# Distinguishing Brill–Noether loci

Asher Auel  A. Auel, Department of Mathematics
Dartmouth College
Kemeny Hall
Hanover, NH 03755<EMAIL_ADDRESS>,  Richard Haburcak  R. Haburcak,
Department of Mathematics
Dartmouth College
Kemeny Hall
Hanover, NH 03755  Max-Planck-Institut für Mathematik
Vivatsgasse 7
53111 Bonn
Germany<EMAIL_ADDRESS>and  Andreas Leopold Knutsen  A. L.
Knutsen, Department of Mathematics, University of Bergen, Postboks 7800, 5020
Bergen, Norway<EMAIL_ADDRESS>
###### Abstract.

We construct curves carrying certain special linear series and not others,
showing many non-containments between Brill–Noether loci in the moduli space
of curves. In particular, we prove the Maximal Brill–Noether Loci conjecture
in full generality.

## Introduction

Whereas classical Brill–Noether theory studies linear systems on general
algebraic curves, _refined Brill–Noether theory_ aims to characterize linear
systems on special curves. A degree $d$ linear system of dimension $r$, called
a $g^{r}_{d}$, corresponds to a non-degenerate morphism $C\to\mathbb{P}^{r}$
of degree $d$ when it is base point free. The Brill–Noether–Petri theorem [19,
22, 30] states that a general smooth curve $C$ of genus $g$ admits a
$g^{r}_{d}$ if and only if the _Brill–Noether number_

$\rho(g,r,d)=g-(r+1)(g-d+r)$

is non-negative. The last few years have seen a flurry of results concerning
refined Brill–Noether theory of curves in a fixed _Brill–Noether locus_

$\mathcal{M}^{r}_{g,d}=\\{C\in{\mathcal{M}}_{g}~{}:~{}C\text{ admits a
}g^{r}_{d}\\}$

when $\rho(g,r,d)<0$. In particular, major advances in refined Brill–Noether
theory for curves of fixed gonality, i.e., when $r=1$, have been made in [11,
25, 28, 29, 39].

Part of a refined Brill–Noether theory concerns the question of whether a
“general” curve with a $g^{r}_{d}$ carries another
$g^{r^{\prime}}_{d^{\prime}}$, which can be reinterpreted in terms of
containments of Brill–Noether loci. For example, Clifford’s theorem implies
that $\mathcal{M}^{r}_{g,2r}\subset\mathcal{M}^{1}_{g,2}$ for every $r\geq 1$.
By adding base points and subtracting non-base points, one obtains the
_trivial containments_
${\mathcal{M}}^{r}_{g,d}\subset{\mathcal{M}}^{r}_{g,d+1}$ and
${\mathcal{M}}^{r}_{g,d}\subset{\mathcal{M}}^{r-1}_{g,d-1}$ between
Brill–Noether loci. The _expected maximal_ Brill–Noether loci are those that
do not admit further trivial containments, see Definition 1.1 for the precise
definition.

The interaction between various Brill–Noether loci is useful in the study of
the birational geometry of ${\mathcal{M}}_{g}$, see [17, 24]. When $\rho=-1$,
the Brill–Noether loci are irreducible divisors, which have been studied by
Harris, Mumford, Eisenbud, and Farkas [14, 15, 17, 18, 24]. A crucial
ingredient in the study of the Kodaira dimension of ${\mathcal{M}}_{23}$ was
the maximality of the Brill–Noether divisors.

Inspired by the lifting of line bundles on curves on K3 surfaces, the
Donagi–Morrison conjecture, building on work of Farkas and Lelli-Chiesa [17,
18, 31], and classical results in Brill–Noether theory, the first two authors
[2] formulated a conjecture stating that the expected maximal Brill–Noether
loci are exactly the maximal ones with respect to containment, except
precisely in genus $g=7,8,9$, where there are exceptional cases, see
Conjecture 1.2, and for details on the exceptional genera see Example 1.3
below. In other words, given any two expected maximal Brill–Noether loci
$\mathcal{M}^{r}_{g,d}$ and $\mathcal{M}^{r^{\prime}}_{g,d^{\prime}}$, there
is a curve $C\in{\mathcal{M}}_{g}$ admitting a $g^{r}_{d}$ but not a
$g^{r^{\prime}}_{d^{\prime}}$. An appealing aspect of the conjecture is a
numerical characterization of the maximal elements of the Brill–Noether
stratification of ${\mathcal{M}}_{g}$. There has been recent progress on this
conjecture, referred to as the Maximal Brill–Noether Loci conjecture, in work
of many authors [3, 6, 7, 9, 10, 31, 44], which together show the conjecture
holds in genus $g\leq 23$ and for infinitely many values of $g$.

In this paper, we settle the conjecture in the affirmative. In fact, we prove
something stronger.

######  Theorem 1.

In every genus $g\geq 3$, with $g\neq 7,8,9$, the expected maximal
Brill–Noether loci $\mathcal{M}^{r}_{g,d}$ are all distinct and are the
Brill–Noether loci that are maximal with respect to containment. More
precisely, $\mathcal{M}^{r}_{g,d}$ has a component on which the general curve
does not admit any $g^{r^{\prime}}_{d^{\prime}}$ with $d^{\prime}\leq g-1$,
$\rho(g,r^{\prime},d^{\prime})<0$, and $(r^{\prime},d^{\prime})\neq(r,d)$,
unless $(g,r,d)=(7,2,6),(8,1,4),(9,2,7)$.

We note that this theorem is stronger than the Maximal Brill–Noether Loci
conjecture, as it shows the existence of a component in each expected maximal
$\mathcal{M}^{r}_{g,d}$ demonstrating all non-containments.

The curves in Theorem 1 behave as generally as possible for curves admitting a
$g^{r}_{d}$. Capturing this idea, we say that a curve
$C\in\mathcal{M}^{r}_{g,d}$ is _$g^{r}_{d}$ -general_ if any additional
Brill–Noether special divisors on $C$ are those determined by Serre duality or
by the trivial containments of Brill–Noether loci, see Definition 1.4. We are
naturally led to the following question generalizing the Maximal Brill–Noether
Loci conjecture.

######  Question 1.

Does $\mathcal{M}^{r}_{g,d}$ contain a $g^{r}_{d}$-general curve?

Theorem 1 gives a positive answer for all expected maximal Brill–Noether loci,
unless $g=7,8,9$. Our more general results, see Theorems 2.5 and 5.1, can be
used to address additional cases. Previously, special cases of Question 1 have
been investigated, see [18]. For $r=1$, the refined Brill–Noether theory for
curves of fixed gonality answers this question, see e.g., [2, Proposition
1.6]. Classically, the genus-degree formula for plane curves gives many
examples where $\mathcal{M}^{2}_{g,d}$ does not contain $g^{2}_{d}$-general
curves. More generally, if $C$ can be embedded as a degree $d$ curve in
$\mathbb{P}^{r}$ and admits a $k$-secant $l$-plane, then $C$ also carries a
$g^{r-l-1}_{d-k}$, which, if Brill–Noether special, shows that $C$ is not
$g^{r}_{d}$-general. Thus Theorem 1 shows that curves in maximal Brill–Noether
loci do not admit unexpected secants, cf. [2, Remark 6.5].

######  Corollary 1.

In every genus $g\geq 3$, with $g\neq 7,8,9$, each expected maximal
Brill–Noether locus $\mathcal{M}^{r}_{g,d}$ with $r\geq 3$ has a component for
which a general curve can be embedded as a degree $d$ curve in
${\mathbb{P}}^{r}$ that does not admit any $k$-secant $l$-plane with
$k-(k-l-1)(r-l)<0$.

The main achievement of this paper is to construct smooth curves of genus $g$
carrying a special $g^{r}_{d}$ and no $g^{r^{\prime}}_{d^{\prime}}$ under
certain numerical conditions on the integers $g,r,d,r^{\prime},d^{\prime}$.
Our main result is Theorem 5.1 stating that we can construct such curves on K3
surfaces with the $g^{r}_{d}$ being induced by a line bundle on the surface.
These K3 surfaces live in a divisor $\mathcal{K}^{r}_{g,d}$ in the moduli
space ${\mathcal{K}}_{g}$ of quasi-polarized K3 surface of genus $g$, see
§1.2. Despite the apparent complexity of the numerical conditions in Theorem
5.1, it provides a very efficient tool to prove non-containments of
Brill–Noether loci in a large number of new cases, in particular, allowing us
to deduce Theorem 1. This proof, while inspired by the program initiated in
[2], does not rely on any of the special cases of the Maximal Brill–Noether
Loci conjecture treated in the literature so far, and in particular, shows
that curves on K3 surfaces suffice to distinguish the maximal Brill–Noether
loci.

We briefly explain the main ideas behind the proof of Theorem 5.1. In his
famous proof of the Brill–Noether–Petri theorem in [30], Lazarsfeld showed
that a Brill–Noether special curve on a K3 surface degenerates to a reducible
curve, due to the non-simplicity of the associated Lazarsfeld–Mukai bundle.
For many $(g,r,d)$, we prove, in Proposition 3.4, that for the very general
element $(S,H)\in\mathcal{K}^{g}_{r,d}$, smooth curves $C\in|H|$ degenerate to
a reducible curve $C_{1}\cup C_{2}$ in an essentially unique way, a property
we call _decomposition rigidity_ , cf. Definition 2.3. We prove that along
this unique degeneration, the limit of any Brill–Noether special divisor on
$C$ is highly constrained, allowing us to rule out the existence of additional
Brill–Noether special divisors. More precisely, the genera of the components
$C_{1}$ and $C_{2}$ are $r$ and $g+r-d-1$, respectively, and any Brill–Noether
special divisor $g^{r^{\prime}}_{d^{\prime}}$ on $C$ has “limits”
$g^{r_{1}}_{d_{1}}$ and $g^{r_{2}}_{d_{2}}$ that are Brill–Noether general on
the two components, see Theorem 2.5. This imposes numerical restrictions on
$r_{i}$ and $d_{i}$, and thus on $g,r,d,r^{\prime},d^{\prime}$. In fact, we
show that we can recover a $g^{r^{\prime}}_{d^{\prime}}$ on $C$ if the
numerical constraints are satisfied, so that Theorem 2.5 gives necessary and
sufficient conditions for the existence of a $g^{r^{\prime}}_{d^{\prime}}$ on
a curve in $|C|$. In particular, this can be interpreted as a “regeneration
theorem” for Brill–Noether special linear systems on curves on K3 surfaces.

### Outline

In §1, we recall the definitions of expected maximal Brill–Noether loci, the
history of progress on distinguishing Brill–Noether loci, as well as
background on K3 surfaces and Lazarsfeld–Mukai bundles. In §2, we introduce
the notion of decomposition rigidity, and then prove our main technical
result, Theorem 2.5, in which we study the limiting behavior of Brill–Noether
special divisors on curves on K3 surfaces satisfying decomposition rigidity.
In §3, we give sufficient conditions for K3 surfaces in
${\mathcal{K}}^{r}_{g,d}$ to satisfy decomposition rigidity, which hold for
those associated to expected maximal Brill–Noether loci. In §4, we study the
numerical conditions forced on limits of Brill–Noether special divisors.
Finally, in §5, we prove our main result, Theorem 5.1, and deduce Theorem 1.
We conclude §5 with additional new applications of Theorem 5.1 to non-maximal
Brill–Noether loci.

### Acknowledgments

We would like to thank Steve Fan and Margherita Lelli-Chiesa for helpful
discussions. A.A. received partial support from Simons Foundation grant 712097
and National Science Foundation grant DMS-2200845. R.H. thanks the Hausdorff
Research Institute for Mathematics funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy–EXC-2047/1–390685813, and the Max–Planck Institut für
Mathematik Bonn for their hospitality and financial support. The authors also
thank the Department of Mathematics at Dartmouth College, whose Shapiro
Visitor Program funded a long-term visit by A.L.K. to Dartmouth, where this
work began, as well as the Department of Mathematics at Roma Tre and the Trond
Mohn Foundation’s project “Pure Mathematics in Norway”, which funded,
respective visits by R.H. and A.L.K. to Rome, where this work was continued.

##  1\. Background on Brill–Noether loci and K3 surfaces

###  1.1. Brill–Noether loci

Surprisingly little is known about the geometry of Brill–Noether loci in
general. The expected codimension of $\mathcal{M}^{r}_{g,d}$ in
${\mathcal{M}}_{g}$ is $-\rho(g,r,d)$, and it is known that the codimension of
any component of $\mathcal{M}^{r}_{g,d}$ is at most $-\rho(g,r,d)$. More is
known about the existence of components of expected dimension, see for example
[7, 27, 40, 42, 44]. However, equidimensionality of components is only known
when $-3\leq\rho(g,r,d)\leq-1$ (additionally assuming $g\geq 12$ when
$\rho(g,r,d)=-3$) [13, 43]. Complicating the picture, components of larger
than expected dimension can exist, examples include Castelnuovo curves, see
for example [40, Remark 1.4]. As mentioned, irreducibility is known for all
Brill–Noether loci with $\rho=-1$ by [14] as well as all
$\mathcal{M}^{2}_{g,d}$ with $\rho=-2$ by [9].

There are various containments known among Brill–Noether loci. For example,
Clifford’s theorem implies that
$\mathcal{M}^{r}_{g,2r}\subset\mathcal{M}^{1}_{g,2}$. There are _trivial
containments_ $\mathcal{M}^{r}_{g,d}\subseteq\mathcal{M}^{r}_{g,d+1}$ obtained
by adding a basepoint to a $g^{r}_{d}$ on $C$; and
$\mathcal{M}^{r}_{g,d}\subseteq\mathcal{M}^{r-1}_{g,d-1}$ when $r\geq 2$
obtained by subtracting a non-basepoint, cf. [17, 32]. Modulo these trivial
containments, the first two authors defined in [2] the _expected maximal_
Brill–Noether loci as follows.

######  Definition 1.1.

A Brill–Noether locus ${\mathcal{M}}^{r}_{g,d}$ is said to be expected maximal
if $2\leq d\leq g-1$, $\rho(g,r,d)<0$, $\rho(g,r,d+1)\geq 0$, and
$\rho(g,r-1,d-1)\geq 0$ if $r\geq 2$.

We will say that a triple $(g,r,d)$ is _associated to an expected maximal
Brill–Noether locus_ if $\mathcal{M}^{r}_{g,d}$ is expected maximal.

We remark that, after accounting for Serre duality which gives
$\mathcal{M}^{r}_{g,d}=\mathcal{M}^{g-d+r-1}_{g,2g-2-d}$, every Brill–Noether
locus is contained in at least one expected maximal Brill–Noether locus.

The first two authors then posed a conjecture identifying the maximal
Brill–Noether loci.

