}\times U^{\prime}$ we get a
map $\phi_{D_{1},D_{2}}=(f_{D_{1}},f_{D_{2}}):X\to Y_{D_{1}}\times Y_{D_{2}}$.
Let $Z_{D_{1},D_{2}}$ denote its image. By semicontinutity, $Z_{D_{1},D_{2}}$
will have a constant bidegree $(e_{1},e_{2})$ on an open dense subset of
$U^{\prime}\times U^{\prime}$, and comparing the degrees of $Z_{D_{1},D_{2}}$
and $Z_{D_{2},D_{1}}$ we see that the bidegree is necessarily symmetric:
$e_{1}=e_{2}=e$. To simplify notation, fix a general pair $(D_{1},D_{2})\in
U^{\prime}\times U^{\prime}$ and write $Y_{1}=Y_{D_{1}}$, $Y_{2}=Y_{D_{2}}$,
$f_{1}=f_{D_{1}}$, $f_{2}=f_{D_{2}}$, $\phi=\phi_{D_{1},D_{2}}$, and
$Z=Z_{D_{1},D_{2}}$. In what follows, we will show that $e>1$, so that the map
$\phi\colon X\to Z$ has degree $d/e\leqslant d/2$, and that the images
$\phi(E)$ of divisors $E\in U(k)$ have low enough degree to force us to be in
case (1).

Since the line bundles
$\mathcal{O}_{X}(D_{1})=f_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)$ and
$\mathcal{O}_{X}(D_{2})=f_{2}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)$ are
distinct, there does not exist an automorphism of $\mathbb{P}^{1}$ bringing
one to the other and thus $Z$ cannot be a $(1,1)$-divisor. Hence, the degree
$e$ of the projection from $Z$ to $Y_{1}$ and $Y_{2}$ is at least $2$. For a
general divisor $E\in U(k)$, the divisors $2D_{i}-E$ are effective by the
abelian translate property. Therefore $f_{i}(E)$ is contained in the
hyperplane section of a (pointless) conic and hence $\deg f_{1}(E)=\deg
f_{2}(E)=2$, and so $\deg\phi(E)\leqslant 4$. Since the map $f_{1}$ factors
through $\phi$ and $\deg f_{1}(E)=2$, the degree of $\phi(E)$ is either $2$ or
$4$.

Case 1: $\deg\phi(E)=2$ for infinitely many $E\in U(k)$. Since
$\deg(\phi)\leqslant d/2$, we are in evidently in case (1), unless $d=2$ and
the map $\phi$ is birational onto its image. However an integral $(2,2)$-curve
(which necessarily has geometric genus $0$ or $1$) with infinitely many degree
$2$ points is always a degree $2$ cover of $\mathbb{P}^{1}$, so we are again
in case (1).

Case 2: $\deg\phi(E)=4$ for general $E\in U$. In this case, we will show that
$e\geqslant 4$, and hence we are in case (1) unless $d=4$ and the map $\phi$
is birational onto its image. Consider a general $E\in A$ and the divisors
$f_{2}(E)$ and $f_{2}(2D_{1}-E)$ on $Y_{2}$. Each one is a degree $2$ point on
$Y_{2}$, and as $D_{1}$ varies the point $f_{2}(2D_{1}-E)$ will vary as well.
Since $D_{1},D_{2}$ are a general pair, we can assume that the divisors
$f_{2}(E)$ and $f_{2}(2D_{1}-E)$ are disjoint for general $E\in A$, and hence
the divisors $\phi(E),\phi(2D_{1}-E)\in Z$ are necessarily disjoint degree $4$
divisors. Since $f_{1}(E)=f_{1}(2D_{1}-E)$, the image
$f_{1}(\phi(E)+\phi(2D_{1}-E))$ of the degree $8$ divisor
$\phi(E)+\phi(2D_{1}-E)$ is supported on a degree $2$ point $f_{1}(E)$.
Therefore the degree $e$ of the projection $Z\to Y_{1}$ is at least $4$, and
hence the degree of $\phi:X\to Z$ is at most $\deg f_{1}/4=d/4$. If $\phi$ is
not birational onto its image, then since $\deg\phi\leqslant d/4$ and $Z$ has
infinitely many degree $4$ points, case (1) holds.

It remains to consider the case when $d=4$ and $X$ is birational to a $(4,4)$
curve $Z$ on $Y_{1}\times Y_{2}$. If $Z$ is smooth, then $X=Z$ and the
projections onto $Y_{1}$ and $Y_{2}$ give the only two degree $4$ maps from
$\bar{X}$ to $\mathbb{P}^{1}$ [ACGH85, Chapter IV, Exercise F-2]. This is a
contradiction, since the infinite family of $D_{i}$ define distinct maps.
Therefore $Z$ is singular. Since $Y_{1}$ is pointless, the singular locus of
$Z$ has to have cardinality $2$ or $\geqslant 4$, for otherwise the projection
of the singular locus would be a zero-cycle of odd degree on $Y_{1}$.
Therefore the genus of $X$ is either $7$ or at most $5$. Since $\dim|2D|=2$,
the Riemann-Roch theorem implies that $g(X)=7$. Consider now the geometric
curve $\bar{X}$. By Mumford’s extension of Marten’s theorem (see [ACGH85,
Chapter IV, Theorem 5.2]), since the curve $\bar{X}$ has a positive-
dimensional family of $g^{1}_{4}$’s, it is either hyperelliptic, trigonal,
bielliptic, or a smooth plane quintic. Since $g(X)=7$, it is not isomorphic to
a smooth plane quintic. If $\bar{X}$ is hyperellitpic, then $X$ is a degree
$2$ cover of a conic, which has infinitely many degree $2$ points, and we are
in case (1). If $\bar{X}$ is trigonal, then the associated $(3,4)$ map onto
$\mathbb{P}^{1}\times Y_{2}$ has to be birational onto its image (since $4$
and $3$ are relatively prime), contradicting $g(X)=7$. Therefore $\bar{X}$ is
bielliptic, so there is a degree $2$ covering $\phi:\bar{X}\to C$, for an
elliptic curve $C$. Consider the composite map $\bar{X}\to C\times Y_{2}$.
Since $X$ has genus $7$, it cannot be birational to a $(2,4)$ curve (genus
would be at most $5$) on $C\times Y_{2}$, therefore the morphism $\bar{X}\to
Y_{2}$ factors through $\phi$. Similarly, the morphism $\bar{X}\to Y_{1}$
factors through $\phi$, contradicting the birationality of $X\to Z$. ∎

Motivated by Theorem 3.5 we make the following definition.

######  Definition 3.6.

Suppose $X/k$ is a curve with $\operatorname{a.\\!irr}_{k}X=d$. We say that
$X$ is _$d$ -minimal_ if there does not exist a covering of curves $\pi:X\to
Y$ of degree at least $2$ such that $\operatorname{a.\\!irr}_{k}Y\deg\pi=d.$

The problem of understanding arithmetic irrationality is reduced, by Theorem
3.5, to analyzing the geometry of $d$-minimal curves. Note that Theorem 3.5
already gives us some control over this geometry: since a $d$-minimal curve
$X$ has a birational embedding of degree $2d$, the genus of $X$ is bounded by
$(d-1)(2d-1)$. We will prove a stronger genus bound in Theorem 4.10 below.

In our analysis of $d$-minimal curves $X$ we occasionally need to use
hyperbolicity of $X$; the following lemma allows us to do so.

######  Lemma 3.7.

Suppose $X$ is a $d$-minimal curve with $d\geqslant 2$. Then the genus of $X$
is at least $2$.

###### Proof.

If $X$ has genus zero, then it is isomorphic to a plane conic. If
$X(k)=\emptyset$, then projection from a rational point on the plane defines a
degree $2$ map $X\to\mathbb{P}^{1}$ and $\operatorname{a.\\!irr}_{k}X=2$. If
$X(k)\neq\emptyset$, then $X=\mathbb{P}^{1}$ and $X$ is $1$-minimal.

If $X$ has genus $1$ and infinitely many rational points, then $X$ is
$1$-minimal. Otherwise, by Riemann-Roch, if $D$ is a rational degree
$d\geqslant 2$ divisor on $X$, then $H^{0}(X,\mathcal{O}(D))\geqslant 1$, and
so $X$ is a degree $d$ cover of $\mathbb{P}^{1}$. ∎

The birationality of $|2D|$ on a $d$-minimal curve implies a classification of
$2$-minimal curves. The resulting theorem is an arithmetic strengthening of
[HS91, Corollary 3] proves that any $2$-minimal curve is a degree $2$ cover of
a genus $0$ or $1$ curve.

######  Theorem 3.8.

There are no $2$-minimal curves over any number field.

###### Proof.

Suppose to the contrary that we are in Setup 2.1 and $X$ is a $2$-minimal
curve. By Theorem 3.5, for a general divisor $D\in A$ the linear system $|2D|$
is birationally very ample. A nondegenerate degree $4$ curve in
$\mathbb{P}^{n}$ for $n\geqslant 3$ has genus at most $1$. By Lemma 3.7 we can
assume that for a general $D$ the linear system $2D$ realizes $X$ as a plane
quartic $Y_{D}\subset\mathbb{P}^{2}$.

If $Y_{D}$ is smooth, then $X=Y_{D}$ is a canonical genus $3$ curve. In
particular $2D=K_{X}$. Since $D$ was general, we can assume $2D\neq K_{X}$.

If $Y_{D}$ is singular, and has a rational singular point $P$, then projection
from $P$ defines a degree $2$ covering $X\to\mathbb{P}^{1}$, thus $X$ is not
$2$-minimal.

If $Y_{D}$ is singular and has at least two singular points, then the
geometric genus of $Y_{D}$ is at most $1$. By Lemma 3.7 curves of genus $0$ or
$1$ are not $2$-minimal. ∎

##  4\. Subspace configurations

We will analyze the geometry of $d$-minimal curves by studying structures
(“subspace configurations”) associated to the birational linear systems
$|nD|$, where $n\geqslant 2$ is an integer and $D$ is a degree $d$ point on
$X$. We first establish notation for and basic properties of these objects,
building to a proof of Theorem 1.4. We summarize the discrete-geometric
structure of these subspace configurations in Section 4.2.

From now on we use notation of Setup 2.1 and assume additionally that $X$ is
$d$-minimal.

###  4.1. Geometric considerations

Given an abelian translate $A\hookrightarrow W_{d}X$, the tensor product map
on line bundles gives a map

$\underbrace{A\times A\times\cdots\times A}_{n}\to W_{nd}X,$

whose image $A^{(n)}$ is (noncanonically) isomorphic to $A$ (since we assumed
$A$ is a trivial torsor). Every divisor in $A^{(n)}$ is (geometrically) of the
form $nD$ for some $D\in A$. By Theorem 3.5 the linear system $|nD|$, for
$n\geqslant 2$, is birationally very ample. By upper-semicontinuity of
dimensions of global sections, there is an open subset of $D$ in $A$ with the
same (minimal) value of $\dim|nD|$; we denote this minimal value by $r(n)$ (so
in fact $A^{(n)}\subset W^{r(n)}_{nd}X$.)