######  Conjecture 1.2 ([2, Conjecture 1]).

In every genus $g\geq 3$, the maximal Brill–Noether loci are the expected
maximal loci, except when $g=7,8,9$.

In other words, the expected maximal Brill–Noether loci should be maximal with
respect to containment, except when $g=7,8,9$. In particular, being maximal
with respect to containment asks that for every pair of expected maximal
Brill–Noether loci $\mathcal{M}^{r}_{g,d}$ and
$\mathcal{M}^{r^{\prime}}_{g,d^{\prime}}$, there exists a curve
$C\in\mathcal{M}^{r}_{g,d}$ such that
$C\notin\mathcal{M}^{r^{\prime}}_{g,d^{\prime}}$. A priori, the curve $C$
could depend on the pair of expected maximal Brill–Noether loci. However,
Theorem 1 shows the existence of a component whose general member demonstrates
all of the non-containments.

We recall the details of the exceptional cases.

######  Example 1.3 (Unexpected containments of expected maximal Brill–Noether
loci, cf. [2, Props. 6.2-4]).

We have ${\mathcal{M}}^{2}_{7,6}\subset{\mathcal{M}}^{1}_{7,4}$,
${\mathcal{M}}^{1}_{8,4}\subset{\mathcal{M}}^{2}_{8,7}$ and
${\mathcal{M}}^{2}_{9,7}\subset{\mathcal{M}}^{1}_{9,5}$, and these
containments are strict.

The containment ${\mathcal{M}}^{2}_{7,6}\subset{\mathcal{M}}^{1}_{7,4}$
follows from the fact that a $g^{2}_{6}$ on a smooth curve of genus $7$ cannot
be very ample by the genus formula for plane curves, whence the curve must
have a $g^{1}_{4}$. The containment
${\mathcal{M}}^{2}_{9,7}\subset{\mathcal{M}}^{1}_{9,5}$ follows for the same
reason. (These were pointed out by H. Larson.)

The containment ${\mathcal{M}}^{1}_{8,4}\subset{\mathcal{M}}^{2}_{8,7}$ was
proven in [34, Lemma 3.8]. Let $C$ be a smooth curve with a $g^{1}_{4}$. Its
Serre adjoint is a $g^{4}_{10}$. If it is not very ample, $C$ will have a
$g^{3}_{8}$, whose Serre adjoint is a $g^{2}_{6}$, whence $C$ carries a
$g^{2}_{7}$. If the $g^{4}_{10}$ is very ample, it embeds $C$ into
${\mathbb{P}}^{4}$ as a curve of degree $10$, where it has $8$ trisecant lines
by the Berzolari formula. Hence $C$ has a $g^{2}_{7}$.

Prior to the formulation of Conjecture 1.2 various non-containments between
Brill–Noether loci were known. For example, generalizing work of
Eisenbud–Harris [15] and Farkas [17] in genus $23$, Choi, Kim, and Kim [8, 10,
9] showed that Brill–Noether loci with $\rho=-1,-2$ in every genus have
distinct support. Already, this is sufficient to prove Conjecture 1.2 for any
genus $g$ such that $g+1$ or $g+2$ is of the form
$\operatorname{lcm}(1,2,\dots,n)$ for some $n\geq 3$, the first few being
$g=4,5,10,11,58,59,418,419,838,839,2518,2519,\dotsc$. Choi and Kim [9] showed
that Brill–Noether loci with $\rho=-2$ are never contained in each other nor
in certain other Brill–Noether divisors. Lelli-Chiesa [31] and the first two
authors [2] also showed various non-containments via the lifting of line
bundles on curves on K3 surfaces. As conjectured by Pflueger [39] and proved
by Jensen–Ranganathan [25] and Cook-Powell–Jensen [11], the combinatorial
formula for the expected dimension of Hurwitz–Brill–Noether loci implies that
the expected maximal Brill–Noether loci with $r=1$ are not contained in any
other expected maximal loci, except when $g=8$, see [2, Proposition 1.6]. This
result is now part of the full Brill–Noether theory for curves of fixed
gonality, established by Larson, Larson, and Vogt [28, 29]. More recently, the
gonality stratification of ${\mathcal{M}}_{g}$ (work of Larson and the first
two authors [3]), strata of differentials (work of Bud [6]), and limit linear
series (work of Teixidor i Bigas [44] and Bud and the second author [7]) were
also used to show new non-containments between Brill–Noether loci, which all
together was sufficient to prove Conjecture 1.2 for $g\leq 23$.

The idea of Conjecture 1.2 is that curves in expected maximal Brill–Noether
loci should behave as generally as possible, given that they carry a special
linear system. More generally, we are interested in Brill–Noether special
curves that admit no further Brill–Noether special divisors, of course
allowing those forced by trivial containments of Brill–Noether loci and Serre
duality.

######  Definition 1.4.

We say a curve $C\in\mathcal{M}^{r}_{g,d}$ is _$g^{r}_{d}$ -general_ if the
only Brill–Noether special divisors $g^{r^{\prime}}_{d^{\prime}}$ with
$r^{\prime}\geq 1$ and $d^{\prime}\leq g-1$ on $C$ are of the form
$(r^{\prime},d^{\prime})=(r-i,d-i)$ for $0\leq i\leq r-1$ or
$(r^{\prime},d^{\prime})=(r,d+j)$ for $0\leq j\leq g-1-d$.

In this language, Theorem 1 says that every expected maximal Brill–Noether
locus $\mathcal{M}^{r}_{g,d}$ contains a $g^{r}_{d}$-general curve, unless
$(g,r,d)=(7,2,6),(8,1,4),(9,2,7)$. This leads naturally to Question 1,
describing when other Brill–Noether loci contain a $g^{r}_{d}$-general curve.
We note that containing a $g^{r}_{d}$-general curve is an open condition in
$\mathcal{M}^{r}_{g,d}$. Various non-trivial containments of Brill–Noether
loci give examples of $\mathcal{M}^{r}_{g,d}$ not containing
$g^{r}_{d}$-general curves. The genus-degree formula for plane curves gives
containments of the form
$\mathcal{M}^{2}_{g,d}\subseteq\mathcal{M}^{1}_{g,d-2}$, giving many examples
of $\mathcal{M}^{2}_{g,d}$ not containing $g^{2}_{d}$-general curves.
Generalizing this, the existence of a $k$-secant $l$-plane to the image of a
curve under a $g^{r}_{d}$ gives a $g^{r-l-1}_{d-k}$, which, if Brill–Noether
special, gives further examples of curves that are not $g^{r}_{d}$-general.
The exceptional cases in Example 1.3 to Conjecture 1.2 are exactly of this
form.

The refined Brill–Noether theory for curves of fixed gonality provides a
complete answer to Question 1 for $r=1$. As proved in [25], the general smooth
projective $k$-gonal curve $C$ of genus $g$ admits a $g^{r}_{d}$ if and only
if Pflueger’s Brill–Noether number

$\rho_{k}(g,r,d)\coloneqq\rho(g,r,d)+\underset{0\leq\ell\leq\operatorname{min}\\{r,~{}g-d+r+1\\}}{\operatorname{max}}(g-k-d+2r+1)\ell-\ell^{2}$

is non-negative. Hence $\mathcal{M}^{1}_{g,k}$ contains a $g^{1}_{k}$-general
curve if and only if $\rho_{k}(g,r,d)<0$ for all $(g,r,d)$ associated to
expected maximal Brill–Noether loci with $r\geq 2$.

######  Example 1.5.

The expected maximal locus $\mathcal{M}^{1}_{g,k}$ has
$k=\left\lfloor\frac{g+1}{2}\right\rfloor$, and contains a $g^{1}_{k}$-general
curve unless $g=8$, as shown in [2, Proposition 1.6]. For sub-maximal gonality
strata, the situation becomes more complicated. For example, if
$k=\left\lfloor\frac{g-1}{2}\right\rfloor\geq 2$, then one can verify that
$\mathcal{M}^{1}_{g,k}$ contains a $g^{1}_{k}$-general curve unless
$g=6,7,8,10,11,14$.

We recall the numerical classification of expected maximal Brill–Noether loci
from [3].

######  Lemma 1.6 ([3, Lemma 1.1]).

An expected maximal Brill–Noether locus $\mathcal{M}^{r}_{g,d}$ exists for
some $d$ if and only if

(1) $1\leq
r\leq\begin{cases}\left\lfloor\sqrt{g}\right\rfloor&\text{if~{}}g\geq\left\lfloor\sqrt{g}\right\rfloor^{2}+\left\lfloor\sqrt{g}\right\rfloor\\\
\left\lfloor\sqrt{g}\right\rfloor-1&\text{if~{}}g<\left\lfloor\sqrt{g}\right\rfloor^{2}+\left\lfloor\sqrt{g}\right\rfloor.\end{cases}$

Once a rank $r$ satisfying the conditions of Lemma 1.6 is fixed, the unique
degree $d$ that makes ${\mathcal{M}}^{r}_{g,d}$ expected maximal is the
largest $d$ such that $\rho(g,r,d)<0$, namely

(2) $d=d_{max}(g,r)\coloneqq r+\left\lceil\frac{gr}{r+1}\right\rceil-1.$

In other words, Lemma 1.6 says that the expected maximal Brill–Noether loci in
${\mathcal{M}}_{g}$ are precisely the $\mathcal{M}^{r}_{g,d}$ for $r$
satisfying (1) and $d=d_{max}(g,r)$.

We remark for later use that an immediate consequence of (1) is the inequality

(3) $g\geq r^{2}+r$

for $(g,r,d)$ associated to an expected maximal Brill–Noether locus, which can
also directly be deduced from the facts that $d\leq g-1$ and
$\rho(g,r,d+1)\geq 0$.

###  1.2. K3 surfaces

We will work with quasi-polarized K3 surfaces of genus $g$, that is, with
pairs $(S,H)$ where $S$ is a K3 surface and $H\in\operatorname{Pic}(S)$ is a
primitive big and nef line bundle such that $H^{2}=2g-2$, so that all smooth
irreducible curves in $|H|$ have genus $g$. To distinguish Brill–Noether loci,
we want such curves to carry a $g^{r}_{d}$, which we ensure by specifying the
Picard group. In the moduli space ${\mathcal{K}}_{g}$ of primitively quasi-
polarized K3 surfaces of genus $g$, the Noether–Lefschetz locus consists of K3
surfaces with Picard rank $>1$. Via Hodge theory, the Noether–Lefschetz locus
is a union of countably many irreducible Noether–Lefschetz divisors. For
$g\geq 2$, $r\geq 0$, and $d\geq 0$, we denote by $\mathcal{K}^{r}_{g,d}$ the
Noether–Lefschetz divisor parameterizing quasi-polarized K3 surfaces
$(S,H)\in{\mathcal{K}}_{g}$ such that the lattice
$\Lambda^{r}_{g,d}={\mathbb{Z}}[H]\oplus{\mathbb{Z}}[L]$ with intersection
matrix

(4) $\begin{bmatrix}H^{2}&H\cdot L\\\ L\cdot
H&L^{2}\end{bmatrix}=\begin{bmatrix}2g-2&d\\\ d&2r-2\end{bmatrix}$

admits a primitive embedding in $\operatorname{Pic}(S)$ preserving $H$ (using
the notation from [2, 23]).

It is well known that $\mathcal{K}^{r}_{g,d}$ is non-empty if and only if the
discriminant

(5) $\Delta^{r}_{g,d}\coloneqq 4(g-1)(r-1)-d^{2}<0.$

Indeed, the Hodge index theorem implies $\Delta^{r}_{g,d}<0$ for any
$(S,H)\in\mathcal{K}^{r}_{g,d}$. Surjectivity of the period map (see [33, Thm.
2.9(i)] or [36]) shows there exists a K3 surface $S$ with
$\operatorname{Pic}(S)=\Lambda^{r}_{g,d}$. Acting with _Picard–Lefschetz
reflections_ on $\Lambda^{r}_{g,d}$, and using [4, VIII, Prop. 3.9], we may
assume that $H$ is nef. In particular, for $(g,r,d)$ satisfying (5),
$\mathcal{K}^{r}_{g,d}$ is non-empty and the very general
$(S,H)\in\mathcal{K}^{r}_{g,d}$ has $\operatorname{Pic}(S)=\Lambda^{r}_{g,d}$.
We note that when $\mathcal{K}^{r}_{g,d}$ is non-empty, it is an irreducible
divisor of ${\mathcal{K}}_{g}$, see [37]. We remark that the
${\mathcal{K}}^{r}_{g,d}$ with $\rho(g,r,d)<0$ are precisely the
Noether–Lefschetz divisors parameterizing the Brill–Noether special
primitively polarized K3 surfaces in the sense of Mukai [35], see [21, 23].

Up to Serre duality, we can assume that $(g,r,d)$ associated to a
Brill–Noether locus satisfies $d\leq g-1$. Hence, for distinguishing
Brill–Noether loci, we will focus on K3 surfaces in $\mathcal{K}^{r}_{g,d}$
satisfying

(6) $g\geq 3,\;\;r\geq 1,\;\;2\leq d\leq g-1.$

We are interested in ${\mathcal{K}}^{r}_{g,d}$ because curves in $|H|$ carry a
$g^{r}_{d}$. We make this more precise as follows (cf. [2, Lemma 1.1].

######  Lemma 1.7.

Let $(S,H)\in{\mathcal{K}}^{r}_{g,d}$ and assume that (6) holds. Then any
irreducible $C\in|H|$ has genus $g$ and carries a $g^{r}_{d}$, and
$|\mathscr{O}_{C}(L)|$ is a $g^{r}_{d}$ if and only if
$h^{1}(L)=h^{1}(H-L)=0$.

Furthermore, if $\operatorname{Pic}(S)\simeq\Lambda^{r}_{g,d}$, then $|H|$
contains smooth irreducible curves and $|\mathscr{O}_{C}(L)|$ is a base point
free $g^{r}_{d}$ for any smooth irreducible $C\in|H|$.