Given any divisor $D^{\prime}$ on $X$, there is an evaluation map

$H^{0}(X,nD)\xrightarrow{\operatorname{ev}_{D^{\prime}}}\mathcal{O}(nD)_{D^{\prime}},$

whose kernel is identified with the space of sections vanishing along
$D^{\prime}$. If we let $D^{\prime}$ vary among the divisors parameterized by
$A$, the dimension of kernel is an upper-semicontinuous and achieves a generic
value on an open subset of $A$. We write $s(n)$ for $r(n)$ minus this generic
dimension of $h^{0}(X,nD-D^{\prime})$ as $D^{\prime}$ varies over the divisors
parameterized by $A$.

The number $s(n)$ has a geometric interpretation in terms of the map to
projective space given by $|nD|$. We will write
$\operatorname{Span}_{nD}(D^{\prime})$ for the linear span of the images of
the points of $D^{\prime}$ under the map $|nD|$. Then
$\operatorname{Span}_{nD}(D^{\prime})$ is a projective space of dimension at
most $s(n)$. For $D^{\prime}\in A$ general,
$\operatorname{Span}_{nD}(D^{\prime})$ has dimension exactly $s(n)$. (When the
linear system $nD$ is unambiguous, we will implicitly write
$\operatorname{Span}(D^{\prime})$.) The abelian translate property from
Section 2.1 in this geometric language says that for any collection of $n-1$
divisors $D_{1},\dots,D_{n-1}$ from $A$, there exists a divisor $D_{n}$ such
that their spans
$\operatorname{Span}_{nD}(D_{1}),\dots,\operatorname{Span}_{nD}(D_{n})$ in
$|nD|$ are contained in a common hyperplane.

######  Lemma 4.1.

Let $X$ be $d$-minimal. Suppose that $D_{1}$ and $D_{2}$ are general divisors
from $A$ and that $D$ is an independently general divisor from $A$.

  * •

If $n\geqslant 3$, then $X\cap\operatorname{Span}_{nD}D_{1}=D_{1}$.

  * •

If $n=2$, then $X\cap\operatorname{Span}_{2D}D_{1}=D_{1}\sqcup(2D-D_{1})$.

In particular, $D_{2}\cap\operatorname{Span}_{nD}D_{1}=D_{1}\cap D_{2}$.

###### Proof.

First suppose that $n\geqslant 3$. Since $D$ is general, the line bundle $nD-
D_{1}$ is basepoint free by Proposition 3.3. On the other hand, any point of
$(\operatorname{Span}_{nD}D_{1})\cap D_{2}$ that is not a point of $D_{1}$
would be a basepoint of $nD-D_{1}$.

Now suppose $n=2$. Since $D_{1},D$ are a general pair, the space
$\operatorname{Span}_{|2D|}D_{1}$ is a hyperplane, for otherwise the
projection from a codimension $2$ space containing
$\operatorname{Span}_{|2D|}D_{1}$ is a degree $d$ (or less) map from $X$ to
$\mathbb{P}^{1}$. Since $D_{1}+(2D-D_{1})=2D$, the hyperplane section
$X\cap\operatorname{Span}_{|2D|}D_{1}$ equals $D_{1}\sqcup(2D-D_{1}).$ ∎

By definition, $r(n)-r(n-1)=s(n)+1$. The difference $s(n)-s(n-1)$ also has a
geometric interpretation.

######  Lemma 4.2.

We have $s(n)-s(n-1)=\lambda+1$, where
$\lambda=\dim\left(\operatorname{Span}_{nD}(D_{1})\cap\operatorname{Span}_{nD}(D_{2})\right)$,
for general $D_{1},D_{2}\in A$.

###### Proof.

Since $D_{1}$ and $D_{2}$ are general, we have $D_{1}\cap D_{2}=\emptyset$ by
(1), and so Lemma 4.1 gurantees that the projection of
$\operatorname{Span}_{nD}D_{2}$ from $\operatorname{Span}_{nD}D_{1}$ is
$\operatorname{Span}_{nD-D_{1}}D_{2}$. Since $D,D_{1},D_{2}$ are general, we
have $s(n)=\dim\operatorname{Span}_{nD}D_{2}$ and
$s(n-1)=\dim\operatorname{Span}_{nD-D_{1}}D_{2}$. ∎

We want to analyze the geometry of the configuration of
$\operatorname{Span}D^{\prime}$ in $|nD|$ for various $n\geqslant 2$. It will
be convenient to project from a maximal subspace that is common to
$\operatorname{Span}D^{\prime}$ for almost all $D^{\prime}$; to formalize this
we make the following definition.

######  Definition 4.3 (Definition/Notation).

For a dense open subset $W\subset A$, let

$V_{|nD|,W}\colonequals\bigcap_{D^{\prime}\in
W}\operatorname{Span}_{nD}D^{\prime}.$

Let $V_{|nD|}$ be the maximal subspace of the form $V_{|nD|,W}$ as $W$ varies
over dense opens in $A$. Explicitly, $V_{|nD|}=V_{|nD|,W}$ for $W$ the locus
of $D^{\prime}$ where $\operatorname{Span}D^{\prime}$ has the maximal
dimension $s(n)$.

######  Lemma 4.4.

Suppose $X$ is a $d$-minimal curve and $D\in A(k)$ is a general divisor. Then
the codimension of $V=V_{|2D|}$ in $|2D|$ is at least $3$.

###### Proof.

Suppose that to the contrary the codimension of $V$ is equal to $2$. The
projection from $V$ defines a morphism $\pi_{V}:X\to\mathbb{P}^{1}$ of degree
at most $2d$. Since for a general divisor $D^{\prime}$,
$\operatorname{Span}_{2D}D^{\prime}$ is contained in a hyperplane and contains
$V$, a general divisor $D^{\prime}\in A$ is contracted to a point by
$\pi_{V}$. In particular the divisors from $A$ vary in a one-dimensional
family, and so $\dim A=1$. Moreover, since $D^{\prime}$ and $2D-D^{\prime}$
belong to the same hyperplane, $\pi_{V}(D^{\prime})=\pi_{V}(2D-D^{\prime})$.
For general $D^{\prime}\in A$, the divisors $D^{\prime}$ and $2D-D^{\prime}$
don’t share points by (1), and so the degree of $\pi_{V}$ equals $2d$. By
Lemma 3.2 this implies that $X$ is a degree $d$ cover of the elliptic curve
$A$, which contradicts our assumption of $d$-minimality. ∎

In Lemma 2.2 we showed that when $X$ is $d$-minimal, a general point $P$ on
$X$ is contained in at least two distinct divisors from $A$. It will be
convenient for us to consider separately the case when a general point $P$ in
$X$ is the intersection of exactly two divisors from $A$. We refer to this
property as condition ($\dagger$ ‣ 4.1), formalized as follows:

($\dagger$) $\text{For a general point }P\in X\text{ there exists a pair of
divisors }F,F^{\prime}\in A\text{ such that }F\cap F^{\prime}=P$

We do not know of examples in which condition ($\dagger$ ‣ 4.1) fails for a
$d$-minimal curve $X$. We have the following sufficient condition for
($\dagger$ ‣ 4.1):

######  Lemma 4.5.

Suppose $X$ is $d$-minimal and $r(2)=2$. Suppose $D\in A$ is general, $x\in D$
is a point, and $D^{\prime}$ is a general divisor containing $x$. Then $D\cap
D^{\prime}=\\{x\\}$. In particular, condition ($\dagger$ ‣ 4.1) holds.

###### Proof.

Choose a general divisor $E$ disjoint from $D$ and $D^{\prime}$. The pair
$(x,D+E)$ is a general point of $X\times A^{(2)}$. In particular, $x$ is a
smooth point on the image of $X\subset|D+E|\simeq\mathbb{P}^{2}$. The span of
$D$ in $|D+E|$ is a line $\ell$ that intersects the curve in $D\cup E$. The
span of $D^{\prime}$ is a line $\ell^{\prime}$ that does not equal $\ell$
since $D^{\prime}\not\subset D\cup E$. A pair of distinct lines shares exactly
one point, and so $D\cap D^{\prime}\subset\ell\cap\ell^{\prime}=\\{x\\}.$ ∎

Under condition ($\dagger$ ‣ 4.1) the linear configuration of
$\operatorname{Span}D^{\prime}$ in $|nD|$ has interesting incidence structure,
as we show in Proposition 4.7. We first need to prove the following linear
nondegeneracy property of $D\subset\operatorname{Span}D$.

######  Lemma 4.6.

Suppose $n\geqslant 2$ is an integer such that $s(n)\leqslant d-2$. Let $D\in
A$ be a general divisor. Then for a general divisor $D^{\prime}\in A$ and any
point $x\in D^{\prime}$ we have
$\operatorname{Span}_{nD}D^{\prime}\setminus\\{x\\}=\operatorname{Span}_{nD}D^{\prime}$.

###### Proof.

Suppose that for a general divisor $D^{\prime}$ there is a point $x\in
D^{\prime}$ such that the set $D^{\prime}\setminus\\{x\\}$ is contained in a
hyperplane inside $\operatorname{Span}D^{\prime}$. Choose a divisor
$D^{\prime}\in U(k)$ such that the Galois group $G_{k}$ acts transitively on
$D^{\prime}$ and the complement of a point $x\in D^{\prime}$ belongs to a
hyperplane $H\subset\operatorname{Span}D^{\prime}$. By transitivity of the
Galois action, for every $x\in D^{\prime}$ there exists a hyperplane $H_{x}$
that contains $D^{\prime}\setminus\\{x\\}$. Choose points
$x_{1},...,x_{s(n)+1}\in D^{\prime}$ that span
$\operatorname{Span}D^{\prime}$. Since $s(n)\leqslant d-1$ there is a point
$x\in D^{\prime}$ such that $x\neq x_{i}$. Then $H_{x}$ would contain the
points $x_{1},...,x_{s(n)+1}$, so $H_{x}$ contains their span
$\operatorname{Span}D^{\prime}$. This is a contradiction. ∎

######  Proposition 4.7.

Suppose $X$ is a $d$-minimal curve, condition ($\dagger$ ‣ 4.1) holds, and
$n\geqslant 2$ is an integer. Suppose that for a general $D^{\prime}\in A$, we
have $\dim\operatorname{Span}_{nD}D^{\prime}\leqslant d-2$. Then for a general
pair of divisors $D_{1},D_{2}\in A$ we have
$\operatorname{Span}_{nD}D_{1}\cap\operatorname{Span}_{nD}D_{2}\neq V_{|nD|}.$

###### Proof.

If $n=2$, $D^{\prime}\in A$ is general, and
$\operatorname{Span}_{2D}D^{\prime}$ is not a hyperplane, then projection from
$\operatorname{Span}_{2D}D^{\prime}$ defines a morphism from $X$ to a
positive-dimensional projective space of degree at most $d$. Therefore
$\operatorname{Span}_{2D}D^{\prime}$ is a hyperplane, and so for a general
pair $D_{1},D_{2}$, we have that
$\operatorname{Span}_{2D}D_{1}\cap\operatorname{Span}_{2D}D_{2}$ has
codimension $2$. Since $V_{|2D|}$ has codimension at least $3$ by Lemma 4.4,
the conclusion holds.

Assume for the remainder of the proof that $n\geqslant 3$. Consider the linear
system $(n-1)E$ for a general $E\in A$ and another general divisor $F\in A$.
Choose a point $x\in F$ and a divisor $F^{\prime}$ such that $F\cap
F^{\prime}=\\{x\\}$; this is possible since ($\dagger$ ‣ 4.1) holds. Since $F$
was general, $F^{\prime}$ is general as well (although the pair $F,F^{\prime}$
is not general). In particular
$\dim\operatorname{Span}_{(n-1)E}F^{\prime}=s(n-1)$. Consider the linear
system $|(n-1)E+F|$. Since $E$ and $F$ are general, $|(n-1)E+F|=|nD|$ for a
general $D$ and $F,D$ form a general pair. The points of $F$ do not belong to
$V_{|nD|}$ (for example, by Lemma 4.1). Therefore, both
$\operatorname{Span}_{nD}F$ and $\operatorname{Span}_{nD}F^{\prime}$ contain
the point $x$, which is outside of $V_{|nD|}$, so

$\operatorname{Span}_{nD}F\cap\operatorname{Span}_{nD}F^{\prime}\neq
V_{|nD|}.$

We have $F\cap F^{\prime}=x$ by construction. Considering the projection $\pi$
from $\operatorname{Span}_{nD}F$, we have

(2)
$\dim\pi(\operatorname{Span}_{nD}F^{\prime})=\dim\operatorname{Span}_{nD}F^{\prime}-\dim(\operatorname{Span}_{nD}F\cap\operatorname{Span}_{nD}F^{\prime})-1<s(n)-\dim
V_{|nD|}-1.$

On the other hand,

(3) $\displaystyle\pi(\operatorname{Span}_{nD}F^{\prime})$
$\displaystyle=\pi(\operatorname{Span}_{nD}(F^{\prime}\smallsetminus\\{x\\}))$
$\displaystyle(\text{Lemma \ref{transitive-monodromy}})$
$\displaystyle=\operatorname{Span}_{nD-F}\left(F^{\prime}\smallsetminus\\{x\\}\right)$
(Lemma 4.1)
$\displaystyle=\operatorname{Span}_{(n-1)E}\left(F^{\prime}\smallsetminus\\{x\\}\right)$
$\displaystyle=\operatorname{Span}_{(n-1)E}F^{\prime}$
$\displaystyle(\text{Lemma \ref{transitive-monodromy}}).$

Combining (2) and (3) we see

$s(n-1)=\dim\operatorname{Span}_{(n-1)E}F^{\prime}<s(n)-\dim V_{|nD|}-1.$

Therefore, by Lemma 4.2, the intersection of a general pair of divisor spans
is larger than $V_{|nD|}$. ∎

We are now in the position to analyze the geometry of linear systems obtained
by projecting $|nD|$ from the subspace $V_{|nD|}$. To do so we introduce the
following definition.

######  Definition 4.8.

Suppose $n\geqslant 2$ is an integer. We denote by $|nD|^{\prime}$ the linear
system obtained from $|nD|$ by projection from $V_{|nD|}$. Similarly, let
$r^{\prime}(n)=\dim|nD|^{\prime}$ and
$s^{\prime}(n)=\dim_{|nD|^{\prime}}\operatorname{Span}D^{\prime}$ for general
$D,D^{\prime}\in A$.

Proposition 4.7 immediately implies:

######  Corollary 4.9.

Suppose ($\dagger$ ‣ 4.1) holds. Then we have
$s^{\prime}(n)\geqslant\min(s(n-1)+1,d-1)$.

###### Proof.

Suppose $s^{\prime}(n)\leqslant d-2$. By Proposition 4.7, for general
$D_{1},D_{2}\in A$ the spaces $\operatorname{Span}_{|nD|^{\prime}}D_{1}$,
$\operatorname{Span}_{|nD|^{\prime}}D_{2}$ share a point. Since $D_{1},D_{2}$
are general, no point of $D_{2}$ belongs to
$\operatorname{Span}_{|nD|^{\prime}}D_{1}$ by Lemma 4.1. Projecting from
$\operatorname{Span}_{|nD|^{\prime}}D_{1}$ we get

$\dim\operatorname{Span}_{|nD|^{\prime}}D_{2}\geqslant\dim\operatorname{Span}_{|nD-
D_{1}|}D_{2}+1.$

Therefore $s^{\prime}(n)\geqslant s(n-1)+1.$ ∎

######  Theorem 4.10.

Suppose that we are in Setup 2.1 and $X$ is $d$-minimal. Suppose ($\dagger$ ‣
4.1) holds. Then for a general divisor $D\in A(k)$ and every number
$n\leqslant d$ we have

$\dim|nD|\geqslant\dim|nD|^{\prime}\geqslant\frac{n(n+1)}{2}-1.$

###### Proof.

By Lemma 4.4, we have $r^{\prime}(2)\geqslant 2$ and $s^{\prime}(2)\geqslant
1$. Combining $s^{\prime}(2)\geqslant 1$ with Corollary 4.9, we have
$s(n)\geqslant s^{\prime}(n)\geqslant\min(d-1,n-1)$. Therefore for
$2<n\leqslant d$,

$\displaystyle r^{\prime}(n)$
$\displaystyle=(s^{\prime}(n)+1)+r(n-1)=(s^{\prime}(n)+1)+(s(n-1)+1)+\cdots+(s(3)+1)+r(2)$
$\displaystyle\geqslant\frac{n(n+1)}{2}-1.\qed$

Theorem 4.10 can be used to bound the genus of curves for which condition
($\dagger$ ‣ 4.1) holds. When ($\dagger$ ‣ 4.1) does not hold, we use the
following lemma instead.

######  Lemma 4.11.

Suppose we are in Setup 2.1 and $X$ is $d$-minimal. Suppose $r(2)\geqslant 3$
and $d\geqslant 4$. Then $r^{\prime}(3)\geqslant 7$.

###### Proof.

Because $X$ does not admit a degree $d$ map to $\mathbb{P}^{1}$, we have
$s(2)=r(2)-1\geqslant 2$. We also have $s^{\prime}(3)\geqslant s(2)$, and
$r^{\prime}(3)=s^{\prime}(3)+r(2)+1$. Therefore the only case in which
$r^{\prime}(3)=6$ is $r(2)=3$, $s^{\prime}(3)=s(2)=2$.

By Lemma 2.2, for a general point $P\in X$ there exists a pair of divisors
$D_{1},D_{2}\in U$, $D_{1}\neq D_{2}$ both containing $P$. We can assume that
both $D_{1},D_{2}$ are general divisors. If condition ($\dagger$ ‣ 4.1) holds,
then $s^{\prime}(3)>s(2)$ by Corollary 4.9, contradicting our calculation that
$s^{\prime}(3)=s(2)=2$ above. We may therefore assume that $D_{1}\cap D_{2}$
contains at least $2$ points, but that there exists some point $y\in
D_{2}\smallsetminus D_{1}$. Choose a general divisor $D\in U$. By Lemma 4.1,
the point $y$ is in $\operatorname{Span}_{2D}D_{2}$ but not in
$\operatorname{Span}_{2D}D_{1}$. Therefore the planes
$\operatorname{Span}_{2D}D_{1}$ and $\operatorname{Span}_{2D}D_{2}$ intersect
along a line.

By Lemma 4.6 the intersection $D_{1}\cap D_{2}$ contains at most $d-2$ points.
Consider now the embedding of $X$ into $\mathbb{P}^{6}$ given by the linear
system $|2D+D_{2}|^{\prime}$. Since $D$ was general, the spans of $D_{1}$ and
$D_{2}$ in this linear system have dimension $s^{\prime}(3)=2$. However
$\operatorname{Span}_{|2D+D_{2}|^{\prime}}D_{1}$ and
$\operatorname{Span}_{|2D+D_{2}|^{\prime}}D_{2}$ share at least $2$ points,
and therefore intersect along a line, again applying Lemma 4.1. Therefore the
projection from $\operatorname{Span}_{|2D+D_{2}|^{\prime}}D_{2}$ maps all
points of $D_{1}\smallsetminus D_{2}$ onto a single point in $|2D|$. Since
$D_{1}\smallsetminus D_{2}$ contains at least $2$ points, this is a
contradiction. ∎

A similar argument can be used to improve the estimate for the value of
$r(4)$; it will be useful in our considerations of low values of $d$.

######  Lemma 4.12.

Suppose we are in Setup 2.1 and $X$ is $d$-minimal, and $d\geqslant 5$ is odd.
Suppose ($\dagger$ ‣ 4.1) does not hold. Then $r^{\prime}(4)\geqslant 12$

###### Proof.

Since ($\dagger$ ‣ 4.1) does not hold, we have $r(2)\geqslant 3$ by Lemma 4.5.
Because $X$ does not admit a degree $d$ map to $\mathbb{P}^{1}$, we have
$s(2)=r(2)-1$. Furthermore, by considering projections from divisor spans, we
see that $s^{\prime}(n)\geqslant s(2)$ for all $n>2$, and that
$r^{\prime}(4)\geqslant r(2)+2s(2)+2$. Combining these, if $r(2)\geqslant 3$,
then $r^{\prime}(4)\geqslant 3r(2)\geqslant 12$. It therefore suffices to
consider the case $r(2)=3$. In this case $r^{\prime}(3)\geqslant 7$ by Lemma
4.11, and so $s(4)\geqslant s(3)\geqslant 3$. Again considering projection, we
have

$r^{\prime}(4)=s^{\prime}(4)+1+r(3)\geqslant s(3)+1+r^{\prime}(3)\geqslant
3+1+7=11.$

Therefore the only way to have $r^{\prime}(4)<12$ is to have equality
everywhere, and hence $r(2)=3$, $r(3)=r^{\prime}(3)=7$,
$s(3)=s^{\prime}(4)=3$, and $r^{\prime}(4)=11$.

Consider a general pair of divisors $D,D_{1}\in A$. Suppose $D_{2}\in A$ is a
divisor that shares points with $D_{1}$. Note that $D_{2}$ is a general
divisor in $A$ since $D_{1}$ is general, and moreover, since ($\dagger$ ‣ 4.1)
does not hold, we can assume that $D_{1}\cap D_{2}$ contains at least $2$
points.

Consider the linear system $|3D|=|3D|^{\prime}$ and the subspaces
$\operatorname{Span}_{|3D|}D_{1}$ and $\operatorname{Span}_{|3D|}D_{2}$. By
assumption these are both $3$-dimensional. Since
$\operatorname{Span}_{|3D|}D_{1}$ and $\operatorname{Span}_{|3D|}D_{2}$ have
nontrivial intersection but do not coincide by Lemma 4.1, Lemma 4.6 implies
that $D_{1}\setminus D_{2}$ contains at least $2$ points. Suppose
$\dim\operatorname{Span}_{|3D|}D_{1}\cap\operatorname{Span}_{|3D|}D_{2}=2$.
Projecting from $\operatorname{Span}_{|3D|}D_{2}$ we see that in the linear
system $|3D-D_{2}|$, the points of $D_{1}\setminus D_{2}$ map to a single
point. Since $D_{1}$ and $|3D-D_{2}|$ is a general pair of divisors, this is a
contradiction. Therefore
$\dim\operatorname{Span}_{|3D|}D_{1}\cap\operatorname{Span}_{|3D|}D_{2}=1$,
and so all points of $D_{1}\cap D_{2}$ in $|3D|$ belong to a single line.