###### Proof.

We note that $L^{2}\geq 0$, $(H-L)^{2}=2(g-d-2+r)\geq 2(r-1)\geq 0$, $L\cdot
H=d>0$ and $(H-L)\cdot H=2(g-1)-d\geq g-1\geq 2$, whence by Riemann-Roch and
Serre duality, $L$ and $H-L$ are both effective and nontrivial. In particular,
$h^{0}(L-H)=0$ and $\dim|L|=\frac{1}{2}L^{2}+1+h^{1}(L)=r+h^{1}(L)\geq r$. The
first assertion in the lemma then follows by adjunction and the sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{S}(L-H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{S}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{C}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By classical results [41, Cor. 3.2, § 2.7], a line bundle $M$ on a K3 surface
is globally generated if and only if there exists no curve $\Gamma$ such that
$\Gamma^{2}=-2$ and $\Gamma.M<0$ or $\Gamma^{2}=0$ and $\Gamma\cdot M=1$.
Hence, if $\operatorname{Pic}(S)\simeq\Lambda^{r}_{g,d}$ one may explicitly
check, arguing as in e.g. [26], that $L$ and $H-L$ are globally generated,
except in one case: $g=4r-2$, $d=g-1=4r-3$ and $H\sim 2D+\Gamma$, with $D\sim
L$ or $D\sim H-L$ globally generated and $\Gamma$ a $(-2)$-curve such that
$D\cdot\Gamma=1$; in this case, $\Gamma$ is the base divisor of $|H-D|$. (We
leave the details to the reader, since we will not make use of the last
assertion in the lemma in this work.) Since $\Gamma\cdot H=0$, we see that
$\mathscr{O}_{C}(L)\simeq\mathscr{O}_{C}(H-L)\simeq\mathscr{O}_{C}(D)$ for any
$C\in|H|$. Thus, in all cases, $\mathscr{O}_{C}(L)$ is base point free.

Since $L$ and $H-L$ are part of a basis of $\operatorname{Pic}(S)$, they
cannot be a multiple of an elliptic curve. Therefore, again by classical
results [41, Prop. 2.6], $L$ and $H-L$ are represented by an irreducible curve
whenever they are globally generated, giving the vanishings
$h^{1}(L)=h^{1}(H-L)=0$. In the one remaining case, they have the form
$D+\Gamma$ with $D$ represented by an irreducible curve and $\Gamma$ an
irreducible curve intersecting $D$ in one point. Hence, $h^{1}(D+\Gamma)=0$ in
this case as well, due to $1$-connectedness. A similar reasoning shows that
$H$ is globally generated and represented by an irreducible curve. Hence, by
Bertini and the first part of the lemma, there are smooth irreducible curves
$C\in|H|$, and $|\mathscr{O}_{C}(L)|$ is a $g^{r}_{d}$. ∎

In the notation of [1, 23], let $\pi:{\mathcal{P}}_{g}\to{\mathcal{K}}_{g}$ be
the universal smooth hyperplane section, whose fiber above $(S,H)$ is the set
of smooth irreducible curves in $|H|$, and let
$\phi:{\mathcal{P}}_{g}\to{\mathcal{M}}_{g}$ be the forgetful map. Also denote
by $\pi:{\mathcal{P}}^{r}_{g,d}\to{\mathcal{K}}^{r}_{g,d}$ the restriction to
the divisor ${\mathcal{K}}^{r}_{g,d}\subset{\mathcal{K}}_{g}$. By Lemma 1.7,
the image of $\phi$ restricted to ${\mathcal{P}}^{r}_{g,d}$ lies in
$\mathcal{M}^{r}_{g,d}$. We summarize this in the following diagram.

$\textstyle{\mathcal{P}^{r}_{g,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\pi}$$\textstyle{\mathcal{K}^{r}_{g,d}}$$\textstyle{\mathcal{M}^{r}_{g,d}}$

Our proof of Theorem 1 will show that when restricted to a general element of
${\mathcal{K}}^{r}_{g,d}$, the image of $\phi$ consists of $g^{r}_{d}$-general
curves.

###  1.3. Bundles on K3 surfaces

Let $S$ be a K3 surface and $C\subset S$ a smooth irreducible curve of genus
$g$. A base point free $g^{r}_{d}$ on $C$ gives rise to a _Lazarsfeld–Mukai
bundle_ on $S$. Denoting by $(\mathscr{L},V)$ the $g^{r}_{d}$, where
$\mathscr{L}$ is a line bundle of degree $d$ on $C$ and $V\subset
H^{0}(\mathscr{L})$ is an $(r+1)$-dimensional subspace, the Lazarsfeld–Mukai
bundle $E_{C,(\mathscr{L},V)}$ is defined as the dual of the kernel of the
evaluation map $V\otimes\mathscr{O}_{S}\twoheadrightarrow\mathscr{L}$. We will
use the following well-known facts about $E_{C,(\mathscr{L},V)}$, cf. [30].

  * •

$\operatorname{rk}(E_{C,(\mathscr{L},V)})=r+1$,
$c_{1}(E_{C,(\mathscr{L},V)})=[C]$, $c_{2}(E_{C,(\mathscr{L},V)})=d$;

  * •

$h^{2}(E_{C,(\mathscr{L},V)})=0$;

  * •

$E_{C,(\mathscr{L},V)}$ is globally generated off a finite set;

  * •

$\chi(E_{C,(\mathscr{L},V)}^{\vee}\otimes
E_{C,(\mathscr{L},V)})=2(1-\rho(g,r,d))$. In particular, if $\rho(g,r,d)<0$,
then $E_{C,(\mathscr{L},V)}$ is non-simple.

We conclude with a technical lemma which will be useful.

######  Lemma 1.8.

Let $\mathscr{E}$ be a torsion free coherent sheaf on a K3 surface $S$ that is
globally generated off a finite set such that $H^{2}(\mathscr{E})=0$ and
$c_{1}(\mathscr{E})=E$, where $E$ is a smooth elliptic curve. Then
$\operatorname{rk}\mathscr{E}=1$ and
$\mathscr{E}^{\vee\vee}\simeq\mathscr{O}_{S}(E)$.

###### Proof.

The cokernel of the canonical inclusion
$\mathscr{E}\subset\mathscr{E}^{\vee\vee}$ is supported on a finite set,
whence $H^{2}(\mathscr{E}^{\vee\vee})=0$ and $\mathscr{E}^{\vee\vee}$ is
globally generated off a finite set. If $\operatorname{rk}\mathscr{E}=1$,
there is nothing to show. If $\operatorname{rk}\mathscr{E}\geq 2$, a general
subspace $W\subset H^{0}(\mathscr{E}^{\vee\vee})$ of $\dim
W=\operatorname{rk}\mathscr{E}^{\vee\vee}-1$ gives rise to an evaluation
sequence

(7)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\otimes\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}^{\vee\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{S}(E)\otimes\mathscr{I}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $Z$ is a $0$-dimensional subscheme of $S$ of length
$c_{2}(\mathscr{E}^{\vee\vee})$. Since $\mathscr{E}^{\vee\vee}$ is globally
generated off a finite set we must have
$h^{0}(\mathscr{O}_{S}(E)\otimes\mathscr{I}_{Z})\geq 2$, whence $Z=\emptyset$.
As $\mathscr{O}_{S}(E)$ is globally generated, (7) shows that
$\mathscr{E}^{\vee\vee}$ is also. Hence, by a general position argument, for a
general subspace $V\subset H^{0}(\mathscr{E}^{\vee\vee})$ of $\dim
V=\operatorname{rk}\mathscr{E}^{\vee\vee}$ the evaluation map
$V\otimes\mathscr{O}_{S}\to\mathscr{E}^{\vee\vee}$ drops rank along a smooth
member of $|E|$, which we still denote by $E$, and we have a short exact
sequence

(8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\otimes\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}^{\vee\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

with $\mathscr{A}$ a line bundle on $E$. Dualizing, we obtain

(9)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}^{\vee\vee\vee}\simeq\mathscr{E}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\vee}\otimes\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega_{E}\otimes\mathscr{A}^{\vee}\simeq\mathscr{A}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The sequences (8) and (9) show that both $\mathscr{A}$ and
$\mathscr{A}^{\vee}$ are globally generated, whence
$\mathscr{A}\simeq\mathscr{O}_{E}$. Since
$h^{0}(\mathscr{E}^{\vee})=h^{2}(\mathscr{E}^{\vee\vee})=0$, we get from (9)
that $\operatorname{rk}\mathscr{E}^{\vee\vee}=\dim V\leq
h^{0}(\mathscr{O}_{E})=1$, a contradiction. ∎

##  2\. Brill–Noether special curves on K3 surfaces

The aim of this section is to give a necessary and sufficient criterion to
determine whether a curve on a K3 surface with a special linear series induced
by a line bundle on the surface can contain other special linear series as
well. The main result is summarized in Theorem 2.5 below.

A crucial observation of Lazarsfeld [30, Lemma 1.3] is that a Brill–Noether
special curve $C$ on a K3 surface $S$ admits in its complete linear system
$|C|$ a reducible curve. More precisely, as will also be clear from the proof
of Theorem 2.5, one has an effective decomposition

(10) $C\sim A+B+T,$

with $A$ and $B$ globally generated and nontrivial, and $T$ effective (and
possibly zero). Control over the (self-)intersection numbers of components of
this effective decomposition have been crucial technical inputs into a host of
results concerning lifting linear systems to line bundles on K3 surfaces and
distinguishing Brill–Noether loci, see [2, 5, 12, 31].

For our purposes, the following will be a convenient definition restricting
effective decompositions of $H$.

######  Definition 2.1.

Let $H$ be a divisor on a K3 surface $S$. We say that $H=A+B$ is a _flexible
decomposition_ if $h^{0}(A)\geq 2$, $h^{0}(B)\geq 2$, $A^{2}\geq-2$,
$B^{2}\geq-2$, and at least one of $A$ and $B$ has self-intersection $\geq 0$.

Note that on K3 surfaces containing no $(-2)$-curves the flexible
decompositions are nothing but the effective, nontrivial decompositions.

######  Example 2.2.

For K3 surfaces $(S,H)\in{\mathcal{K}}^{r}_{g,d}$ satisfying (6) we have that
$H\sim L+(H-L)$ is a flexible decomposition. Indeed, since
$(H-L)^{2}=2(g+r-d-2)\geq 0$ and $(H-L)\cdot H=2g-2-d>0$, we get by
Riemann–Roch that $h^{0}(H-L)\geq 2$. Similarly for $L$.

We will henceforth be interested in the uniqueness of flexible decompositions.
The following is a weakening of the condition that $|H|$ contains no reducible
curves from [30, Theorem].

######  Definition 2.3.

For a K3 surface $S$ and a divisor $H$, we say that $(S,H)$ satisfies
_decomposition rigidity_ if $H$ admits at most one flexible decomposition.

We will show in Proposition 3.4 that decomposition rigidity is satisfied for
general members $(S,H)\in{\mathcal{K}}^{r}_{g,d}$ for many $(g,r,d)$, in
particular for $(g,r,d)$ associated to expected maximal Brill–Noether loci
(cf. Lemma 5.3 and Remark 5.5).

The following lemma, that will be central in the proof of Theorem 2.5, shows
the connection between flexible decompositions and the decompositions of
Lazarsfeld’s form (10).

######  Lemma 2.4.

Let $(S,H)\in{\mathcal{K}}^{r}_{g,d}$ satisfying (6) and decomposition
rigidity. If we can write $H\sim A+B+R$, with $A,B,R$ effective and
nontrivial, and $A^{2}\geq 0$, $B^{2}\geq 0$, then either

  1. (a)

$g+4r-2d=4$ and $r>1$, or

  2. (b)

$r=1,\;g=2d,\;A=B=L,\;R^{2}=-2$, $h^{0}(R)=1$ and $L+R$ is nef.

###### Proof.

As $(S,H)$ satisfies decomposition rigidity, Example 2.2 shows that $H\sim
L+(H-L)$ is the unique flexible decomposition. Also note that $h^{0}(A)\geq 2$
and $h^{0}(B)\geq 2$.

We first treat the case $R^{2}\geq-2$.

We start by proving that

(11) $R\cdot A\geq-1\;\;\mbox{and}\;\;R\cdot B\geq-1.$

Indeed, say $R\cdot A\leq-2$. Then $R^{2}=-2$ and nefness of $H$ implies that
$R\cdot B\geq 4$ and one easily computes that $(A-jR)^{2}\geq 0$ for $j=1,2$
and $(B+jR)^{2}>0$ for $j=1,2,3$. Hence,

$H\sim A+(B+R)\sim(A-R)+(B+R)\sim(A-2R)+(B+3R)$

are all flexible decompositions, contradicting decomposition rigidity. Hence,
$R\cdot A\geq-1$, and similarly $R\cdot B\geq-1$, proving (11)

By (11), the two decompositions

$H\sim(A+R)+B\sim A+(R+B)$

are flexible, whence decomposition rigidity yields that $A\sim B$, and both
are linearly equivalent to $L$ or $H-L$, so that $R\sim H-2A\sim\pm(H-2L)$. If
$R^{2}\geq 0$ or more generally if $h^{0}(R)\geq 2$, then also $H\sim 2A+R$ is
a flexible decomposition, whence decomposition rigidity yields that $R\sim A$,
so that $H\sim 3A$, contradicting that primitivity of $H$. Hence it remains to
treat the case $h^{0}(R)=1$ and $R^{2}=-2$. The latter is equivalent to
$g+4r-2d=4$. Hence, if $r>1$, we are in case (a). If $r=1$, then $g=2d$ and we
have $H\sim 2A+R$, with $A\sim L$ or $A\sim H-L$. In the latter case we would
have $2L\sim H+R$, whence

$g=2d=2L\cdot H=H^{2}+R\cdot H=2g-2+R\cdot H,$

so that $R\cdot H=2-g<0$ (as $g\geq 3$), contradicting nefness of $H$. Hence,
$A\sim L$ and $H\sim 2L+R$, which is case (b), where we have left to prove
that $L+R$ is nef. If it were not, there would exist an irreducible
$(-2)$-curve $\Gamma$ such that $\Gamma\cdot(L+R)<0$. Since $H$ is nef, we
must have $\Gamma\cdot L>0$ and $\Gamma\cdot R\leq-2$, whence
$(R-\Gamma)^{2}\geq 0$, yielding the contradiction $h^{0}(R)\geq
h^{0}(R-\Gamma)\geq 2$.