Consider the linear system $|3D+D_{2}|^{\prime}$. Since $D$ is general,
$3D+D_{2}$ is a general point of $A^{(4)}$. By assumption the subspaces
$\operatorname{Span}_{|3D+D_{2}|^{\prime}}D_{1}$ and
$\operatorname{Span}_{|3D+D_{2}|^{\prime}}D_{2}$ are $3$-dimensional,
distinct, and meet in at least $2$ points. Therefore the projection of
$D_{1}\setminus D_{2}$ from $\operatorname{Span}_{|3D+D_{2}|^{\prime}}D_{2}$
maps to a space of dimension at most $1$ in $|3D|$. Thus for a general divisor
$D$, the image of $D_{1}$ in $|3D|$ is contained in a pair of skew lines each
containing at least $2$ points (since $D_{1}\setminus D_{2}$ and $D_{1}\cap
D_{2}$ both contain at least $2$ points). A nondegenerate set $S$ of
$d\geqslant 5$ distinct points in $\mathbb{P}^{3}$ is contained in at most one
pair of skew lines with each line containing at least $2$ points. Therefore
the pair of lines $\operatorname{Span}_{3D}(D_{1}\setminus D_{2})$ and
$\operatorname{Span}_{3D}(D_{1}\cap D_{2})$ are preserved by the Galois action
on $D_{1}$, and, in particular, each line has to contain the same number of
points of $D_{1}$. This contradicts the assumption that $d$ is odd. ∎

We now prove Theorem 1.4 from the introduction.

######  Theorem 4.13.

Given an integer $d$, let $m\colonequals\lceil d/2\rceil-1$ and let
$\varepsilon\colonequals 3d-1-6m<6$. Suppose $X$ is a $d$-minimal curve. If
($\dagger$ ‣ 4.1) holds, then the genus of $X$ is bounded by

$d(d-1)/2+1.$

If ($\dagger$ ‣ 4.1) does not hold, then the genus is at most

$3m(m-1)+m\varepsilon.$

###### Proof.

If ($\dagger$ ‣ 4.1) holds, this follows from Theorem 4.10 for $n=d$ and
Castelnuovo’s genus bound (see [ACGH85, Chapter III, page 116] for the
latter). Alternatively, since any nondegenerate special curve in
$\mathbb{P}^{r}$ has degree at least $2r$, the linear system $|dD|$ on $X$ is
nonspecial for $d>2$ and the genus bound follows from Riemann–Roch.

If ($\dagger$ ‣ 4.1) does not hold, this is Castelnuovo’s bound for a degree
$3d$ curve in $\mathbb{P}^{7}$, which applies by Lemma 4.11. ∎

An immediate corollary of this bound is the theorem of Abramovich-Harris on
degree $3$ points on curves.

######  Corollary 4.14.

Suppose $\operatorname{a.\\!irr}_{k}X=3$. Then $\bar{X}$ is a degree $3$ cover
of $\mathbb{P}^{1}$ or an elliptic curve.

###### Proof.

If $X$ is not $3$-minimal the conclusion holds, so we can assume $X$ is
$3$-minimal. Then by Theorem 4.13 the genus of $X$ is at most $4$. The
geometric gonality of a curve of genus $g\leqslant 4$ is at most $3$. ∎

###  4.2. Summary of Setup and Notation

We give a brief summary of the basic structures and properties introduced in
the previous section. We fix a $d$-minimal curve $X$, and let $A\subset
W_{d}X$ be a corresponding abelian variety with dense $k$-points. For every
$D\in A$ and every integer $n\geqslant 2$ we consider linear system $|nD|$ and
the corresponding projective embedding of $X$. Within the projective space
$\mathbb{P}^{|nD|}$, we look at the linear spaces of the form
$\operatorname{Span}_{nD}E$ for all divisors $E\in A$. The resulting system of
subspace configurations enjoys a number of unusual properties. We use
$V=V_{|nD|}$ to denote the maximal subspace shared by all spaces
$\operatorname{Span}_{nD}E$ for a Zariski-open family of $E\in A$. Projecting
from $V$ defines the linear system $|nD|^{\prime}$ on $X$ equipped with a
similar family of linear spaces $\operatorname{Span}_{|nD|^{\prime}}E$. The
basic properties of these structures are the following:

  1. (1)

The dimensions of $|nD|$ and $|nD|^{\prime}$ have fixed values
$r(n),r^{\prime}(n)$ for a generic choice of $D\in A$;

  2. (2)

For general $D,E$ the dimensions of $\operatorname{Span}_{nD}E$ and
$\operatorname{Span}_{nD^{\prime}}E$ have constant values
$s(n),s^{\prime}(n)$;

  3. (3)

If ($\dagger$ ‣ 4.1) holds, then for a general $D$ and a general pair
$E_{1},E_{2}$ the subspaces $\operatorname{Span}_{|nD|^{\prime}}E_{1}$ and
$\operatorname{Span}_{|nD|^{\prime}}E_{2}$ have nonempty intersection;

  4. (4)

The intersection of all subspaces $\operatorname{Span}_{|nD|^{\prime}}E$ as
$E$ varies over any Zariski-open subset in $A$ is empty.

  5. (5)

For any $D\in A$ and any divisors $E_{1},...,E_{n-1}\in A$ there exists a
divisor $E_{n}\in A$ such that the subspaces
$\operatorname{Span}_{|nD|^{\prime}}E_{i}$ all belong to the same hyperplane;

  6. (6)

For general $D,E\in A$ the projection of $X$ in $|nD|$ from
$\operatorname{Span}_{|nD|}E$ is equivalent to the embedding given by
$|nD-E|$; in particular we have the identities
$r(n)-s(n)=r^{\prime}(n)-s^{\prime}(n)=r(n-1)+1.$

The presence of these linear configurations allows us to give various
restrictions on the geometry of the curve $X$. In the next section we will use
this structure to identify the curves with $r(2)=2$ with the curves
constructed by Debarre and Fahlaoui [DF93].

##  5\. Debarre-Fahlaoui curves

Let $A$ be a positive rank elliptic curve over $k$. For all $d\geqslant 4$,
Debarre and Fahlaoui give examples of $d$-minimal curves lying on the smooth
surface $\operatorname{Sym}^{2}A$. We first recall their construction, and
then we show that any $d$-minimal curve with $r(2)=2$ naturally arises in this
way. This shows that the simplest class of $d$-minimal curves is the one
provided by the Debarre-Fahlaoui construction.

We begin by recalling the setup from [DF93, Section 4.1]. The addition law on
$A$ induces a natural map $\pi\colon\operatorname{Sym}^{2}A\to A$. Let $o\in
A(k)$ be the origin, and let ${\mathscr{E}}$ be the unique nonsplit extension

$0\to\mathcal{O}_{A}\to{\mathscr{E}}\to\mathcal{O}_{A}(o)\to 0.$

Then the fibration $\operatorname{Sym}^{2}A\xrightarrow{\pi}A$ is isomorphic
to $\mathbb{P}{\mathscr{E}}\to A$. Let $H$ denote the relative
$\mathcal{O}_{\mathbb{P}{\mathscr{E}}}(1)$. Then we have

$\operatorname{Pic}(\operatorname{Sym}^{2}A)\simeq\pi^{*}\operatorname{Pic}(A)\oplus\mathbb{Z}H.$

We will write $F_{x}$ for the divisor $\pi^{*}\mathcal{O}_{A}(x)$; in terms of
the moduli description of $\operatorname{Sym}^{2}A$, this consists of all
degree $2$ effective divisors on $A$ that sum to $x$ under the group law. The
divisors $F_{x}$ for all $x\in A$ are numerically equivalent, and we simply
write $F$ for this numerical class. Another natural divisor on
$\operatorname{Sym}^{2}A$, which we denote $H_{x}$ consists of all effective
divisors of degree $2$ on $A$ that contain $x$. The rational equivalence class
of this divisor is $H_{x}=H-F_{o}+F_{x}$ [DF93, Section 4.1 (ii)]. (In
particular, $H_{o}=H$.) The numerical classes of divisors are spanned by $H$
and $F$, with the following intersection relations:

$H^{2}=1,\qquad H\cdot F=1,\qquad F^{2}=0.$

The canonical class $K$ on $\operatorname{Sym}^{2}E$ has numerical class
$K=-2H+F$. The Nef and effective cones both consist of all classes $aH+bF$
where $a\geqslant 0$ and $a+2b\geqslant 0$ [Har77, Chapter V, Proposition
2.21].

######  Definition 5.1.

A Debarre–Fahlaoui curve is a geometrically integral curve on
$\operatorname{Sym}^{2}A$ in numerical class

$(d+m)H-mF,$

for some $1\leqslant m\leqslant d$.

This terminology comes from the fact that Debarre and Fahlaoui consider the
case $m=1$ in [DF93] to give counterexamples to the conjecture of
Abramovich–Harris [AH91, page 229]. Let us recall this construction.

Let $X$ be a Debarre–Fahlaoui curve. The family of divisors $H_{x}$ on
$\operatorname{Sym}^{2}A$ restricts to a family of degree $d$ effective
divisors on $X$ parameterized by $A$, since $H\cdot((d+m)H-mF)=d$. This family
gives rise to an embedding

$\psi\colon A\hookrightarrow W_{d}X.$

This family of degree $d$ divisors is _not_ induced by a map $X\to A$: if
$H_{x}\cdot X$ contains the degree $2$ effective divisor $[x+x^{\prime}]$,
then so does $H_{x^{\prime}}\cdot X$; as such these cannot be (the necessarily
disjoint) fibers of a map.

######  Proposition 5.2 ([DF93, Propositions 5.7 and 5.14]).

Let $d\geqslant 4$ and $1\leqslant m\leqslant d$ be integers. Consider the
numerical class $(d+m)H-mF$.

  1. (1)

If $m<d/2$, then for any nice curve $X$ in this class, we have
$\operatorname{gon}\bar{X}>d$.

  2. (2)

If the class of $X$ is very ample (e.g., if $m=1$), then a general curve in
this class admits no nontrivial maps of degree at most $d$ to a non-isomorphic
curve of genus at least $1$.

In particular, under both of these assumptions, such a curve $X$ is
$d$-minimal.

We now turn to $d$-minimal curves with $r(2)=2$. We will analyze the geometry
of such curves by looking at the induced subspace configurations in
$|3D|^{\prime}$. We begin by showing, in Lemma 5.5, that this structure is a
configuration of $2$-planes in a $5$-space; in other words $r^{\prime}(3)=5$
(which forces $s^{\prime}(3)=2$).

######  Lemma 5.3.

Let $V_{i}\subset\mathbb{P}^{n},i\in I$ be a collection of codimension $2$
subspaces of $\mathbb{P}^{n}$ spanning all of $\mathbb{P}^{n}$. Suppose that
for any $i,j\in I$, the subspaces $V_{i},V_{j}$ belong to a common hyperplane.
Then there is a codimension $3$ subspace $\Lambda$ that belongs to $V_{i}$ for
all $i\in I$.