We next treat the case $R^{2}<-2$. Since $R$ is effective, it must contain a
$(-2)$-curve $\Gamma$ in its support such that $\Gamma\cdot R<0$, and
$R-\Gamma$ is still effective and nontrivial. Since $H$ is nef, we must have
$\Gamma\cdot(A+B)\geq 1$, and we can without loss of generality assume that
$\Gamma\cdot A\geq 1$. Then $(A+\Gamma)^{2}\geq 0$ and we may replace the
decomposition $H\sim A+B+R$ with the decomposition $(A+\Gamma)+B+(R-\Gamma)$.
Repeating the process we will eventually reach a decomposition of the form
$H\sim A^{\prime}+B^{\prime}+R^{\prime}$, with
$A^{\prime},B^{\prime},R^{\prime}$ effective and nontrivial,
${A^{\prime}}^{2}\geq 0$, ${B^{\prime}}^{2}\geq 0$ and ${R^{\prime}}^{2}=-2$.
This reduces us to the case treated above, which means that we are either in
case (a) again, or $r=1$, $A^{\prime}\sim B^{\prime}\sim L$ and
$h^{0}(R^{\prime})=1$. We claim that $A\sim A^{\prime}$ and $B\sim B^{\prime}$
(so that we end up in case (b) again). Indeed, assume by abuse of notation
that at the last but one step of the procedure we have $H\sim A+B+R$, and
there is a $(-2)$-curve $\Gamma$ such that $\Gamma\cdot R<0$, $R-\Gamma$ is
still effective and nontrivial and $\Gamma\cdot A>0$. Then $A^{\prime}\sim
A+\Gamma$, $B^{\prime}\sim B$ and $R^{\prime}\sim R-\Gamma$. Since
$0=L^{2}={A^{\prime}}^{2}=(A+\Gamma)^{2}=A^{2}+2(\Gamma\cdot A-1)$, we must
have $A^{2}=0$ and $\Gamma\cdot A=1$. Hence $\Gamma\cdot A^{\prime}=-1$, so
that

$\Gamma\cdot
H=\Gamma\cdot(2A^{\prime}+R^{\prime})=\Gamma\cdot(2A^{\prime}+R-\Gamma)=\Gamma\cdot
R<0,$

contradicting nefness of $H$. ∎

The main result of this section is the following.

######  Theorem 2.5.

Let $(S,H)\in{\mathcal{K}}^{r}_{g,d}$ satisfying (6) and decomposition
rigidity. Assume further that if $r>1$, then $g+4r-2d\neq 4$. Then $|H|$
contains smooth irreducible curves of genus $g$, and for any smooth
irreducible $C\in|H|$, the linear system $|\mathscr{O}_{C}(L)|$ is a base
point free complete $g^{r}_{d}$, which is very ample if $r\geq 3$. Moreover,
there exists a smooth irreducible curve in $|H|$ carrying a
$g^{r^{\prime}}_{d^{\prime}}$ with $\rho(g,r^{\prime},d^{\prime})<0$ if and
only if there exist non-negative integers $r_{1},r_{2},d_{1},d_{2}$ such that

(12) $\displaystyle r_{1}+r_{2}$ $\displaystyle=$ $\displaystyle r^{\prime}-1$
(13) $\displaystyle d_{1}+d_{2}$ $\displaystyle\leq$ $\displaystyle
d^{\prime}-d+2r-2$ (14) $\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\rho(r,r_{1},d_{1})<r$ (15) $\displaystyle 0$
$\displaystyle\leq$ $\displaystyle\rho(g+r-d-1,r_{2},d_{2})<g+r-d-1.$

###### Proof.

As $(S,H)$ satisfies decomposition rigidity, Example 2.2 shows that $H\sim
L+(H-L)$ is the unique flexible decomposition. Lemma 2.4 and our assumptions
imply that $L$ and $H-L$ admit no nontrivial effective decomposition with at
least one summand of non-negative self-intersection except in the case $r=1$,
$g=2d$, $H-L\sim L+R$, with $R^{2}=-2$, $h^{0}(R)=1$ and $L+R$ nef. Therefore,
well-known results on linear systems on K3 surfaces [41, Corollary 3.2, §2.7]
imply that $L$ and $H-L$ are globally generated and the general members of
$|L|$ and $|H-L|$ are smooth irreducible curves of genus $g(L)=r\geq 1$ and
$g(H-L)=g+r-d-1\geq 1$, respectively, and that $L$ is even very ample if
$r\geq 3$. Similarly, the general member of $|H|$ is a smooth, irreducible
curve of genus $g$. In particular, $h^{1}(L)=h^{1}(H-L)=0$, whence
$|\mathscr{O}_{C}(L)|$ is a base point free complete $g^{r}_{d}$ for any
smooth $C\in|H|$ by Lemma 1.7, and it is even is very ample if $r\geq 3$ (as
$L$ is).

We have left to prove the last statement of the proposition. We first prove
the “only if”-part.

Assume that there exists a smooth curve $C\in|H|$ carrying a
$g^{r^{\prime}}_{d^{\prime}}$ with $\rho(g,r^{\prime},d^{\prime})<0$. Its base
point free part is a $g^{r^{\prime}}_{d^{\prime\prime}}$ with
$d^{\prime\prime}\leq d^{\prime}$. Let $\mathscr{E}$ be the Lazarsfeld–Mukai
bundle associated to the base point free part. The fact that
$\rho(g,r^{\prime},d^{\prime\prime})\leq\rho(g,r^{\prime},d^{\prime})<0$
implies that $\mathscr{E}$ is non-simple, whence we can by standard arguments
find an endomorphism $\varphi:\mathscr{E}\to\mathscr{E}$ dropping rank
everywhere (see [30, p. 302]). Set
$\mathscr{F}\coloneqq\operatorname{im}\varphi$ and
$\mathscr{G}\coloneqq\operatorname{coker}\varphi$, which both have positive
ranks. Being both a quotient sheaf and subsheaf of $\mathscr{E}$, we have that
$\mathscr{F}$ is torsion free, globally generated off a finite set and with
$h^{2}(\mathscr{F})=0$. In particular, $\mathscr{F}$ is nontrivial, whence
$c_{1}(\mathscr{F})$ is represented by an effective nonzero divisor $F$ (see,
e.g., [30, Fact at the bottom of p. 302]). Since $F$ must be globally
generated off a finite set, we must have $F^{2}\geq 0$. Let $\mathscr{T}$ be
the (possibly zero) torsion subsheaf of $\mathscr{G}$ and
$\overline{\mathscr{G}}\coloneqq\mathscr{G}/\mathscr{T}$. Then
$\overline{\mathscr{G}}$ is torsion free, and, being a quotient sheaf of
$\mathscr{E}$, it is globally generated off a finite set and satisfies
$h^{2}(\overline{\mathscr{G}})=0$. As above, this implies that
$c_{1}(\overline{\mathscr{G}})$ is represented by an effective nonzero divisor
$G$, with $G^{2}\geq 0$. We thus have

$H\sim c_{1}(\mathscr{E})\sim c_{1}(\mathscr{F})+c_{1}(\mathscr{G})\sim
c_{1}(\mathscr{F})+c_{1}(\overline{\mathscr{G}})+c_{1}(\mathscr{T})\sim
F+G+T,$

with $T$ an effective (possibly zero) divisor supported on the support of
$\mathscr{T}$.

Letting $\overline{\mathscr{F}}$ denote the kernel of the surjection
$\mathscr{E}\to\mathscr{G}\to\overline{\mathscr{G}}$, we have a commutative
diagram

(16) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 33.11111pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.14584pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.64584pt\raise-14.80556pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.47224pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
110.97224pt\raise-15.9611pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
144.98615pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 181.09726pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-24.63887pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.61111pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.38194pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.38194pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.64584pt\raise-39.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
105.47224pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.47224pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{\mathscr{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
110.97224pt\raise-39.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
141.37503pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
141.37503pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
178.59726pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
178.59726pt\raise-24.63887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-3.0pt\raise-49.47218pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
33.11111pt\raise-49.47218pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
67.24306pt\raise-49.47218pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.64584pt\raise-64.47218pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern
104.56946pt\raise-49.47218pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
110.97224pt\raise-65.62773pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
144.98615pt\raise-49.47218pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
181.09726pt\raise-49.47218pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-5.5pt\raise-74.3055pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.72223pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.72223pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.64584pt\raise-89.49992pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
105.47224pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.47224pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{\mathscr{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
110.97224pt\raise-89.49992pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
142.48615pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
142.48615pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern
181.09726pt\raise-74.3055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-98.94437pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
33.11111pt\raise-98.94437pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
68.14584pt\raise-98.94437pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern
105.47224pt\raise-98.94437pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern
144.98615pt\raise-98.94437pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
181.09726pt\raise-98.94437pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

Note that $\overline{\mathscr{F}}$ is torsion free, being a subsheaf of
$\mathscr{E}$. The rightmost vertical sequence in the diagram, together with
the facts that $\mathscr{E}$ is locally free and $\overline{\mathscr{G}}$ is
torsion free, yields that $\overline{\mathscr{F}}$ is _normal_ (cf., e.g.,
[38, II, Lemma 1.1.16]), whence _reflexive_ (cf., e.g., [38, II, Lemma
1.1.12]), whence locally free.

Also note that $h^{2}(\overline{\mathscr{F}})=0$, as $h^{2}(\mathscr{F})=0$.

Consider the standard exact sequence

(17)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathscr{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathscr{G}}^{\vee\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau_{\overline{\mathscr{G}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\overline{\mathscr{G}}^{\vee\vee}$ is locally free and
$\tau_{\overline{\mathscr{G}}}$ has finite support. In particular, also
$\overline{\mathscr{G}}^{\vee\vee}$ is globally generated off a finite set and
$h^{2}(\overline{\mathscr{G}}^{\vee\vee})=0$. Combining the rightmost vertical
sequence in (16) with (17), we obtain

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathscr{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathscr{G}}^{\vee\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau_{\overline{\mathscr{G}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\overline{\mathscr{F}}$ and $\overline{\mathscr{G}}^{\vee\vee}$ are
both locally free, $\overline{\mathscr{G}}^{\vee\vee}$ is globally generated
off a finite set,
$h^{2}(\overline{\mathscr{F}})=h^{2}(\overline{\mathscr{G}}^{\vee\vee})=0$,
and $\tau_{\overline{\mathscr{G}}}$ supported on a finite set.

######  Claim 2.6.

Also $\overline{\mathscr{F}}$ is globally generated off a finite set.
Moreover, $c_{1}(\overline{\mathscr{F}})$ and
$c_{1}(\overline{\mathscr{G}}^{\vee\vee})$ are linearly equivalent to $L$ and
$H-L$.

###### Proof of claim.

If $T=0$, then $\mathscr{T}$ is supported on a finite set. Since $\mathscr{F}$
is also globally generated off finitely many points, the upper horisontal
vertical sequence in (16) shows that $\overline{\mathscr{F}}$ is again
globally generated off a finite set. Moreover, $c_{1}(\overline{\mathscr{F}})$
and $c_{1}(\overline{\mathscr{G}}^{\vee\vee})$ are linearly equivalent to $L$
and $H-L$ by decomposition rigidity.

If $T\neq 0$, then the hypotheses and Lemma 2.4 yield that $r=1$, $g=2d$,
$F\sim G\sim L$ and $T\sim R\sim H-2L$, with $R^{2}=-2$. In particular, $L$ is
represented by a smooth elliptic curve. By Lemma 1.8 we obtain that
$\operatorname{rk}\overline{\mathscr{G}}^{\vee\vee}=\operatorname{rk}\mathscr{G}=1$
and $\operatorname{rk}\overline{\mathscr{F}}=\operatorname{rk}\mathscr{F}=1$,
whence $r^{\prime}=\operatorname{rk}\mathscr{E}-1=1$. Hence
$\overline{\mathscr{G}}^{\vee\vee}\simeq L$ and $\overline{\mathscr{F}}\simeq
H-L$. In particular, $\overline{\mathscr{F}}$ is globally generated and the
claim follows.∎

To simplify notation, we can therefore assume that we have an exact sequence

(18)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with $\mathscr{F}$ and $\mathscr{G}$ both locally free of positive ranks and
globally generated off finite sets, such that
$h^{2}(\mathscr{F})=h^{2}(\mathscr{G})=0$ and $c_{1}(\mathscr{F})$ and
$c_{1}(\mathscr{G})$ are represented by $L$ and $H-L$, and $\tau$ is supported
on a finite set.

We set $r_{\mathscr{F}}\coloneqq\operatorname{rk}\mathscr{F}-1$ and
$r_{\mathscr{G}}\coloneqq\operatorname{rk}\mathscr{G}-1$,
$d_{\mathscr{F}}\coloneqq c_{2}(\mathscr{F})$, $d_{\mathscr{G}}\coloneqq
c_{2}(\mathscr{G})$,
$g_{\mathscr{F}}\coloneqq\frac{1}{2}c_{1}(\mathscr{F})^{2}+1$ and
$g_{\mathscr{G}}\coloneqq\frac{1}{2}c_{1}(\mathscr{G})^{2}+1$. Then
$r_{\mathscr{F}}\geq 0$ and $r_{\mathscr{G}}\geq 0$. Since $\mathscr{F}$ and
$\mathscr{G}$ are globally generated off finite sets, we also have
$d_{\mathscr{F}}\geq 0$ and $d_{\mathscr{G}}\geq 0$. Since
$\\{c_{1}(\mathscr{F}),c_{1}(\mathscr{G})\\}=\\{L,H-L\\}$, we have

(19)
$\\{g_{\mathscr{F}},g_{\mathscr{G}}\\}=\\{g(L),g(H-L)\\}=\\{r,g+r-d-1\\}.$

From (18) we find

$r^{\prime}+1=\operatorname{rk}\mathscr{E}=\operatorname{rk}\mathscr{F}+\operatorname{rk}\mathscr{G}=r_{\mathscr{F}}+1+r_{\mathscr{G}}+1,$

that is,

(20) $r_{\mathscr{F}}+r_{\mathscr{G}}=r^{\prime}-1;$

moreover,

$\displaystyle d^{\prime}$ $\displaystyle\geq$ $\displaystyle
d^{\prime\prime}=c_{2}(\mathscr{E})=c_{2}(\mathscr{F})+c_{2}(\mathscr{G})+c_{1}(\mathscr{F})\cdot
c_{1}(\mathscr{G})+\operatorname{length}\tau\geq
d_{\mathscr{F}}+d_{\mathscr{G}}+L\cdot(H-L)$ $\displaystyle=$ $\displaystyle
d_{\mathscr{F}}+d_{\mathscr{G}}+d-2r+2,$

in other words

(21) $d_{\mathscr{F}}+d_{\mathscr{G}}\leq d^{\prime}-d+2r-2.$

If $r_{\mathscr{F}}=0$, then $\operatorname{rk}\mathscr{F}=1$, so that
$d_{\mathscr{F}}=c_{2}(\mathscr{F})=0$. This implies that
$\rho(g_{\mathscr{F}},r_{\mathscr{F}},d_{\mathscr{F}})=\rho(g_{\mathscr{F}},0,0)=0$.
If $r_{\mathscr{F}}>0$, then, as $\mathscr{F}$ is globally generated off a
finite set, we have a short exact sequence

(22)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}^{r_{\mathscr{F}}+1}\otimes\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}_{\mathscr{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

with $\mathscr{A}_{\mathscr{F}}$ a torsion free rank-one sheaf on a reduced
and irreducible member $D\in|c_{1}(\mathscr{F})|$. As $h^{2}(\mathscr{F})=0$,
the sheaf $\mathscr{F}$ is nontrivial, whence
$h^{0}(\mathscr{F})>\operatorname{rk}(\mathscr{F})=r_{\mathscr{F}}+1$, so that