###### Proof.

Since $V_{i}$ have codimension $2$, if $V_{i},V_{j}$ are two distinct
subspaces, then $\dim\operatorname{Span}(V_{i},V_{j})=n-1$, and so $\dim
V_{i}\cap V_{j}=n-3$. Choose a subspace $V_{k}$ such that $V_{k}$ does not
belong to $\operatorname{Span}(V_{i},V_{j})$. Then $\dim
V_{k}\cap\operatorname{Span}(V_{i},V_{j})=n-3$, and on the other hand $V_{k}$
intersects each of $V_{i}$ and $V_{j}$ in a subspace of dimension $n-3$.
Therefore $V_{k}$ contains $V_{i}\cap V_{j}$. Finally take any subspace
$V_{\ell}$, $\ell\neq i,j,k$. Then $V_{\ell}$ does not belong to
$\operatorname{Span}(V_{u},V_{w})$ for some $u,w\in\\{i,j,k\\}$ and by the
previous argument $V_{\ell}$ contains $V_{u}\cap V_{w}=V_{i}\cap V_{j}$. Thus
$\Lambda=V_{i}\cap V_{j}$ satisfies the conclusion of the lemma. ∎

######  Lemma 5.4.

Let $X$ be a $d$-minimal curve with $r(2)=2$. Suppose that $P\in X$ is a
general point, and $D\in A$ is general. Then there exists a pair of distinct
divisors $D_{1},D_{2}$ from $A$ through $P$ such that
$\operatorname{Span}_{|3D|^{\prime}}D_{1}$ and
$\operatorname{Span}_{|3D|^{\prime}}D_{2}$ span a hyperplane in
$|3D|^{\prime}$.

###### Proof.

By Lemma 2.2 there exists a pair of distinct divisors $D_{1},D_{2}$ through a
general point of $X$, and moreove each $D_{i}$ individually is general in $A$.
Choose such a pair. Recall that since $r(2)=2$ and
$r^{\prime}(3)-r(2)=s^{\prime}(3)+1$, the spaces
$\operatorname{Span}_{|3D|^{\prime}}D_{i}$ have codimension $3$ in
$|3D|^{\prime}$. By Lemma 4.5, we may assume that $D_{1}$ and $D_{2}$ meet
only at $P$.

Consider the divisor $E\in A$ such that $3D-D_{1}=2E$; such $E$ is a general
point of $A$, and in particular $|2E|$ is an embedding along each $D_{i}$.
Note that $\operatorname{Span}_{|3D|^{\prime}}(D_{1}+D_{2})$ has codimension
either $1$ or $2$. If $\operatorname{Span}_{|3D|^{\prime}}(D_{1}+D_{2})$ has
codimension $2$, then the projection $\pi$ from
$\operatorname{Span}_{|3D|^{\prime}}D_{1}$ sends the points of
$D_{2}\smallsetminus D_{1}$ to the same point in $|2E|$. The set
$D_{2}\smallsetminus D_{1}$ contains at least $d-1$ points, contradicting the
assumption that $|2E|$ is an embedding along $D_{2}$. ∎

######  Lemma 5.5.

Suppose $r(2)=2$. Then $r^{\prime}(3)=5$.

###### Proof.

We will consider the configuration of divisor spans in $|3D|^{\prime}$. Since
$r(2)=2$, we have $s^{\prime}(3)=r^{\prime}(3)-r(2)-1=r^{\prime}(3)-3$. Since
two general divisor spans span a hyperplane by Lemma 5.4, their intersection
has codimension $5$. Furthermore, $r^{\prime}(3)\geqslant 5$ by Lemma 4.5 and
Theorem 4.10. Suppose $r^{\prime}(3)\geqslant 6.$

Let $E$ be a general divisor in $A$. Consider a general collection of divisors
$D_{1},\dots,D_{N}$ in $A$. The spaces
$W_{i}\colonequals\operatorname{Span}D_{i}\cap\operatorname{Span}E$ form a
collection of distinct codimension $2$ subspaces of $\operatorname{Span}E$,
any two of which span a hyperplane
$\operatorname{Span}(D_{i}+D_{j})\cap\operatorname{Span}E$. For $N$ large
enough, the intersection of all $W_{i}$ is empty (since we are working in
$|3D|^{\prime}$). Thus by Lemma 5.3, there is a hyperplane $\Lambda$ in
$\operatorname{Span}E$ containing all $W_{i}$. Since $N$ was arbitrarily
large, for a general $D^{\prime}\in A$ we have
$\operatorname{Span}D^{\prime}\cap\operatorname{Span}E\subset\Lambda.$ Being
contained in $\Lambda$ is a closed condition; therefore every divisor
$D^{\prime}$ for which the codimension of
$\operatorname{Span}D^{\prime}\cap\operatorname{Span}E$ in
$\operatorname{Span}E$ is $2$ satisfies
$\operatorname{Span}D^{\prime}\cap\operatorname{Span}E\subset\Lambda$. This
contradicts Lemma 5.4 applied to $D$, $D_{1}=E$ and a point $P$ of $E$ outside
$\Lambda$. ∎

We will relate curves with $r(2)=2$ to Debarre-Fahlaoui curves as follows. If
$r^{\prime}(3)=5$, then the resulting configuration of divisor spans in
$\mathbb{P}^{5}$ is a family of $2$-planes, parametrized by $A$, pairwise
sharing points. This naturally gives rise to a rational map
$\psi\colon\operatorname{Sym}^{2}A\to\mathbb{P}^{5}$ sending a pair of
divisors to the intersection of their spans. Since there are at least two
divisors from $A$ through every point on $X$ we expect $X$ to be in
$\psi(\operatorname{Sym}^{2}A)$. In this way $X$ “wants to be” a curve on
$\operatorname{Sym}^{2}A$.

To realize this idea, we first need to reduce to the case $\dim A=1$ (in Lemma
5.6) and establish a nondegeneracy property of our configuration (Lemma 5.7).

######  Lemma 5.6.

Suppose that $X$ is $d$-minimal and $\dim A>1$. Then $r(2)>2$.

###### Proof.

By Proposition 3.3 we have $r(2)\geqslant\dim A$. Thus it suffices to show
that the case $\dim A=r(2)=2$ does not occur. We consider the cases $d=3$ and
$d\geqslant 4$ separately.

Suppose $d\geqslant 4$. Choose a general divisor $D\in A$ and consider the
rational map $\phi:A\to(\mathbb{P}^{2})^{\vee}$ that sends a divisor class $E$
to the line $\operatorname{Span}_{2D}E$.

The set of $2d$ points on a general linear section of $X\subset\mathbb{P}^{2}$
is thereby equipped with a nonempty collection of $d$-element subsets coming
from $A$. But the monodromy of the linear section is the symmetric group (see,
for example, [ACGH85, Lemma, Chapter III, page 111]), and so every $d$-element
subset of a general linear section of $X$ is a divisor from $A$. Therefore,
for a general divisor $E\in A$, there exists another divisor $E^{\prime}\in A$
such that both $E\cap E^{\prime}$ and $E^{\prime}\smallsetminus E$ consist of
at least two points. Choose a general $E$ and consider the linear system
$|2D+E|^{\prime}$. By Lemma 5.5, we have $\dim|2D+E|^{\prime}=5$ and
$\dim\operatorname{Span}_{|2D+E|^{\prime}}E=\dim\operatorname{Span}_{|2D+E|^{\prime}}E^{\prime}=r^{\prime}(3)-r(2)-1=2$.
Since $E$ and $E^{\prime}$ share at least two points, the $2$-planes
$\operatorname{Span}_{|2D+E|^{\prime}}E$ and
$\operatorname{Span}_{|2D+E|^{\prime}}E^{\prime}$ share a line $\ell$.
Therefore the projection of $\operatorname{Span}_{|2D+E|^{\prime}}E^{\prime}$
from $E$ is a single point. Therefore all the points of
$E^{\prime}\smallsetminus E$ are mapped to the same point under $2D$, which is
a contradiction.

Suppose now that $d=3$. Consider the linear system $|3D|^{\prime}$ and the
associated embedding of $X$ in $\mathbb{P}^{5}$. Consider a general point
$P\in X$ and a general pair of divisors $D_{P},D_{P}^{\prime}$ through $P$.
Such a pair does not share any points on $X$ except for $P$ by Lemma 4.5. By
Lemma 5.4 we have

$\dim\operatorname{Span}_{|3D|^{\prime}}D_{P}\cap\operatorname{Span}_{|3D|^{\prime}}D_{P}^{\prime}=0.$

Choose a pair of general points $P,Q\in X$. Since the pair is general,
projection from the line $\ell=PQ$ realizes $X$ as a degree $7$ curve in
$\mathbb{P}^{3}$. The projection from $\ell$ maps the divisor spans that
contain $P$ or $Q$ to lines in $\mathbb{P}^{3}$. Since $P,Q$ are a general
pair of points, a general divisor $D_{P}$ containing $P$ and a general divisor
$D_{Q}$ containing $Q$ form a general pair of divisors (as $P,Q$ vary), and so
$\operatorname{Span}_{|3D|^{\prime}}D_{P}$ and
$\operatorname{Span}_{|3D|^{\prime}}D_{Q}$ intersect at a point. By the above
description of generic intersections, we can choose an infinite collection of
divisors $D_{P}^{1},D_{P}^{2},\dots$ and $D_{Q}^{1},D_{Q}^{2},\dots$ such that
the projections
$\ell_{i}=\pi_{\ell}(\operatorname{Span}_{|3D|^{\prime}}D_{P}^{i})$ and
$\ell_{i}^{\prime}=\pi_{\ell}(\operatorname{Span}_{|3D|^{\prime}}D_{Q}^{i})$
form two families of lines with lines in each family pairwise skew, and
$\ell_{i}\cap\ell_{j}^{\prime}\neq\emptyset$ for all $i,j$. Such a pair of
families is always contained in a smooth quadric. Since every line
$\ell_{i},\ell_{i}^{\prime}$ contains points from $\pi_{\ell}(X)$, the curve
$\pi_{\ell}(X)$ shares infinitely many points with the quadric, and so belongs
to the quadric. Therefore projection from $PQ$ realizes $X$ as a degree $7$,
$(e_{1},e_{2})$-curve on the quadric. As $(P,Q)$ varies, the value of
$(e_{1},e_{2})$ achieves a generic value on an open subset of
$\operatorname{Sym}^{2}X$; by monodromy, for this generic value $e_{1}=e_{2}$.
However, $X$ has degree $7=e_{1}+e_{2}$, contradiction. ∎

######  Lemma 5.7.

Suppose we are in Setup 2.1, $X$ is $d$-minimal, and $r(2)=2$. Consider a
general triple of divisors $D,D_{1},D_{2}\in A$ and let
$D_{3}=3D-D_{1}-D_{2}$. Then

$\bigcap_{i}\operatorname{Span}_{|3D|^{\prime}}D_{i}=\emptyset.$

###### Proof.

Suppose the spaces $\operatorname{Span}_{|3D|^{\prime}}D_{i}$ share a common
point $P$. Consider a divisor $E$ such that $D,D_{1},D_{2},E$ is a general
quadruple; then $D_{1},D_{2},D_{3},3D-E$ is a general quadruple as well and
$P$ does not belong to $\operatorname{Span}_{|3D|^{\prime}}E$. Consider the
projection $\pi_{E}$ from $\operatorname{Span}_{|3D|^{\prime}}E$. We have

$\pi_{E}(P)\in\pi_{E}(\operatorname{Span}_{|3D|^{\prime}}D_{i})=\operatorname{Span}_{3D-E}D_{i}.$

By the generality of the quadruple $D_{1},D_{2},D_{3},3D-E$, the lines
$\operatorname{Span}_{3D-E}D_{i}$ do not share a point, contradiction. ∎

We now prove the main result of this section.

######  Theorem 5.8.

Suppose we are in Setup 2.1 and $X$ is $d$-minimal. If $r(2)=2$, then $X$ is
birational to a Debarre-Fahlaoui curve.