(23) $h^{0}(\mathscr{A}_{\mathscr{F}})>0.$

Dualizing (22), we obtain

(24)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}^{r_{\mathscr{F}}+1}\otimes\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{B}_{\mathscr{F}}\coloneqq\operatorname{\mathscr{E}\\!\mathit{xt}}^{1}(\mathscr{A}_{\mathscr{F}},\mathscr{O}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

again with $\mathscr{B}_{\mathscr{F}}$ a torsion free rank-one sheaf on $D$,
which is globally generated by construction and satisfies
$\deg(\mathscr{B}_{\mathscr{F}})=c_{2}(\mathscr{F})=d_{\mathscr{F}}$. Since
$h^{0}(\mathscr{F}^{\vee})=h^{2}(\mathscr{F})=0$, we see that
$h^{0}(\mathscr{B}_{\mathscr{F}})=r_{\mathscr{F}}+1+h^{1}(\mathscr{F})$. Now
choose $h^{1}(\mathscr{F})$ general points
$x_{1},\ldots,x_{h^{1}(\mathscr{F})}$ in the smooth locus of $D$ and set
$\mathscr{B}^{\prime}_{\mathscr{F}}\coloneqq\mathscr{B}_{\mathscr{F}}(-x_{1}-\cdots-
x_{h^{1}(\mathscr{F})})$. Then

$\displaystyle h^{0}(\mathscr{B}^{\prime}_{\mathscr{F}})$ $\displaystyle=$
$\displaystyle
h^{0}(\mathscr{B}_{\mathscr{F}})-h^{1}(\mathscr{F})=r_{\mathscr{F}}+1,$
$\displaystyle h^{1}(\mathscr{B}^{\prime}_{\mathscr{F}})$ $\displaystyle=$
$\displaystyle
h^{1}(\mathscr{B}_{\mathscr{F}})=h^{0}(\operatorname{\mathscr{E}\\!\mathit{xt}}^{1}(\mathscr{B}_{\mathscr{F}},\mathscr{O}_{S}))=h^{0}(\mathscr{A}_{\mathscr{F}})>0,$

using (23) (cf. [20, Lemmas 2.1 and 2.3]). Moreover,
$\mathscr{B}^{\prime}_{\mathscr{F}}$ is still globally generated, so we have a
short exact sequence

(25)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{F}^{\prime}}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}^{r_{\mathscr{F}}+1}\otimes\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{B}^{\prime}_{\mathscr{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

defining a new Lazarsfeld–Mukai bundle $\mathscr{F}^{\prime}$. We have
$\operatorname{rk}(\mathscr{F}^{\prime})=\operatorname{rk}(\mathscr{F})=r_{\mathscr{F}}+1$,
$c_{1}(\mathscr{F}^{\prime})=c_{1}(\mathscr{F})$ and
$d^{\prime}_{\mathscr{F}}\coloneqq
c_{2}(\mathscr{F}^{\prime})=\deg(\mathscr{B}^{\prime}_{\mathscr{F}})=d_{\mathscr{F}}-h^{1}(\mathscr{F})\leq
d_{\mathscr{F}}$. Dualizing (25) we obtain

(26)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}^{r_{\mathscr{F}}+1}\otimes\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{F}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathscr{E}\\!\mathit{xt}}^{1}(\mathscr{B}^{\prime}_{\mathscr{F}},\mathscr{O}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

and since
$h^{0}(\operatorname{\mathscr{E}\\!\mathit{xt}}^{1}(\mathscr{B}^{\prime}_{\mathscr{F}},\mathscr{O}_{S}))=h^{1}(\mathscr{B}^{\prime}_{\mathscr{F}})>0$
(cf. [20, Lemma 2.3]), we see that $\mathscr{F}^{\prime}$ is globally
generated off a finite set. Recall that $D\sim L$ or $H-L$, and, as a
consequence of what we said in the first lines of the proof, $L$ and $H-L$
admit no decompositions in moving classes. Hence

$\rho(g_{\mathscr{F}},r_{\mathscr{F}},d^{\prime}_{\mathscr{F}})\geq 0,$

as in the proof of [30, Lemma 1.3]. At the same time, since
$h^{0}(\mathscr{B}^{\prime}_{\mathscr{F}})>0$ and
$h^{1}(\mathscr{B}^{\prime}_{\mathscr{F}})>0$, we also have

$\rho(g_{\mathscr{F}},r_{\mathscr{F}},d^{\prime}_{\mathscr{F}})=g(D)-h^{0}(\mathscr{B}^{\prime}_{\mathscr{F}})h^{1}(\mathscr{B}^{\prime}_{\mathscr{F}})<g(D)=g(\mathscr{F}).$

To summarize, we have in any event found an integer $d^{\prime}_{\mathscr{F}}$
such that $0\leq d^{\prime}_{\mathscr{F}}\leq d_{\mathscr{F}}$ and

(27)
$0\leq\rho(g_{\mathscr{F}},r_{\mathscr{F}},d^{\prime}_{\mathscr{F}})<g_{\mathscr{F}}.$

Similarly, considering $\mathscr{G}$, we find an integer
$d^{\prime}_{\mathscr{G}}$ such that $0\leq
d^{\prime}_{\mathscr{G}^{\prime}}\leq d_{\mathscr{G}}$ and

(28)
$0\leq\rho(g_{\mathscr{G}},r_{\mathscr{G}},d^{\prime}_{\mathscr{G}})<g_{\mathscr{G}}.$

By (21) we see that

(29) $d^{\prime}_{\mathscr{F}}+d^{\prime}_{\mathscr{G}}\leq
d^{\prime}-d+2r-2.$

Recalling (19), we see that (20), (29) and (27)-(28) imply (12), (13) and
(14)-(15), with $\\{r_{1},r_{2}\\}=\\{r_{\mathscr{F}},r_{\mathscr{G}}\\}$ and
$\\{d_{1},d_{2}\\}=\\{d^{\prime}_{\mathscr{F}},d^{\prime}_{\mathscr{G}}\\}$.

We then prove the “if”-part.

Assume the existence of the integers as stated in the theorem. Assume first
that $r_{1},r_{2}>0$. Then, by (14)-(15), on any smooth curve in $|L|$ and
$|H-L|$ there exist a complete special $g^{r_{1}}_{d_{1}}$ and
$g^{r_{2}}_{d_{2}}$, respectively. Denote by $\mathscr{A}_{i}$, $i=1,2$, the
corresponding line bundles, and by $D_{i}$ the corresponding curves. Define
$\mathscr{A}^{\prime}_{i}$ to be the base point free part of
$\mathscr{A}_{i}$, that is, the image of the evaluation map
$H^{0}(\mathscr{A}_{i})\otimes\mathscr{O}_{D_{i}}\to\mathscr{A}_{i}$, by
$\mathscr{B}_{i}$ the base point free part of
$\omega_{D_{i}}\otimes{\mathscr{A}^{\prime}}_{i}^{-1}$, and set
$\overline{\mathscr{A}}_{i}\coloneqq\omega_{D_{i}}\otimes\mathscr{B}_{i}^{-1}$.
Then $h^{0}(\overline{\mathscr{A}}_{i})=h^{0}(\mathscr{A}_{i})+l_{i}$ and
$\deg(\overline{\mathscr{A}}_{i})=\deg(\mathscr{A}^{\prime}_{i})+l_{i}\leq
d_{i}+l_{i}$, where $l_{i}$ is the length of the base locus of
$|\omega_{D_{i}}\otimes{\mathscr{A}^{\prime}}_{i}^{-1}|$. One easily checks,
as e.g. in [20, Lemma 3.1], that both $|\overline{\mathscr{A}}_{i}|$ and
$|\omega_{D_{i}}\otimes\overline{\mathscr{A}}_{i}^{-1}|$ are base point free.
Hence, we have found, on (all) smooth curves in $|L|$ and $|H-L|$, a base
point free complete $g^{r_{i}+l_{i}}_{d^{\prime}_{i}+l_{i}}$, with
$d^{\prime}_{i}\leq d_{i}$ and $l_{i}\geq 0$, for $i=1,2$, respectively, and
such that its adjoint is also base point free.

Let $\mathscr{F}$ and $\mathscr{G}$ be the associated Lazarsfeld–Mukai
bundles, of ranks $r_{1}+l_{1}+1$ and $r_{2}+l_{2}+1$, respectively, which are
globally generated by the assumption on the base point freeness of the adjoint
linear systems. Set $\mathscr{E}\coloneqq\mathscr{F}\oplus\mathscr{G}$. Then

$\operatorname{rk}(\mathscr{E})=r_{1}+l_{1}+r_{2}+l_{2}+2=r^{\prime}+1+l_{1}+l_{2}$

by (12) and

$\displaystyle c_{2}(\mathscr{E})$ $\displaystyle=$ $\displaystyle
c_{2}(\mathscr{F})+c_{2}(\mathscr{G})+c_{1}(\mathscr{F})\cdot
c_{1}(\mathscr{G})=d^{\prime}_{1}+l_{1}+d^{\prime}_{2}+l_{2}+L\cdot(H-L)$
$\displaystyle=$ $\displaystyle
d^{\prime}_{1}+l_{1}+d^{\prime}_{2}+l_{2}+d-2r+2\leq
d_{1}+l_{1}+d_{2}+l_{2}+d-2r+2$ $\displaystyle\leq$ $\displaystyle
d^{\prime}+l_{1}+l_{2},$

by (13). Then, as is well-known, the evaluation map
${\mathbb{C}}^{r^{\prime}+1}\otimes\mathscr{O}_{S}\to\mathscr{E}$ drops rank
along a smooth curve $C\in|H|$ and the cokernel is a line bundle $\mathscr{A}$
on $C$, such that $|\omega_{C}-\mathscr{A}|$ is a
$g^{r^{\prime}+l_{1}+l_{2}}_{c_{2}(\mathscr{E})}$; in particular $C$ carries a
$g^{r^{\prime}+l_{1}+l_{2}}_{d^{\prime}+l_{1}+l_{2}}$, whence also a
$g^{r^{\prime}}_{d^{\prime}}$.

Assume now that $r_{1}=0$ and $r_{2}>0$. Then by (15) there exist a complete
special $g^{r_{2}}_{d_{2}}$ on any smooth curve in $|H-L|$, and as before, we
find a complete, base point free $g^{r_{2}+l_{2}}_{d^{\prime}_{2}+l_{2}}$ with
$d^{\prime}_{2}\leq d_{2}$ and $l_{2}\geq 0$, such that its adjoint is again
globally generated. Letting $\mathscr{F}$ be its associated Lazarsfeld–Mukai
bundle of rank $r_{2}+l_{2}+1$, which again is globally generated, we set
$\mathscr{E}\coloneqq L\oplus\mathscr{F}$, and argue as above. Similarly if
$r_{2}=0$ and $r_{1}>0$.

Finally, if $r_{1}=r_{2}=0$, we set $\mathscr{E}=L\oplus(H-L)$ and argue as
before. ∎

######  Remark 2.7.

A careful look at the proofs of Lemma 2.4 and Theorem 2.5 shows that the
condition “$(S,H)\in{\mathcal{K}}^{r}_{g,d}$” can be replaced by “$(S,H)$ a
quasi-polarized K3 surface of genus $g$ with a line bundle
$L\in\operatorname{Pic}(S)$ satisfying $L^{2}=2r-2$, $L.H=d$ and $H\not\sim
3L$”. Also note that whenever $H$ is $n$-divisible in $\operatorname{Pic}(S)$
for an $n\geq 4$, then $(S,H)$ does not satisfy decomposition rigidity.

##  3\.  Flexible decompositions on K3s of Picard number two

We want to find K3 surfaces satisfying the conditions in Theorem 2.5, in
particular decomposition rigidity. Throughout the section we will fix integers
$(g,r,d)$ satisfying (5) and (6), and a pair $(S,H)\in{\mathcal{K}}^{r}_{g,d}$
with $\operatorname{Pic}(S)=\Lambda^{r}_{g,d}$. For simplicity we write
$\Delta\coloneqq\Delta^{r}_{g,d}$.

Assume now that $H\sim A+B$ is a flexible decomposition of $H$. We may write

$A=xH-yL\;\;\mbox{and}\;\;B=(1-x)H+yL\;\;\mbox{for
some}\;\;x,y\in{\mathbb{Z}}.$

We may and will assume that $A^{2}\geq 0$ and $B^{2}\geq-2$. The next two
lemmas give necessary conditions on $x$ and $y$.

######  Lemma 3.1.

We have

(30) $\displaystyle(g-1)x^{2}-dxy+(r-1)y^{2}\geq 0,$ (31)
$\displaystyle(g-1)(1-x)^{2}+d(1-x)y+(r-1)y^{2}\geq-1,$ (32)
$\displaystyle\frac{2(g-1)}{d}(x-1)<y<\frac{2(g-1)}{d}x.$

###### Proof.

The two first conditions are equivalent to $\frac{1}{2}A^{2}\geq 0$ and
$\frac{1}{2}B^{2}\geq-1$, respectively. The last condition is equivalent to
$A\cdot H>0$ and $B\cdot H>0$, which are satisfied since $h^{0}(A)\geq 2$ and
$h^{0}(B)\geq 2$ and $H$ is big and nef. ∎

######  Lemma 3.2.

If $r=1$, then

$x=0\;\;\mbox{and}\;\;-\frac{g}{d}\leq
y<0,\;\;\;\;\;\mbox{or}\;\;x=1\;\;\mbox{and }\;\;0<y\leq\frac{g-1}{d}.$

If $r\geq 2$, then either

(33) $\displaystyle
0<x\leq\operatorname{max}\left\\{1+\frac{d}{\sqrt{|\Delta|}\sqrt{g-1}}~{},~{}\frac{2(r-1)g}{(d-\sqrt{|\Delta|})\sqrt{|\Delta|}}\right\\}\;\;\mbox{and}$
$\displaystyle\frac{2(g-1)}{d}(x-1)<y\leq\operatorname{min}\left\\{\frac{d-\sqrt{|\Delta|}}{2(r-1)}x\;,\;\frac{g}{\sqrt{|\Delta|}}\right\\}$

or

(34) $\displaystyle-\frac{2(r-1)g}{(d+\sqrt{|\Delta|})\sqrt{|\Delta|}}\leq
x\leq 0\;\;\mbox{and}$
$\displaystyle\operatorname{max}\left\\{-\frac{g}{\sqrt{|\Delta|}}\;,\;\frac{2(g-1)}{d}(x-1)\right\\}\leq
y\leq\frac{d+\sqrt{|\Delta|}}{2(r-1)}x,\;\;y\neq
0,\;\;y\neq\frac{2(g-1)}{d}(x-1).$

###### Proof.

We first treat the case $r=1$.