###### Proof.

By Lemma 5.6, we have $\dim A=1$. By Lemma 5.5, we have $r^{\prime}(3)=5$. Let
$D$ be a point of $A(k)$ achieving these generic values, so that the linear
systems $|nD|$ are basepoint-free for $n\geqslant 2$, $\dim|2D|=2,$ and
$\dim|3D|^{\prime}=5$.

Write $\varphi\colon X\to\mathbb{P}^{5}$ for the morphism associated to
$|3D|^{\prime}$. We will now define a rational map from
$\operatorname{Sym}^{2}A$ to $\mathbb{P}^{5}$, whose image contains $X$ in its
closure. Since $s^{\prime}(3)=r^{\prime}(3)-r(2)-1=2$, a general divisor
$[D_{1}]\in A$ has $2$-dimensional span. For a general pair of divisors
$[D_{1}],[D_{2}]\in A$, we have
$\dim\operatorname{Span}_{|3D|^{\prime}}D_{1}\cap\operatorname{Span}_{|3D|^{\prime}}D_{2}=0$
by Lemma 5.4. This yields a rational map

$\displaystyle\psi\colon\operatorname{Sym}^{2}A$
$\displaystyle\dashrightarrow\mathbb{P}^{5}$
$\displaystyle\qquad(D_{1},D_{2})$
$\displaystyle\mapsto\operatorname{Span}D_{1}\cap\operatorname{Span}D_{2}.$

If $\operatorname{Span}D_{1}\cap\operatorname{Span}D_{2}$ has dimension $1$,
then projecting from $\operatorname{Span}(D_{1}+D_{2})$ yields a (geometric)
degree $d$ map $X\to\mathbb{P}^{1}$; in particular, by Setup 2.1, the divisor
$[3D-D_{1}-D_{2}]\in A$ lies in a proper Zariski closed (dimension at most
$0$) locus. From this we observe

(4) $\text{For general $D_{1}$, there exist finitely many lines $\Sigma$ in
$\operatorname{Span}D_{1}$, such that for any $P\not\in\Sigma$},\\\ \text{if
$D_{2}\neq D_{1}$ and
$P\in\operatorname{Span}D_{1}\cap\operatorname{Span}D_{2}$, then
$P=\operatorname{Span}D_{1}\cap\operatorname{Span}D_{2}$.}$

If the divisors $D_{1}$ and $D_{2}$ share a point, then the intersection of
$\operatorname{Span}D_{1}$ and $\operatorname{Span}D_{2}$ necessarily contains
that point. The closure of the image of $\psi$ contains the image of $X$ under
$\varphi$ since by Lemma 5.4 for a general point $P$ of $X(\bar{k})$, there
exist divisors $D_{1}$ and $D_{2}$ such that

$\operatorname{Span}D_{1}\cap\operatorname{Span}D_{2}=P.$

Our goal is to show that a general point $P\in X$ is contained in _exactly_
two divisors. Once we do so, we will have a natural map
$X\to\operatorname{Sym}^{2}A$, and we will then show via a simple argument
that the image is indeed a Debarre–Fahlaoui curve. Fix a general divisor class
$D_{1}$. We first analyze the image of the morphism
$\eta:A\to\operatorname{Span}D_{1}$ that sends a divisor $D_{2}$ to
$\operatorname{Span}D_{2}\cap\operatorname{Span}D_{1}$. This map is
nondegenerate by Lemma 5.4. We will show that it must be the inclusion of $A$
as a plane cubic curve by considering several cases based on the possible
degrees of the image of $\eta$.

Case 1: $\deg\eta(A)=2$. Consider divisors $D_{2},D_{3}$ such that
$D_{1},D_{2},D_{3}$ form a general triple. Let
$P\colonequals\operatorname{Span}D_{2}\cap\operatorname{Span}D_{1}$ and
$Q\colonequals\operatorname{Span}D_{3}\cap\operatorname{Span}D_{1}$. Consider
the divisor $D_{4}\colonequals 3D-D_{2}-D_{3}$; since $D_{1},D_{2},D_{3}$ are
a general triple we can assume that $D_{4}$ does not pass through $P$ or $Q$.

$P$$Q$$D_{1}$$D_{2}$$D_{3}$$D_{4}$

On the other hand, $\operatorname{Span}(D_{2},D_{3})$ is a hyperplane that
contains $\operatorname{Span}D_{4}$ and intersects $\operatorname{Span}D_{1}$
in the line $\overline{PQ}$. Since $\overline{PQ}$ meets $\eta(A)$ only at $P$
and $Q$, this is a contradiction.

Case 2: $\deg\eta(A)\geqslant 3$ and $\eta$ multiple-to-one onto its image.

Choose a general line $\ell$ in $\operatorname{Span}D_{1}$ that intersects the
image of $\eta$ in at least three smooth points $P_{2},P_{3},P_{4}$. Through
each of the points $P_{2},P_{3},P_{4}$ passes at least two divisor-spans
$\operatorname{Span}D_{i},\operatorname{Span}D_{i}^{\prime}$, $i=2,3,4$. Since
the line $\ell$ is general, the pair $D_{1},D_{2}$ is a general pair of
divisors in $A\times A$. Hence, using the fact that $\eta$ is nondegenerate,
we can assume that the point where they meet is not in the finite set $\Sigma$
guaranteed by (4) on either $\operatorname{Span}D_{1}$ or
$\operatorname{Span}D_{2}$. In particular

$\operatorname{Span}D_{1}\cap\operatorname{Span}D_{2}=\operatorname{Span}D_{1}\cap\operatorname{Span}D_{2}^{\prime}=\operatorname{Span}D_{2}\cap\operatorname{Span}D_{2}^{\prime}=P_{2}.$

By symmetry the same holds for $i=3$ and $i=4$.

$P_{2}$$P_{3}$$P_{4}$$D_{2}$$D_{2}^{\prime}$$\eta(A)$$\ell$$D_{1}$

By Lemma 5.7 applied to the triple $D,D_{1},D_{i}$ we may choose
$D_{i}^{\prime}\neq 3D-D_{1}-D_{i}$. Since $D_{1},D_{2}$ is general, for every
possible $D_{2}^{\prime}$ through $P_{2}$, the divisor $D_{2}-D_{2}^{\prime}$
is not a $2$-torsion point on $A^{0}$, as we now explain. Equivalently, for a
general $D_{2}$, and each of the finitely many possible $2$-torsion points $T$
on $A^{0}$ giving rise to $D_{2}^{\prime}=D_{2}-T$, a general divisor span
$\operatorname{Span}D_{1}$ does not meet
$\operatorname{Span}D_{2}\cap\operatorname{Span}D_{2}^{\prime}$, which is
clear since the the map $\eta$ associated to $D_{2}$ is nondegenerate.

We may further assume that the $15$ pairwise intersection points of
$\operatorname{Span}D_{i},\operatorname{Span}D_{j}^{\prime}$ do not belong to
$X$ (if $i\neq j$ then $D_{i},D_{j}^{\prime}$ are a general pair of divisors;
if $i=j$, then $P_{i}$ is a general point on $\eta(A)$ which contains only
finitely many points of $X$).

Projection from $\ell$ maps each $D_{i}$ and $D_{i}^{\prime}$ for $i=2,3,4$ to
a line. Since $D_{i}^{\prime}\neq 3D-D_{1}-D_{i}$, we have that
$\operatorname{Span}(D_{i}+D_{i}^{\prime})$, for $i=2,3,4$, does not
contain$D_{1}$. Further, since $\ell$ is a general line in
$\operatorname{Span}(D_{1})$ through $P_{i}$, we also have that
$\operatorname{Span}(D_{i}+D_{i}^{\prime})$ does not contain $\ell$. Since in
addition $\operatorname{Span}D_{i}$ and $\operatorname{Span}D_{i}^{\prime}$
meet only at $P_{i}$, their images under projection from $\ell$ are skew.
Moreover, since
$\operatorname{Span}D_{i}\cap\operatorname{Span}D_{j}^{\prime}$ is a point off
of $\ell$ for $i\neq j$, any two such lines meet.

The only such configuration of a triple of pairs of skew lines
$\pi_{\ell}(\operatorname{Span}D_{i}),\pi_{\ell}(\operatorname{Span}D_{i}^{\prime})$
is the configuration of edges of a tetrahedron.

$P_{2}$$P_{3}$$P_{4}$$D_{2}$$D_{2}^{\prime}$$\eta(A)$$\ell$$D_{1}$$D_{2}$$D_{2}^{\prime}$$D_{4}$$D_{4}^{\prime}$$D_{3}^{\prime}$$D_{3}$

The triples of divisors corresponding to faces of the tetrahedron sum to $3D$,
since they are coplanar. Summing the faces containing a shared edge and
subtracting the two faces containing the opposite edge, we get
$2(D_{i}-D_{i}^{\prime})=0$ for all $i$, contradicting the generality
assumption $D_{2}-D_{2}^{\prime}\not\in A[2]$.

Case 3: $\deg\eta(A)\geqslant 4$ and $\eta$ birational onto its image. We
proceed as before by choosing a general line $\ell\in\operatorname{Span}D_{1}$
and analyzing the projection from $\ell$. Let $P_{2},...,P_{n}$, $n\geqslant
5$ be the points of $\ell\cap\eta(A)$. Since $\ell$ is general there is a
unique divisor-span $\operatorname{Span}D_{i}$ through $P_{i}$. Projection
from $\ell$ maps the $\operatorname{Span}(D_{i})$ into a collection of lines
in $\mathbb{P}^{3}$ pairwise sharing points. Since they cannot all belong to
the same plane, by Lemma 5.3 they have to share a point $P$.

Consider two divisors $D_{i},D_{j}$ for $i,j>2$ and let $D_{ij}\colonequals
3D-D_{i}-D_{j}$ be the remaining divisor contained in the hyperplane
$\operatorname{Span}(D_{i},D_{j})$. Since $(D_{1},D_{i},D_{j})$ is general,
$D_{ij}\neq D_{1}$; equivalently, $D_{1}$ is not contained in
$\operatorname{Span}(D_{i},D_{j})$. In particular
$\operatorname{Span}(D_{i},D_{j})\cap\operatorname{Span}D_{1}=\ell$, and so
$\operatorname{Span}D_{ij}\cap\operatorname{Span}(D_{1})\in\ell$. Thus
$D_{ij}=D_{k}$ for some index $k$. Projecting the configuration of lines
$\pi_{\ell}(\operatorname{Span}(D_{i}))$ from $P$ gives a configuration of
$n-1\geqslant 4$ points $Q_{2},...,Q_{n}\in\mathbb{P}^{2}$ with the following
two properties:

  1. (1)

for any two distinct points $Q_{i},Q_{j}$ there exists a point $Q_{k},$ $k\neq
i,j$ collinear with $Q_{i},Q_{j}$;

  2. (2)

no more than $3$ of $Q_{i}$ are collinear.

Configurations of points satisfying Property (1) are known as Sylvester–Gallai
configurations; see [KN73, Theorems 3.1 – 3.6] for classification results for
small values of $n$. In particular, either $n-1=9$ and the configuration is
Hesse configuration (and the points are a base locus for a pencil of cubic
curves) or $n-1\geqslant 12.$

We first show that for general choices of $D_{1}$ and $\ell$, the $2$-plane
$\Lambda\colonequals\pi_{\ell}^{-1}(P)$ does not meet $X$. Indeed, suppose to
the contrary that $\Lambda$ meets $X$ in $\lambda$ points for a general choice
of $D_{1}$ and $\ell$. Fixing $D_{1}$ and varying $\ell$, we obtain a map

$(\mathbb{P}^{2})^{\vee}\dashrightarrow\operatorname{Sym}^{\lambda}X.$

If this map is nonconstant, then $\lambda\geqslant d+1$ since $X$ is
$d$-minimal. Since $\operatorname{Span}D_{1}$ meets $X$ in $d$ points, none of
which are on $\ell=\Lambda\cap\operatorname{Span}D_{1}$, we must have that
$\operatorname{Span}(D_{1},\Lambda)$ meets $X$ in at least $2d+1$ points. If
$D_{1}$ and $\ell$ are defined over the ground field, then so is the unique
point $P$, and hence so is the $3$-plane $\operatorname{Span}(D_{1},\Lambda)$.
Projection from this plane defines a degree at most $d-1$ map to
$\mathbb{P}^{1}$, contradicting $d$-minimality.

We may therefore assume that the plane $\Lambda$ meets $X$ in points
independent of $\ell$. If it meets $X$ in at least $2$ distinct points in
$\mathbb{P}^{5}$, then their span meets $\operatorname{Span}D_{1}$ in a unique
point, which does not lie on a general line $\ell$. Hence $\Lambda$ meets the
image of $X$ in $\mathbb{P}^{5}$ in a unique point (possibly with
multiplicity). Varying $D_{1}$, and noting that a $d$-minimal curve cannot
have genus at most $1$, we see that this unique point must also be independent
of $D_{1}$. This is a contradiction, since a general pair of divisors
$D_{1},D_{2}$ can be chosen so that their span
$\operatorname{Span}(D_{1},D_{2})$ misses any specific point.

Consider the projection $\pi_{\Lambda}\colon X\to\mathbb{P}^{2}$, and suppose
that it factors as $X\to Y\to Y^{\prime}\subset\mathbb{P}^{2}$, where $X\to Y$
is a finite degree $s$ morphism of smooth curves and $Y\to Y^{\prime}$ is
birational. The plane curve $Y^{\prime}$ has degree $3d/s$ and every one of
the points $Q_{2},...,Q_{n}$ is a singular point of multiplicity at least
$d/s$. Moreover, the point
$Q_{1}\colonequals\pi_{\Lambda}(\operatorname{Span}D_{1}))$ is also a singular
point of $Y^{\prime}$ of multiplicity at least $d/s$.

Suppose $n-1=9$. Then the configuration of points $Q_{2},...,Q_{10}$ is a
Hesse configuration, in particular there is a pencil of cubics through
$Q_{2},...,Q_{10}$. We may therefore choose $Q$ to be a cubic though
$Q_{2},...,Q_{10}$ and $Q_{1}$. The curve $Y^{\prime}$ is not a cubic, since
it has at least $10$ singular points, and so $Y^{\prime}\cap Q$ is a finite
scheme. However $Q$ intersects $Y^{\prime}$ in at least $10$ points of
multiplicity $d/s$, thus the total multiplicity of the intersection is at
least $10d/s>9d/s=\deg Q\deg Y^{\prime}$, contradiction.

Suppose $n-1\geqslant 12$. Then the geometric genus of $Y^{\prime}$ is at most

$g_{Y^{\prime}}\leqslant\frac{(3d/s-1)(3d/s-2)}{2}-13\frac{d/s(d/s-1)}{2}=1+2\frac{d}{s}-2\frac{d^{2}}{s^{2}}.$

Since $Y^{\prime}$ has at least $13$ singular points, the degree of
$Y^{\prime}$ is at least $5$, and so $g_{Y^{\prime}}<0$, contradiction.

Thus the map $\eta\colon A\to\operatorname{Span}D_{1}$ is an isomorphism onto
a plane cubic curve. This means that for every point $P\in D_{1}$ there is
exactly one divisor $D_{2}\neq D_{1}$, with $[D_{2}]\in A$ that contains $P$.
Since $D_{1}$ was an arbitrary general divisor, we conclude that for a general
point $P\in X$ there exist exactly two divisors $D_{1},D_{2}$ from $A$ that
contain $P$. Therefore we can define a birational morphism $\mu\colon
X\to\operatorname{Sym}^{2}A$ that sends a point $P\in X$ to the unique pair of
divisors $(D_{1},D_{2})\in\operatorname{Sym}^{2}A$ that contain $P$. We claim
that $\mu$ is the birational equivalence of $X$ with a Debarre-Fahlaoui curve
that we seek. To do this we need to identify the numerical class of $\mu(X)$
on $\operatorname{Sym}^{2}A$.

For $x\in A$, recall that $H_{x}$ is the set of all pairs of divisors in
$\operatorname{Sym}^{2}A$ that contain $x$. Hence the intersection
$H_{x}\cap\mu(X)$ is supported on the points $\mu(\operatorname{supp}(x))$.
Since a general divisor $x\in A$ is a degree $d$ point (and hence a single
monodromy orbit), $H_{x}\cap\mu(X)$ is a multiple of
$\mu(\operatorname{supp}(x))$. Consider a general point $(x,x^{\prime})$ on
$\mu(X)$. Since the divisors $H_{x}$, and $H_{x^{\prime}}$ for $x\neq
x^{\prime}$ intersect transversely at
$(x,x^{\prime})\in\operatorname{Sym}^{2}A$, the intersections
$H_{x}\cap\mu(X)$ and $H_{x^{\prime}}\cap\mu(X)$ cannot both be nontransverse
at $(x,x^{\prime})$. Hence the generic intersection $H_{x}\cap\mu(X)$ cannot
consist of multiple points. In other words, for a general $x\in A$ the
intersection $H_{x}\cap\mu(X)\subset\operatorname{Sym}^{2}A$ is smooth and
$H_{x}\cap\mu(X)=\mu(\operatorname{supp}(x))$, so $[\mu(X)]\cdot H=d$.
Therefore, numerically, $[\mu(X)]=aH+(d-a)F$. Since $[\mu(X)]$ is effective,
by the description of the effective cone we have $a\geqslant 0$ and
$2d-a\geqslant 0$. The fibers of the addition map $\operatorname{Sym}^{2}A\to
A$ have numerical class $F$, and since $X$ does not admit maps of degree less
than $d$ to $A$, we conclude that $d<[\mu(X)]\cdot F=a$. Thus
$[\mu(X)]=(d+m)H-mF$ for some $m$ between $1$ and $d$ as claimed. ∎

######  Remark 5.9.

As observed in Remark 3.4, the geometric gonality of a curve with
$\operatorname{a.\\!irr}_{k}X=d$ which is not a degree $d$ cover of an
elliptic curve is at most $2d-1$. Theorem 5.8 implies that if, in addition,
such curves are not Debarre-Fahlaoui curves, then their geometric gonality is
at most $2d-2$.

##  6\. Applications and Extensions

###  6.1. Low degree points on projective curves

Our main strategy can be applied to study low degree points on “special”
curves. The geometry of configurations of divisor spans as summarized in
Section 4.2 can be used to estimate the dimensions of various linear systems
from below. Combining this with Castelnuovo’s bound yields a bound on the
genus of the curve. We now recall Castelnuovo’s theorem [ACGH85, Chapter III,
page 116]. Given positive integers $\delta,n$, write

$\delta-1=m(n-1)+\varepsilon,$

for integers $m$ and $0\leqslant\varepsilon<n-1.$ Then the genus of a
nondegenerate curve of degree $\delta$ in $\mathbb{P}^{n}$ is bounded by

$\pi(\delta,n)=\frac{m(m-1)}{2}(n-1)+m\varepsilon.$

For fixed $n$ and large $\delta$, the genus bound $\pi(\delta,n)$ is roughly
$\delta^{2}/(2n-2)$.

######  Theorem 6.1.

Suppose $X\subset\mathbb{P}^{r}$ is an irreducible (possibly singular) curve
of degree $e$ and genus $g$. Suppose $X$ has infinitely many points of degree
$d$ not contained in hyperplanes of $\mathbb{P}^{r}$. Then

$g\leqslant\pi(e+2d,2r+1).$

###### Proof.

By the Mordell–Lang Conjecture, as explained in Section 2, for all but
finitely may degree $d$ points $D$ on $X$, either $D$ moves in a pencil, or
the class of $D$ in $W_{d}X$ belongs to a translate of an abelian subvariety
in $\operatorname{Pic}^{d}X$. In either of those cases, the class $2D$ is
basepoint-free.

Let $[H]$ denote the divisor class corresponding to the embedding
$X\subset\mathbb{P}^{r}$, choose a degree $d$ point $D$ for which $2D$ is
basepoint-free and such that $D$ is not contained in divisors from $|H|$.
Consider the linear system $|2D+H|$. Since $2D$ is basepoint-free, for a
divisor $H^{\prime}\in|H|$ we have
$X\cap\operatorname{Span}_{2D+H}H^{\prime}=H^{\prime}$. Suppose the dimension
of $\operatorname{Span}_{2D+H}D$ is equal to $s$ and choose a set $S\subset D$
of size $s+1$ such that
$\operatorname{Span}_{2D+H}S=\operatorname{Span}_{2D+H}D$. If $s<r$, then
there exists a divisor $H^{\prime}\in|H|$ that contains $S$. Then
$\operatorname{Span}_{2D+H}H^{\prime}\supset\operatorname{Span}_{2D+H}S=\operatorname{Span}_{2D+H}D$,
and so the points of $D$ are contained in
$X\cap\operatorname{Span}_{2H+D}H^{\prime}=H^{\prime}$, contradicting our
assumption. Therefore $s\geqslant r$. Since the projection from
$\operatorname{Span}_{2D+H}D$ maps to a space of dimension at least $r$, we
have $\dim|2D+H|\geqslant 2r+1.$ By Castelnuovo’s theorem applied to the
embedding $|2D+H|$, the genus of $X$ is at most $\pi(e+2d,2r+1)$. ∎

###  6.2. Genus estimate for non-Debarre-Fahlaoui curves

The argument of Theorem 4.13 with the added assumption $r(2)>2$ gives a better
genus bound for currves that satisfy ($\dagger$ ‣ 4.1). Together Theorems
4.13, 5.8, 3.5 and 6.2 below yield Theorem 1.5 from the introduction.

######  Theorem 6.2.

Suppose $X$ is a $d$-minimal curve with $r(2)>2$. Suppose that condition
($\dagger$ ‣ 4.1) holds. Then the genus $g$ of $X$ satisfies

$g\leqslant\frac{(d-1)(d-2)}{2}+2$

.

###### Proof.

The argument is identical to the proof of Theorem 4.13. We have $r(2)\geqslant
3$, $s(2)\geqslant 2$ by assumption and $s(n)\geqslant\min(d-1,s(n-1)+1)$ by
Proposition 4.7. Therefore for $n\leqslant d-1$ we have

$r(n)\geqslant\frac{(n+1)(n+2)}{2}-3.$

By Castelnuovo’s theorem applied to the linear system $|(d-1)D|$ we get the
desired genus bound. ∎

###  6.3. Classification results for low values of $d$

We now summarize what the main results say about curves with small arithmetic
degree of irrationality.