If $x<0$, then (30) and the right inequality in (32) yield

$(g-1)x\leq dy<2(g-1)x,$

whence the contradiction $(g-1)x>0$. If $x>0$, then (30) and the left
inequality in (32) yield

$2(g-1)(x-1)<dy\leq(g-1)x,$

which implies $(g-1)x<2(g-1)$, whence $x=1$, and thus $0<y\leq\frac{g-1}{d}$.

If $x=0$, then (31) and the left inequality in (32) yield $-\frac{g}{d}\leq
y<0$.

The rest of the proof will deal with the case $r\geq 2$.

We first note that (32) implies that

(35) either $x\leq 0$ and $y<0$, or $x>0$ and $y>0$.

Letting $a_{\pm}\coloneqq\frac{d\pm\sqrt{|\Delta|}}{2(r-1)}$, we note that
(30) factors as

(36) $(r-1)(y-a_{+}x)(y-a_{-}x)\geq 0.$

Define the following lines in the plane:

$\ell_{+}:\;y=a_{+}x,\;\;\;\;\ell_{-}:\;y=a_{-}x,\;\;\;\;\ell:\;y=\frac{2(g-1)}{d}(x-1).$

Since $a_{+}>\frac{2(g-1)}{d}>a_{-}>0$, the conditions (36) and (32) yield,
also recalling (35), that either

(37) $x>0\;\;\mbox{and}\;\;0\leq\frac{2(g-1)}{d}(x-1)<y\leq a_{-}x$

(which determines a triangle bounded by the $x$-axis and the lines $\ell$ and
$\ell_{-}$ in the first quadrant) or

(38) $x\leq 0\;\;\mbox{and}\;\;\frac{2(g-1)}{d}(x-1)<y\leq a_{+}x,\;\;y<0$

(which determines a triangle bounded by the $y$-axis and the lines $\ell$ and
$\ell_{+}$ in the third quadrant). Furthermore, condition (31) implies that
the points $(x,y)$ must lie in the region containing the origin determined by
the hyperbola with two branches

$\mathfrak{c}:\;(g-1)(1-x)^{2}+d(1-x)y+(r-1)y^{2}=-1,$

as shown in the following picture, which shows the case $(g,r,d)=(14,3,13)$.

We compute the intersection points between the curves, as shown in the
picture:

$\displaystyle Q_{+}=(x_{+},y_{+})\coloneqq\ell\cap\ell_{-}$ $\displaystyle=$
$\displaystyle\left(1+\frac{d}{\sqrt{|\Delta|}},\frac{2(g-1)}{\sqrt{|\Delta|}}\right),$
$\displaystyle Q_{-}=(x_{-},y_{-})\coloneqq\ell\cap\ell_{-}$ $\displaystyle=$
$\displaystyle\left(1-\frac{d}{\sqrt{|\Delta|}},-\frac{2(g-1)}{\sqrt{|\Delta|}}\right),$
$\displaystyle P_{1}=(x_{1},y_{1})\coloneqq\mathfrak{c}\cap\ell_{-}$
$\displaystyle=$
$\displaystyle\left(\frac{2(r-1)g}{(d-\sqrt{|\Delta|})\sqrt{|\Delta|}}~{},~{}\frac{g}{\sqrt{|\Delta|}}\right),$
$\displaystyle P_{4}=(x_{4},y_{4})\coloneqq\mathfrak{c}\cap\ell_{+}$
$\displaystyle=$
$\displaystyle\left(-\frac{2(r-1)g}{(d+\sqrt{|\Delta|})\sqrt{|\Delta|}}~{},~{}-\frac{g}{\sqrt{|\Delta|}}\right),$
$\displaystyle P_{2}=(x_{2},y_{2})\coloneqq\mathfrak{c}\cap\ell\cap\\{y\geq
0\\}$ $\displaystyle=$
$\displaystyle\left(1+\frac{d}{\sqrt{g-1}\sqrt{|\Delta|}}~{},~{}\frac{2\sqrt{g-1}}{\sqrt{|\Delta|}}\right),$
$\displaystyle P_{3}=(x_{3},y_{3})\coloneqq\mathfrak{c}\cap\ell\cap\\{y\leq
0\\}$ $\displaystyle=$
$\displaystyle\left(1-\frac{d}{\sqrt{g-1}\sqrt{|\Delta|}}~{},~{}-\frac{2\sqrt{g-1}}{\sqrt{|\Delta|}}\right).$

We first consider case (37). We note that $x_{2}\leq x_{+}$ and $y_{+}\geq
y_{1}\geq y_{2}$. Thus, the points $(x,y)$ lie in the region bounded by the
$x$-axis, the lines $\ell_{-}$ and $\ell$, and the segment of $\mathfrak{c}$
between $P_{1}$ and $P_{2}$. Since $\mathfrak{c}$ is convex, the points
$(x,y)$ are contained in the pentagon bounded by the $x$-axis, the lines
$\ell_{-}$ and $\ell$, and the line segment $P_{1}P_{2}$. In particular,
$x\leq\operatorname{max}\\{x_{1},x_{2}\\}$ and $y\leq y_{1}$. Together with
(37), this yields (33).

We next consider case (38). We note that $x_{-}\leq x_{3}$ and $y_{-}\leq
y_{4}\leq y_{3}$. The latter implies that the intersection points $P_{4}$ and
$P_{3}$ occur in the region where $\ell_{+}$ lies to the left of $\ell$,
whence it follows that also $x_{4}\leq x_{3}$. (Note, however, that contrary
to the impression one may get by the picture, we could have $x_{3}<0$.) The
points $(x,y)$ lie in the region bounded by the $y$-axis, the lines $\ell_{+}$
and $\ell$ and the the segment of $\mathfrak{c}$ between $P_{4}$ and $P_{3}$.
Since $\mathfrak{c}$ is convex, the points $(x,y)$ are contained in the
pentagon bounded by the $y$-axis, the lines $\ell_{+}$ and $\ell$, and the
line segment $P_{4}P_{3}$. In particular, $x\geq x_{4}$ and $y\leq y_{4}$.
Together with (38), this yields (34). ∎

######  Remark 3.3.

A direct computation shows that

$\frac{2(r-1)g}{(d-\sqrt{|\Delta|})\sqrt{|\Delta|}}-\frac{2(r-1)g}{(d+\sqrt{|\Delta|})\sqrt{|\Delta|}}=1+\frac{1}{g-1}.$

In particular, setting
$m\coloneqq\left\lfloor\operatorname{max}\left\\{1+\frac{d}{\sqrt{|\Delta|}\sqrt{g-1}}~{},~{}\frac{g(2r-2)}{(d-\sqrt{|\Delta|})\sqrt{|\Delta|}}\right\\}\right\rfloor$,
Lemma 3.2 shows that $x\in\\{-m+1,\ldots,m\\}$.

As a consequence, we can deduce decomposition rigidity under certain
assumptions.

######  Proposition 3.4.

Assume that (6) holds,

(39) $d\geq\begin{cases}\frac{g-3}{2}+2r,&\mbox{if}\;\;r\geq 2,\\\
\frac{g}{2},&\mbox{if}\;\;r=1,\end{cases}$

and

(40) $(r-1)(3g-4)^{2}<2d^{2}(g-2).$

Then any $(S,H)\in{\mathcal{K}}^{r}_{g,d}$ with
$\operatorname{Pic}(S)=\Lambda^{r}_{g,d}$ satisfies decomposition rigidity.

######  Remark 3.5.

Condition (39) together with the assumption that $d\leq g-1$ (cf. (6)) imply
that $g+1\geq 4r$.

######  Remark 3.6.

If (39) is not fulfilled, then
$(H-2L)^{2}\geq\begin{cases}-2,&\mbox{if}\;\;r\geq 2,\\\
0,&\mbox{if}\;\;r=1,\end{cases}$ whence either $H\sim 2L+(H-2L)$ is another
flexible decomposition, or the assumption in Theorem 2.5 that $g+4r-2d\neq 4$,
which is equivalent to $(H-2L)^{2}\neq-2$, is not satisfied. Thus (39) is the
optimal bound to be able to apply Theorem 2.5.

######  Remark 3.7.

Condition (40) is automatically fulfilled if $r=1$. Moreover, under assumption
(39), condition (40) is for instance implied by the simplified assumption

(41)
$g\geq\operatorname{min}\left\\{r(r+1),\;10(r-1)\right\\}=\begin{cases}r(r+1),&\mbox{if}\;\;2\leq
r\leq 7,\\\ &\\\ 10(r-1),&\mbox{if}\;\;r\geq 8.\end{cases}$

Indeed, because of (39), condition (40) is implied by

$(r-1)(3g-4)^{2}<2(g-2)\left(\frac{g-3}{2}+2r\right)^{2},$

equivalently

(42) $g^{3}+g^{2}(-10r+10)+g(16r^{2}+8r-27)-32r^{2}+16r+14>0.$

As $g(16r^{2}+8r-27)-32r^{2}+16r+14>0$ for $r\geq 2$ and $g\geq 3$, we see
that (42) holds if $g\geq 10(r-1)$. If $r\leq 7$, one can check that (42)
holds when $g\geq r(r+1)$. Hence (40) holds.

###### Proof of Proposition 3.4.

We want to prove that $H\sim L+(H-L)$ is the unique flexible decomposition of
$H$. To this aim, let $H\sim A+B$ where $A=xH-yL$ and $B=(1-x)H+yL$ be a
flexible decomposition of $H$. We will prove that $(x,y)=(1,1)$ or $(0,-1)$,
which both correspond to the decomposition $H\sim(H-L)+L$.

If $r=1$, assumption (39) and Lemma 3.2 show that the only possibilities are
$(x,y)=(1,1)$, $(0,-1)$ or $(0,-2)$. In the latter case we have $B\sim H-2L$
and $B^{2}=-2$. We claim that $B$ is the only $(-2)$-divisor on $S$, which
will prove that $B$ is irreducible, giving the contradiction $h^{0}(B)=1$.
Indeed, if $\Gamma$ is a $(-2)$-divisor on $S$, we may write $\Gamma\sim aH-
bL$, with $a>0$, since multiples of $L$ can never contain $H$. Then
$-1=\frac{1}{2}\Gamma^{2}=a[a(g-1)-bd]$, whence $a=1$ and $g=bd$, so $b=2$ and
$\Gamma\sim H-2L\sim B$. This gives the desired contradiction, showing that
the case $(x,y)=(0,-2)$ does not occur, as desired.

Now suppose $r\geq 2$.

We first claim that if $-3\leq y\leq 3$ and $0\leq x\leq 1$, then again
$(x,y)=(1,1)$ or $(0,-1)$. Indeed, Lemma 3.2 shows that it suffices to rule
out the cases $(x,y)=(0,-2),(0,-3),(1,2)$ and $(1,3)$. In these cases one of
$A$ or $B$ is $H-2L$ or $H-3L$. By (39) we have

$(H-2L)^{2}=2g+8r-10-4d\leq 2g+8r-10-4\left(\frac{g-3}{2}+2r\right)=-4,$

a contradiction. Similarly, we have

$-2\leq(H-3L)^{2}=2g+18r-20-6d\leq
2g+18r-20-6\left(\frac{g-3}{2}+2r\right)=6r-11-g,$

whence $g\leq 6r-9$. But then, using this and (39) once more, we obtain the
contradiction

$(H-3L)\cdot H=2g-2-3d\leq
2g-2-3\left(\frac{g-3}{2}+2r\right)=\frac{g+5}{2}-6r\leq\frac{1}{2}\left(6r-4\right)-6r=-3r-2<0.$

It remains to show that $-3\leq y\leq 3$ and $0\leq x\leq 1$. By Remark 3.3
and Lemma 3.2, to show that $0\leq x\leq 1$ it suffices to show that

(43) $1+\frac{d}{\sqrt{|\Delta|}\sqrt{g-1}}<2$

and

(44) $\frac{2(r-1)g}{(d-\sqrt{|\Delta|})\sqrt{|\Delta|}}<2.$

Having done this, the case $x=1$ yields, because of the right inequality in
(32), that

$0<y<\frac{2(g-1)}{d}\leq\frac{2(g-1)}{\frac{g-3}{2}+2r}\leq\frac{4(g-1)}{g+5}<4,$

and similarly the case $x=0$ yields, because of the left inequality in (32),
that $0<-y<4$. We have therefore left to prove (43) and (44).

Inequality (44) is equivalent to

$d^{2}-(r-1)(3g-4)<d\sqrt{|\Delta|}.$

The latter is implied by the inequality obtained by squaring both sides, which
is (40). Thus, (44) is satisfied.

Inequality (43) is equivalent to

(45) $4(g-1)^{2}(r-1)<(g-2)d^{2}.$

Our assumptions yield that $g\geq 4r-1\geq 7$, cf. Remark 3.5, and then one
readily checks that $4(g-1)^{2}\leq\frac{1}{2}\left(3g-4\right)^{2}$, so that
(45) is implied by (40). Thus, (43) is also satisfied. ∎

##  4\. Bounds on the limits of Brill–Noether special divisors

The aim of this section is to give sufficient conditions so that (12)-(15) in
Theorem 2.5 cannot be verified, thus ruling out the existence of a
$g^{r^{\prime}}_{d^{\prime}}$ as in that theorem.

######  Lemma 4.1.

Assume that $r^{\prime}\geq 1$, $2\leq d^{\prime}\leq g-1$, (6) is satisfied,
and that

(46)
$\begin{cases}d-r+r^{\prime}-d^{\prime}>\frac{r}{r^{\prime}}-1,&\mbox{if}\;\;r^{\prime}\leq
r,\\\
g+r^{\prime}-1-d^{\prime}-\frac{g+r-d-1}{r^{\prime}-r+1}>0,&\mbox{if}\;\;r<r^{\prime}.\end{cases}$

Then there are no non-negative integers $r_{1},r_{2},d_{1},d_{2}$ satisfying
(12)–(15).

###### Proof.

Assume, to get a contradiction, that $r_{1},r_{2},d_{1},d_{2}$ are non-
negative integers satisfying (12)–(15).

We have

(47) $0\leq r_{1}\leq r^{\prime}-1.$

Condition (14) can be rewritten as

(48) $r_{1}+\frac{r_{1}r}{r_{1}+1}\leq d_{1}<r_{1}+r.$

From the left hand inequality of (15) combined with (12) and (13), along with
the fact that $\rho(g,r,-)$ is an increasing function, we obtain

$\rho(g+r-d-1,r^{\prime}-1-r_{1},d^{\prime}-d+2r-2-d_{1})\geq 0,$

which can be rewritten as

(49) $d_{1}\leq
d^{\prime}+r+r_{1}-r^{\prime}-g+\frac{g+r-d-1}{r^{\prime}-r_{1}}.$

Grant for the moment the following:

######  Claim 4.2.

If

  * •

$r^{\prime}\leq r$, or

  * •

$r_{1}<r<r^{\prime}$,

then

(50)
$d^{\prime}+r+r_{1}-r^{\prime}-g+\frac{g+r-d-1}{r^{\prime}-r_{1}}<r_{1}+\frac{r_{1}r}{r_{1}+1}.$

We show how the lemma follows from the claim.

Assume that $r^{\prime}\leq r$. Then Claim 4.2 shows that (49) and the left
hand inequality of (48) are incompatible. We are therefore done in this case.

Assume, finally, that $r^{\prime}>r$. Then Claim 4.2 shows that (49) and the
left hand inequality of (48) are compatible only for $r_{1}>r-1$. This latter
inequality is on the other hand equivalent to

$r+r_{1}-\left(r_{1}+\frac{rr_{1}}{r_{1}+1}\right)<1.$

Thus there is no integer $d_{1}$ satisfying (48). This concludes the proof of
the lemma.

We have left to prove the claim:

###### Proof of Claim 4.2.

Since the left hand side of (50) is a convex function of $r_{1}$ and the right
hand side is a concave function of $r_{1}$, it suffices to prove the
inequality for the two boundary values $r_{1}=0$ and

$r_{1}=\begin{cases}r^{\prime}-1,&\mbox{if}\;\;r^{\prime}\leq r,\\\
r-1&\mbox{if}\;\;r^{\prime}>r,\end{cases}$

cf. (47), where it reads, respectively,

(51) $d^{\prime}+r-r^{\prime}-g+\frac{g+r-d-1}{r^{\prime}}<0,$ (52)
$d^{\prime}-d+2r-2<r^{\prime}-1+\frac{(r^{\prime}-1)r}{r^{\prime}},\;\;\mbox{if}\;\;r^{\prime}\leq
r,$

and

(53)
$d^{\prime}+1-r^{\prime}-g+\frac{g+r-d-1}{r^{\prime}-r+1}<0,\;\;\mbox{if}\;\;r^{\prime}>r.$

One easily verifies that (52) is equivalent to the upper inequality in (46)
and implies (51). We are therefore done in the case $r^{\prime}\leq r$.

Assume now that $r^{\prime}>r$. If $r\geq
r^{\prime}-\frac{g-d-1}{r^{\prime}+1}$, one verifies that (53) implies (51).
If $r<r^{\prime}-\frac{g-d-1}{r^{\prime}+1}$, then
$g-d-1<(r^{\prime}-r)(r^{\prime}+1)$. Using this together with $d^{\prime}\leq
g-1$, one checks that (51) is satisfied. Thus, (51) is redundant, and we are
left with (53), which can be rewritten as the lower inequality in (46). This
concludes the proof of the claim. ∎

Having proved the claim, the lemma follows. ∎

##  5\. Main result and applications

We summarize the results of the previous sections as the main result of the
paper.

######  Theorem 5.1.

Let $g,r,d,r^{\prime},d^{\prime}$ be integers satisfying

(54) $\displaystyle g\geq 3,\;\;r\geq 1,\;\;r^{\prime}\geq 1,\;\;2\leq d\leq
g-1,\;\;2\leq d^{\prime}\leq g-1,$ (55) $\displaystyle
d\geq\begin{cases}\frac{g-3}{2}+2r,&\mbox{if}\;\;r\geq 2,\\\
\frac{g}{2},&\mbox{if}\;\;r=1,\end{cases}$ (56)
$\displaystyle(r-1)(3g-4)^{2}<2d^{2}(g-2),$ (57)
$\displaystyle\rho(g,r^{\prime},d^{\prime})<0,$ (58)
$\displaystyle\begin{cases}d-r+r^{\prime}-d^{\prime}>\frac{r}{r^{\prime}}-1,&\mbox{if}\;\;r^{\prime}\leq
r,\\\
g+r^{\prime}-1-d^{\prime}-\frac{g+r-d-1}{r^{\prime}-r+1}>0,&\mbox{if}\;\;r<r^{\prime}.\end{cases}$

Then for any K3 surface $(S,H)\in{\mathcal{K}}^{r}_{g,d}$ with
$\operatorname{Pic}(S)=\Lambda^{r}_{g,d}$, we have that $|H|$ contains smooth
irreducible curves, and for any such $C\in|H|$ we have:

  1. (i)

$C$ has genus $g$,

  2. (ii)

$|\mathscr{O}_{C}(L)|$ is a base point free complete $g^{r}_{d}$, which is
very ample if $r\geq 3$,

  3. (iii)

$C$ carries no $g^{r^{\prime}}_{d^{\prime}}$.

###### Proof.

Condition (56) implies (5), which allows us to construct a K3 surface
$(S,H)\in{\mathcal{K}}^{r}_{g,d}$ with
$\operatorname{Pic}(S)=\Lambda^{r}_{g,d}$ as in §3. Conditions (54)-(56) are
the conditions in Proposition 3.4, which guarantee that $(S,H)$ satisfies
decomposition rigidity. For $r\geq 2$, inequality (55) can be rewritten as
$g+4r-2d\leq 3$, so that also the assumption that $g+4r-2d\neq 4$ in Theorem
2.5 is satisfied. Thus, by the latter, properties $(i)$ and $(ii)$ in the
theorem are fulfilled. If $(iii)$ is not fulfilled, then, due to (57), there
would be integers $r_{1},r_{2},d_{1},d_{2}$ satisfying (12)-(15). But
condition (58) is (46) of Lemma 4.1, which then tells us that such integers
cannot exist. Thus, $(iii)$ is fulfilled. ∎

######  Remark 5.2.

Remark 3.7 tells us that condition (56) can be replaced by the simplified
assumption

(59)
$g\geq\operatorname{min}\left\\{r(r+1),\;10(r-1)\right\\}=\begin{cases}r(r+1),&\mbox{if}\;\;2\leq
r\leq 7,\\\ &\\\ 10(r-1),&\mbox{if}\;\;r\geq 8.\end{cases}$

We will now deduce Theorem 1 in the introduction from Theorem 5.1. To this
aim, we first need two lemmas.

######  Lemma 5.3.

Assume that the triple $(g,r,d)$ is associated to an expected maximal
Brill–Noether locus, $g\geq 3$ and $(g,r,d)\neq(6,2,5)$. Then conditions
(55)-(56) in Theorem 5.1 are satisfied.

###### Proof.

If $r=1$, then $d=\left\lceil\frac{g}{2}\right\rceil$ by (2), so (55) is
satisfied. If $r\geq 2$, we have $g\geq r(r+1)$ by (3). Remark 5.2 yields that
(56) is satisfied. Using $(g,r)\neq(6,2)$, one can check that the inequality

$r-1+\left\lceil\frac{gr}{r+1}\right\rceil\geq\frac{g-3}{2}+2r$

holds, whence (2) implies that (55) is satisfied. Hence, (55)-(56) in Theorem
5.1 are satisfied. ∎

######  Lemma 5.4.

Assume that the triple $(g,r,d)$ is associated to an expected maximal
Brill–Noether locus, with $2\leq d\leq g-1$, and $(r^{\prime},d^{\prime})$ a
pair of integers such that $r^{\prime}\geq 1$, $2\leq d^{\prime}\leq g-1$,
$\rho(g,r^{\prime},d^{\prime})<0$, and either

  1. (a)

$(g,r^{\prime},d^{\prime})$ is associated to an expected maximal Brill–Noether
locus, $r^{\prime}\neq r$, or

  2. (b)

$r^{\prime}=r$ and $d^{\prime}\leq d-1$, or

  3. (c)

$r^{\prime}=r+1$ and $d^{\prime}\leq d+1$.

Then, except for the cases

(60)
$(g,r,d,r^{\prime},d^{\prime})\neq(6,2,5,1,3),(7,2,6,1,4),(8,1,4,2,7),(9,2,7,1,5),$

condition (58) in Theorem 5.1 is satisfied.

###### Proof.

It is straightforward to check that (58) is satisfied under assumptions (b)
and (c). We are therefore left with case (a), where $r^{\prime}\neq r$ and
$(g,r^{\prime},d^{\prime})$ is associated to an expected maximal Brill–Noether
locus.

We first prove the lower inequality in (58). After substituting expressions
for $d$ and $d^{\prime}$ given by (2) the inequality is equivalent to

(61)
$\left\lceil\frac{gr^{\prime}}{r^{\prime}+1}\right\rceil\left(r^{\prime}+1-r\right)<\left\lceil\frac{gr}{r+1}\right\rceil+g(r^{\prime}-r).$

The latter is implied by

$\left(\frac{gr^{\prime}}{r^{\prime}+1}+1\right)\left(r^{\prime}+1-r\right)\leq\frac{gr}{r+1}+g(r^{\prime}-r),$

which is equivalent to

(62) $r^{\prime}+1-r\leq\frac{gr(r^{\prime}-r)}{(r+1)(r^{\prime}+1)}.$

Since $g\geq r^{\prime}(r^{\prime}+1)\geq(r+1)(r^{\prime}+1)$ by (3), the
latter is implied by

$r^{\prime}+1-r\leq r(r^{\prime}-r),$

which is easily seen to hold as soon as $r\geq 2$. This leaves the case $r=1$,
where (62) reads

$g\geq\frac{2r^{\prime}(r^{\prime}+1)}{r^{\prime}-1}.$

Using the fact that $g\geq r^{\prime}(r^{\prime}+1)$ by (3), we see that the
latter is satisfied as long as $r^{\prime}\geq 3$. Recalling that
$r^{\prime}\geq 2$ (as $r^{\prime}>r$), this leaves the case
$(r,r^{\prime})=(1,2)$, in which case (61) reads

(63)
$2\left\lceil\frac{2g}{3}\right\rceil<g+\left\lceil\frac{g}{2}\right\rceil.$

One easily checks that this is satisfied, since we are assuming
$(g,r,d,r^{\prime},d^{\prime})\neq(8,1,4,2,7)$. This finishes the proof of the
lower inequality in (58).

We finally prove the upper inequality in (58). After substituting the
expressions for $d$ and $d^{\prime}$ given by (2), the inequality is
equivalent to

(64)
$\left\lceil\frac{r^{\prime}g}{r^{\prime}+1}\right\rceil+\frac{r}{r^{\prime}}<\left\lceil\frac{rg}{r+1}\right\rceil+1.$

The latter inequality is implied by

$\frac{r^{\prime}g}{r^{\prime}+1}+\frac{r}{r^{\prime}}\leq\frac{rg}{r+1},$

which can be rewritten as

$\frac{r}{r^{\prime}}\leq\frac{g(r-r^{\prime})}{(r^{\prime}+1)(r+1)}.$

By (3) the latter inequality is implied by

$\frac{1}{r^{\prime}}\leq\frac{r-r^{\prime}}{r^{\prime}+1},$

which is equivalent to

$\frac{r^{\prime}+1}{r^{\prime}}\leq r-r^{\prime}.$

Since $r^{\prime}<r$, the latter is satisfied unless $r^{\prime}=r-1$. We have
therefore left to prove (64) when $r^{\prime}=r-1$, in which case it reads

(65)
$\left\lceil\frac{(r-1)g}{r}\right\rceil+\frac{1}{r-1}<\left\lceil\frac{rg}{r+1}\right\rceil.$

Recall that $r\geq 2$ (as $r\geq r^{\prime}+1$). If $r\geq 3$, then (65) is
implied by

$\frac{(r-1)g}{r}+1\leq\frac{rg}{r+1},$

which can be rewritten as $g\geq r^{2}+r$, which holds by (3). Thus, (65) is
proved if $r\geq 3$. If $r=2$, then (65) reads

$\left\lceil\frac{g}{2}\right\rceil+1<\left\lceil\frac{2g}{3}\right\rceil.$

Using our assumptions that
$(g,r,d,r^{\prime},d^{\prime})\neq(6,2,5,1,3),(7,2,6,1,4),(9,2,7,1,5)$, one
verifies that the latter inequality is satisfied. Thus, (65) is proved if
$r=2$. This finishes the proof of the upper inequality in (58). ∎

######  Remark 5.5.

While the hypotheses of Theorem 5.1 do not hold in the case $(g,r,d)=(6,2,5)$,
we still have the existence of a polarized K3 surface
$(S,H)\in{\mathcal{K}}^{2}_{6,5}$ such that $|H|$ contains smooth irreducible
curves of genus 6 and $|\mathscr{O}_{C}(L)|$ is a base point free very ample
complete $g^{2}_{5}$. Indeed, the existence of $(S,H)$ with
$\operatorname{Pic}(S)=\Lambda^{2}_{6,5}$ such that the conclusions
$(i)$–$(ii)$ of Theorem 5.1 are satisfied follows as above; this has, in fact,
already been done (in a more general setting) in [16, Thm. 4.3]. In
particular, the curves are isomorphic to smooth plane quintics, whence do not
carry $g^{1}_{3}$s, so they again do not carry any
$g^{r^{\prime}}_{d^{\prime}}$ with $(r^{\prime},d^{\prime})\neq(2,5)$. One may
also check that $(S,H)$ satisfies decomposition rigidity.

Finally, applying all this to the expected maximal Brill–Noether loci, we can
now give a proof of Theorem 1.

###### Proof of Theorem 1.

Lemma 5.3 tells us that $(g,r,d)$ satisfies conditions (54)-(56) in Theorem
5.1. In particular, by Proposition 3.4, the hypotheses in Theorem 2.5 are
satisfied by any K3 surface $S$ with
$\operatorname{Pic}(S)=\Lambda^{r}_{g,d}$, as in §1.2. Let $C\in|H|$ be any
smooth curve and assume it carries a $g^{r^{\prime}}_{d^{\prime}}$ with
$\rho(g,r^{\prime},d^{\prime})<0$ and $(r^{\prime},d^{\prime})\neq(r,d)$,
hence $C$ also carries a $g^{r^{\prime\prime}}_{d^{\prime\prime}}$ with
$(g,r^{\prime\prime},d^{\prime\prime})$ associated to an expected maximal
Brill–Noether locus, obtained using the trivial containments
${\mathcal{M}}^{r}_{g,d}\subset{\mathcal{M}}^{r}_{g,d+1}$ and
${\mathcal{M}}^{r}_{g,d}\subset{\mathcal{M}}^{r-1}_{g,d-1}$, along with Serre
duality.