######  Proposition 6.3.

The following Table 1 summarizes the classification of curves $X$ of genus $g$
with small values of $\operatorname{a.\\!irr}_{k}X$ or
$\operatorname{a.\\!irr}_{\bar{k}}X$. We use the following shorthand:

  * •

“covers”: a degree $d$ cover of $\mathbb{P}^{1}$ or a (positive rank) elliptic
curve

  * •

“DF”: a normalization of a Debarre–Fahaloui curve

###### Proof.

If $X$ is not $d$-minimal, then it is a cover of an $s$-minimal curve for some
$s\mid d$. Since there are no $2$-minimal curves by Theorem 3.8, and in our
cases $d\leqslant 5$, we can assume that $X$ is either $d$-minimal or a cover
of a $1$-minimal curve (i.e., $\mathbb{P}^{1}$ or an elliptic curve of
positive rank). From now on, we assume that $X$ is $d$-minimal. If $X$ is not
a Debarre–Fahlaoui curve, but satisfies the condition ($\dagger$ ‣ 4.1), then
by Theorem 6.2, the genus of $X$ is at most $(d-1)(d-2)/2+2$. If ($\dagger$ ‣
4.1) does not hold, then the genus bound from Theorem 4.13 applies. Finally,
when $d=5$ the genus of $X$ is bounded by the Castelnuovo function
$\pi(20,12)=8$ by Lemma 4.12.

Any curve of genus $g$ has geometric gonality at most $\lfloor(g+3)/2\rfloor$
and gonality at most $2g-2$. Combining this with the genus bounds described
above gives the result. ∎

$d$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---
$\operatorname{a.\\!irr}_{\bar{k}}=d$ | covers | covers | covers + DF | covers + DF
$\operatorname{a.\\!irr}_{k}=d$ | covers | covers + DF + $g=3$ | covers + DF + $g=4,5$ | covers + DF + $g=5,6,7,8$

Table 1. The classification of curves with small arithmetic degree of
irrationality

In the case $d=3$ and $g=3$, a $3$-minimal curve $X$ cannot be hyperelliptic,
since any conic with a degree $3$ point is isomorphic to $\mathbb{P}^{1}$.
Hence a $3$-minimal curve $X$ of genus $3$ is isomorphic to a smooth plane
quartic. Since the gonality of a plane quartic is $3$ if and only if it has a
rational point, we see that $X$ must be pointless. In this case,
$\operatorname{Sym}^{3}X(k)=\operatorname{Pic}^{3}X(k)$, and so the Jacobian
of $X$ must have positive rank. Conversely, any such curve with a single
degree $3$ which is not a degree $3$ cover of an elliptic curve (i.e., with
simple Jacobian) is $3$-minimal. Combining this with Proposition 6.3 proves
Theorem 1.3.

##  7\. Questions and Problems

###  7.1. Geometric questions

All of the questions we consider have a geometric analogue, that applies to
curves $X$ over any field $k$, and concerns the existence of abelian
translates in $W_{d}\bar{X}$. The resulting geometric questions are usually
slightly easier then the arithmetic ones.

Given any curve $X$ over any field $k$, we can define

$\overline{\operatorname{a.\\!irr}}\ X=\min\big{(}\operatorname{gon}\bar{X},\
\min(d\colon\exists A\subset W_{d}\bar{X},\ \dim A\geqslant 1)\big{)},$

where (as usual) $A$ denotes an abelian translate. When $k$ is a number field,
the Mordell–Lang conjecture guarantees that
$\overline{\operatorname{a.\\!irr}}\ X=\operatorname{a.\\!irr}_{\bar{k}}X$;
however, this more general definition makes sense for complex curves. It is
thus natural to define the locus $Z_{d}$ of curves $[X]\in M_{g}(\mathbb{C})$
with $\overline{\operatorname{a.\\!irr}}\ X=d$. For
$d\geqslant\lfloor(g+3)/2\rfloor$, the locus $Z_{d}$ is all of
$M_{g}(\mathbb{C})$. If the Main Problem 1.2 has a satisfactory solution, then
the following question should have a negative answer:

######  Question 7.1.

For $d<\lfloor(g+3)/2\rfloor$, can $Z_{d}$ be Zariski-dense in $M_{g}$?

Since any curve with $\overline{\operatorname{a.\\!irr}}\
X<\operatorname{gon}X$ has nonsimple Jacobian, the locus $Z_{d}$ is
necessarily contained in a countable union of subvarieties of $M_{g}$. For $d$
small compared to $g$, the locus $Z_{d}$ can never be Zariski-dense. For
example, the gonality of any curve in $Z_{d}$ is less than $2d$, and so
$Z_{d}$ is not dense when $d<\frac{g}{4}$. One can hope that having a subspace
configuration of the type described in Section 4 is a sufficiently strong
property to force $Z_{d}$ to not be Zariski dense. However, Brill–Noether
theory for multisecants of special linear series is not well-understood. Since
Debarre-Fahlaoui curves are not Zariski dense in $M_{g}$ we can as well study
the density question for the set of non-Debarre-Fahlaoui curves
$Z_{d}^{\prime}\subset Z_{d}$. For a curve in $Z_{d}^{\prime}$ the linear
system $|3D|$ has dimension at least $7$ and has many $d$-secants of
codimension at least $4$. Applying [Far08, Theorem 1] shows that $Z_{d}$ is
not Zariski dense for $d\leq\frac{g}{3}$, improving on the gonality bound.

A geometrically $d$-minimal curve $X$ is a curve with
$\overline{\operatorname{a.\\!irr}}\ X=d$ such that there does not exist a
degree $s\geqslant 2$ covering of curves $X\to Y$ with
$\overline{\operatorname{a.\\!irr}}\ Y=d/s$. The next specific case in which
we don’t know the classification of geometrically $d$-minimal curves is $d=6$
and $g=11$.

######  Question 7.2.

Do there exist geometrically $6$-minimal curves of genus $11$?

Abramovich and Harris claimed [AH91][Theorem 2] that the genus of a
geometrically $d$-minimal curve is bounded by $d(d-1)/2$; however the proof
presented there is incomplete. The key to obtaining this bound is the
ineqality $s(n)\geqslant\min(s(n-1)+1,d-1)$, which we only prove in Corollary
4.9 under the assumption ($\dagger$ ‣ 4.1). It seems likely that the genus
bound holds without additional assumptions.

######  Question 7.3.

Do there exist geometrically $d$-minimal curves with genus larger than
$d(d-1)/2$?

In the geometric situation, it is natural to treat $a=\dim A$ as an extra
parameter together with $d$ and $g$. The key to the proof of the Main Theorem
is Proposition 4.7 concerning the difference between the dimensions $s(n)$ and
$s(n-1)$ of divisor spans in $|nD|$ and $|(n-1)D|$. It seems likely that such
a bound can be strengthened to depend on $a$.

######  Question 7.4.

If $\dim A=a$, is it true that when $s(n)\leqslant d-1$, we have
$s(n)-s(n-1)\geqslant a$ for $n\geqslant 3$?

If Question 7.4 has a positive answer, then one can obtained significant
improvements on the genus bound for $d$-minimal curves with $a>1$. In
particular it would imply that the $a=2$-family constructed by Debarre and
Fahlaoui [DF93] achieves the largest possible genus. Such an estimate is
claimed in [AH91], but the proof has a gap (as remarked in [DF93]).

The problem of classifying curves with $\overline{\operatorname{a.\\!irr}}\
X=d$ is interesting over any field $k$. The results of this paper use the
assumption $\operatorname{char}k=0$ in a few places. For instance, Kani’s
version of de Franchis theorem (Theorem 3.1) requires separability assumption
in positive characteristic, the classification of small Sylvester-Gallai
configurations used in the proof of Theorem 5.8 is more complicated when
$\operatorname{char}k=2,3$, and the monodromy argument in Lemma 5.6 only works
in characteristic zero.

######  Question 7.5.

Do the geometric analogues of our main Theorems 4.13, 5.8 hold in positive
characteristic?

###  7.2. Arithmetic Questions

The smallest cases for which some questions are still left open concern
$d$-minimal curves with $d=3$ and $g=3,4$. In both of these cases we expect
that $3$-minimal curves exist. The case $d=g=3$ is discussed in Problem 7.7.
In the case $d=3,g=4$, Proposition 6.3 implies that such a curve is a Debarre-
Fahlaoui curve of class $4H-F$. However, a genus $4$ curve $X$ over
${\bar{k}}$ generically admits two maps of degree $3$ to $\mathbb{P}^{1}$: the
canonical embedding realizes $X$ as a complete intersection of a qudric and a
cubic, and projections from rulings on the quadric give a pair of
$g^{1}_{3}$’s. We expect that there exist Debarre-Fahlaoui curves in class
$4H-F$ for which these two maps are Galois conjugate, and that such Debarre-
Fahlaoui curves are $3$-minimal. It is in principle possible to verify this
claim (if true) by exhibiting a specific curve on $\operatorname{Sym}^{2}A$
and checking that the unique quadric containing the canonical curve is
nonsplit (and independently verifying that it is not a triple cover of an
elliptic curve), but such a computation is nontrivial in practice. We thus
leave this question as a problem.

######  Problem 7.6.

Show that a general curve in numerical class $4H-F$ on
$\operatorname{Sym}^{2}A$ is $3$-minimal.

There is another natural source of low degree points on curves of genus $g$,
as we now describe. Consider a general (in a non-technical sense) genus $g$
curve $X$ over a number field equipped with a degree $g$ point. The abelian
variety $\operatorname{Pic}^{g}X$ may have positive rank, at the same time it
appears that there is no clear reason for $X$ to have maps to other curves or
low gonality. If this is the case, then $\operatorname{Sym}^{g}X$, which is
birational to $\operatorname{Pic}^{g}X$, will have an infinite family of
rational points. While we expect such curves to be abundant, we do not know if
examples can be proved to exist, and thus leave this as a problem.

######  Problem 7.7.

Show that for every $d\geqslant 3$ there exists a $d$-minimal curve of genus
$d.$

The problem for $d=3$ (i.e., smooth plane quartics) is already interesting and
should be computationally feasible.

Theorem 4.13 shows that, under condition ($\dagger$ ‣ 4.1), the genus of a
$d$-minimal curve is bounded by $\frac{d(d-1)}{2}+1$; curiously, this number
is exactly the (maximal) genus of a Debarre-Fahlaoui curve, and in Theorem 5.8
we explained this conincidence by identifying curves with $r(2)=2$ with
Debarre-Fahlaoui curves. Question 7.4 predicts a similar situation for $\dim
A=2$: the maximal genus for such a curve is $d^{2}/4+1$, which is exactly the
genus of $\dim A=2$ examples constructed in [DF93]. We hope the this can also
be explained in terms of geometry of configurations.

######  Question 7.8.

Is it true that a $d$-minimal curve with $\dim A=2$ and $r(2)=3$ is birational
to one of the curves constructed in [DF93]?

It would be very interesting to obtain better results for special curves; in
particular, we do not expect Theorem 6.1 to be close to optimal. One way to
test optimality of such a genus bound is to compare it to known results in low
dimension.

######  Problem 7.9.

Suppose $X$ is a curve equipped with a $g^{r}_{e}$ linear system. Show that
for a certain function $g(r,e,d)$ the following holds: if the genus of $X$ is
larger than $g(r,e,d)$, and $X$ has infinitely many points of degree $d$, then
all but finitely many of those points are contained in the divisors from
$g^{r}_{e}$. Can the function $g(r,e,d)$ be such that the value of $g(1,e,d)$
implies Vojta’s estimate [Voj92] for low degree covers of $\mathbb{P}^{1}$,
and the value of $g(2,e,d)$ implies the Debarre-Klassen theorem [DK94] on
smooth plane curves?

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