Assume that $(r^{\prime\prime},d^{\prime\prime})\neq(r,d)$. Then, if
$(g,r,d)\neq(6,2,5)$, Lemma 5.4 tells us that all the remaining conditions in
Theorem 5.1 are satisfied for
$(r^{\prime},d^{\prime})=(r^{\prime\prime},d^{\prime\prime})$ (recall that we
are assuming $(g,r,d)\neq(7,2,6),(8,1,4),(9,2,7)$). Hence, by the same
theorem, we get the desired contradiction, that is, that the
$g^{r^{\prime\prime}}_{d^{\prime\prime}}$ cannot exist.

Assume that $(r^{\prime\prime},d^{\prime\prime})=(r,d)$. Then the $g^{r}_{d}$
is obtained from the $g^{r^{\prime}}_{d^{\prime}}$ by a series of trivial
containments and Serre duality, the last of which is Serre duality or one of
the trivial containments $\mathcal{M}^{r}_{g,d-1}\subset\mathcal{M}^{r}_{g,d}$
or $\mathcal{M}^{r+1}_{g,d+1}\subset\mathcal{M}^{r}_{g,d}$. However, if Serre
duality were the last step, then the $g^{g-d+r-1}_{2g-2-d}$ was obtained using
trivial containments from the Serre duals of a $g^{r+1}_{d+1}$ or a
$g^{r}_{d-1}$. Thus, in any case, $C$ carries a $g^{r}_{d-1}$ or a
$g^{r+1}_{d+1}$. Again if $(g,r,d)\neq(6,2,5)$, Lemma 5.4 tells us that all
the remaining conditions in Theorem 5.1 are satisfied for
$(r^{\prime},d^{\prime})=(r,d-1)$ or $(r+1,d+1)$, and we get a contradiction
again.

The remaining case of $(g,r,d)=(6,2,5)$ has already been handled by a direct
geometric argument in [2, Proposition 6.1]. One can, in fact, also handle this
case using K3 surfaces, as in Remark 5.5. ∎

######  Remark 5.6.

Theorem 5.1 directly shows that the general curve in the Brill–Noether loci
$\mathcal{M}^{1}_{7,4}$, $\mathcal{M}^{2}_{8,7}$, and $\mathcal{M}^{1}_{9,5}$
is $g^{r}_{d}$-general for $(r,d)=(1,4),(2,7),(1,5)$, respectively, which are
the maximal Brill–Noether loci in the exceptional genera, as in Example 1.3.
Hence Theorem 1 says that any expected maximal Brill–Noether locus
$\mathcal{M}^{r}_{g,d}$ contains a $g^{r}_{d}$-general curve, unless
$g=7,8,9$.

######  Remark 5.7.

This also answers the question of containments of Brill–Noether loci of the
form $\mathcal{M}^{r}_{g,d}\subset\mathcal{M}^{r^{\prime}}_{g,d^{\prime}}$
with $\rho(g,r,d)=-2$ and $\rho(g,r^{\prime},d^{\prime})=-1$, studied in [3,
9, 44]. Indeed, as observed in [7, Lemma 1.2], Brill–Noether loci with
$\rho=-1,-2$ are expected maximal.

We give a few examples in genus $7,8,9$ of distinguishing Brill–Noether loci
using Theorem 5.1.

######  Example 5.8.

In genus $7$, Theorem 5.1 yields that the locus $\mathcal{M}^{2}_{7,6}$ is not
contained in any Brill–Noether locus except $\mathcal{M}^{1}_{7,4}$, which is
the unexpected containment of Example 1.3.

In genus $8$, we obtain the non-containment
$\mathcal{M}^{1}_{8,4}\nsubseteq\mathcal{M}^{2}_{8,6}$, thus the locus
$\mathcal{M}^{1}_{8,4}$ is not contained in any Brill–Noether locus except
$\mathcal{M}^{2}_{8,7}$, again an unexpected containment of Example 1.3. The
containment $\mathcal{M}^{2}_{8,6}\subset\mathcal{M}^{1}_{8,4}$ is obtained in
[34, Lemma 3.4].

In genus $9$, we obtain that the locus $\mathcal{M}^{2}_{9,7}$ is not
contained in any Brill–Noether locus except $\mathcal{M}^{1}_{9,5}$.

Theorem 5.1 also gives non-containments of non-maximal Brill–Noether loci.

######  Example 5.9.

In genus $11$, we obtain the non-containments
$\mathcal{M}^{3}_{11,10}\nsubseteq\mathcal{M}^{2}_{11,8}$ and
$\mathcal{M}^{2}_{11,8}\nsubseteq\mathcal{M}^{3}_{11,10}$.

In fact, Theorem 5.1 gives infinitely many non-containments of Brill–Noether
loci that are not maximal. As a sample, let us see what happens when we lower
the maximal degrees a little. Recall the definition of $d_{max}(g,r)$ from
(2).

######  Proposition 5.10.

Let $\mathcal{M}^{r}_{g,d_{max}(g,r)}$ and
$\mathcal{M}^{r^{\prime}}_{g,d_{max}(g,r^{\prime})}$ be expected maximal loci
with $r\geq 2$, $r^{\prime}\geq 2$ and $r\neq r^{\prime}$. We have a non-
containment
$\mathcal{M}^{r}_{g,d}\nsubseteq\mathcal{M}^{r^{\prime}}_{g,d_{max}(g,r^{\prime})-1}$
in any of the following cases:

  * •

$d\geq d_{max}(g,r)-1$ and

    * $\circ$

$r^{\prime}<r$, $(r,r^{\prime},g)\neq(3,2,12)$, or

    * $\circ$

$(r,r^{\prime})=(2,3)$, $g\in\\{13,14,15,16,18\\}$,

  * •

$d\geq d_{max}(g,r)-2$ and $(r,r^{\prime})=(2,4)$, $g\in\\{20,21,22,24\\}$,

  * •

$d\geq d_{max}(g,r)-3$ and $(r,r^{\prime})=(2,5)$, $g=30$,

  * •

$d\geq d_{max}(g,r)-(r^{\prime}-r+1)$ and

    * $\circ$

$r^{\prime}>r\geq 3$, or

    * $\circ$

$r^{\prime}\geq 6$, $r=2$, or

    * $\circ$

$(r,r^{\prime})=(2,5)$ and $g\geq 31$, or

    * $\circ$

$(r,r^{\prime})=(2,4)$ and $g\geq 25$, or $g=23$, or

    * $\circ$

$(r,r^{\prime})=(2,3)$ and $g\geq 19$, or $g=17$.

###### Proof.

We have proved that(58) holds for $d=d_{max}(g,r)$ and for
$d^{\prime}=d_{max}(g,r^{\prime})$. Setting instead
$d^{\prime}=d_{max}(g,r^{\prime})-1$, we therefore see that the upper
inequality in (58) holds for $d\geq d_{max}(g,r^{\prime})-1$, whereas the
lower one hold for $d\geq d_{max}(g,r)-(r^{\prime}-r+1)$. Since $g\geq r(r+1)$
by maximality of $\mathcal{M}^{r}_{g,d_{max}(g,r)}$ (cf. (3)), Remark 5.2
tells us that (56) is redundant. Hence, it remains to check (55) by
substituting for the values of $d$ and using $g\geq r(r+1)$ and $g\geq
r^{\prime}(r^{\prime}+1)$. We leave the details to the reader. ∎

######  Example 5.11.

In genus $14$, Proposition 5.10 gives the non-containments
$\mathcal{M}^{3}_{14,12}\nsubseteq\mathcal{M}^{2}_{14,10}$ and
$\mathcal{M}^{2}_{14,10}\nsubseteq\mathcal{M}^{3}_{14,12}$.

We also give an example where Theorem 5.1 does not suffice to prove a non-
containment.

######  Example 5.12.

Let $(g,r,d)=(9,2,6)$ and $(g,r^{\prime},d^{\prime})=(9,3,8)$. We cannot apply
Theorem 2.5, as on the lattice $\Lambda^{2}_{9,6}$, we have $(H-2L)^{2}=0$,
hence $(S,H)$ does not satisfy decomposition rigidity. However, from [3,
Proposition 2.5], we obtain
$\mathcal{M}^{2}_{9,6}\nsubseteq\mathcal{M}^{3}_{9,8}$.

## References

  * [1] E. Arbarello, A. Bruno, E. Sernesi, Mukai’s program for curves on a K3 surface, Algebr. Geom. 1 (2014), 532–557.
  * [2] A. Auel, R. Haburcak, Maximal Brill–Noether loci via K3 surfaces, arXiv:2206.04610.
  * [3] A. Auel, R. Haburcak, H. Larson, Maximal Brill–Noether loci via the gonality stratification, arXiv:2310.09954.
  * [4] W. Barth, K. Hulek, C. Peters, A.van de Ven, Compact Complex Surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 4) 2nd ed. 2004.
  * [5] A. Bayer, A. Kuznetsov, E. Macrì, Mukai bundles on Fano threefolds, arXiv:2402.07154.
  * [6] A. Bud, Brill–Noether loci and strata of differentials, arXiv:2402.11599.
  * [7] A. Bud, R. Haburcak, Maximal Brill–Noether loci via degenerations and double covers, arXiv:2404.15066.
  * [8] Y. Choi, S. Kim, Y. R. Kim, Remarks on Brill–Noether divisors and Hilbert schemes, J. Pure Appl. Algebra, 216(2) (2012), 377–384.
  * [9] Y. Choi, S. Kim, Linear series on a curve of compact type bridged by a chain of elliptic curves, Indagationes Mathematicae, 33 (2022), 844–860.
  * [10] Y. Choi, S. Kim, Y. R. Kim, Brill–Noether divisors for even genus, Journal of Pure and Applied Algebra, 218 (2014), 1458–1462.
  * [11] K. Cook-Powell, D. Jensen, Components of Brill–Noether loci for curves with fixed gonality, Michigan Math. Journal, 71 (2022), 19–45.
  * [12] R. Donagi, D. Morrison, Linear systems on K3-sections, J. Differential Geom. 29 (1989), 49–64.
  * [13] D. Edidin, Brill–Noether theory in codimension-two, Journal of Algebraic Geometry, 2 (1993), 25–67.
  * [14] D. Eisenbud, J. Harris, Irreducibility of some families of linear series with Brill-Noether number -1, Annales scientifiques de l’École Normale Supérieure, 22 (1989), 33–53.
  * [15] D. Eisenbud, J. Harris, The Kodaira dimension of the moduli space of curves of genus $\geq$ 23, Inventiones Mathematicae, 90(2) (1987), 359–387.
  * [16] D. Eisenbud, H. Lange, G. Martens; F.–O. Schreyer, The Clifford dimension of a projective curve, Compos. Math. 72 (1989), 173–204.
  * [17] G. Farkas, The geometry of the moduli space of curves of genus 23, Mathematische Annalen, 318 (2000), 43–65.
  * [18] G. Farkas, Brill–Noether loci and the gonality stratification of $\mathcal{M}_{g}$, J. reine angew. Math., 539 (2001), 185–200.
  * [19] D. Gieseker, Stable curves and special divisors: Petri’s conjecture, Invent. Math. 66 (1982), 251–275.
  * [20] T.L. Gómez, Brill–Noether theory on singular curves and torsion-free sheaves on surfaces, Comm. Analysis and Geometry 9 (2001), 725–756.
  * [21] F. Greer, Z. Li, Z. Tian, Picard groups on moduli of K3 surfaces with Mukai models, Int. Math. Res. Not. IMRN 16 (2015), 7238–7257.
  * [22] P. Griffiths, J. Harris, On the variety of special linear systems on a general algebraic curve, Duke Mathematical Journal, 47 (1980), 233–272.
  * [23] R. Haburcak, Curves on Brill–Noether special K3 surfaces, arXiv:2306.11664.
  * [24] J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves, Inventiones Mathematicae, 67(1) (1982), 23–86.
  * [25] D. Jensen, D. Ranganathan, Brill–Noether theory for curves of a fixed gonality, Forum of Mathematics. Pi, 9:e1, 2021.
  * [26] A. L. Knutsen, Smooth curves on projective K3 surfaces, Math. Scand. 90 (2002), 215–231.
  * [27] A. L. Knutsen, M. Lelli-Chiesa, G. Mongardi, Severi varieties and Brill-Noether theory of curves on abelian surfaces, J. Reine Angew. Math. 749 (2019), 161–200.
  * [28] H. Larson, A refined Brill–Noether theory over Hurwitz spaces, Inventiones Mathematicae, 224 (2021), 767–790.
  * [29] E. Larson, H. Larson, I. Vogt, Global Brill–Noether theory over the Hurwitz space, arXiv:2008.10765, to appear in Geometry & Topology.
  * [30] R. Lazarsfeld, Brill–Noether–Petri without degenerations, J. Differential Geom. 23 (1986), 299–307.
  * [31] M. Lelli-Chiesa, Stability of rank-3 Lazarsfeld–Mukai bundles on K3 surfaces, Proc. Lond. Math. Soc. (3), 107 (2013), no.2, 451–479.
  * [32] M. Lelli-Chiesa, The Gieseker–Petri divisor in $\mathcal{M}_{g}$ for $g\leq 13$, Geometriae Dedicata, 158 (2012), 149–165.
  * [33] D. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105–121.
  * [34] S. Mukai, Curves and Grassmannians. In Algebraic geometry and related topics (Inchon, 1992), Conf. Proc. Lecture Notes Algebraic Geom., I, pages 19–40. Int. Press, Cambridge, MA, 1993.
  * [35] S. Mukai, New developments in the theory of Fano threefolds: vector bundle method and moduli problems, Sūgaku 47 (1995), no. 2, 125–144.
  * [36] V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR- Izv. 14 (1980), 103–167.
  * [37] K. G. O’Grady, Moduli of Abelian and K3 Surfaces, PhD thesis, Brown University, 1986, p. 118.
  * [38] C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective Spaces, Progress in Mathematics, Birkhäuser Boston, 1980.
  * [39] N. Pflueger, Brill–Noether varieties of $k$-gonal curves, Advances in Mathematics, 312 (2017), 46–63.
  * [40] N. Pflueger, Linear series with $\rho<0$ via thrifty lego-building, J. Reine Angew. Math. 797 (2023), 193–228.
  * [41] B. Saint-Donat, Projective models of K-3 surfaces, Amer. J. Math. 96 (1974), 602–639.
  * [42] E. Sernesi, On the existence of certain families of curves, Invent. Math. 75 (1984), 25–57.
  * [43] F. Steffen, A generalized principal ideal theorem with an application to Brill–Noether theory, Inventiones Mathematicae, 132 (1998), 73–89.
  * [44] M. Teixidor i Bigas, Brill–Noether loci, arXiv:2308.10581.