The Strong Maximal Rank Conjecture and higher rank Brill--Noether theory
Ethan Cotterill, Adri\'an Alonso Gonzalo, and Naizhen Zhang
TL;DR
This paper verifies the non-emptiness of special maximal-rank loci in algebraic geometry, providing new evidence for the Strong Maximal Rank Conjecture and higher-rank Brill--Noether theory.
Contribution
It computes cohomology classes of special loci and confirms their non-vanishing, advancing understanding of the Strong Maximal Rank Conjecture and higher-rank Brill--Noether theory.
Findings
Non-zero cohomology classes of special loci are established.
Non-emptiness of these loci in many cases is verified.
Supports existence conjectures in higher-rank Brill--Noether theory.
Abstract
In this paper, we compute the cohomology class of certain "special maximal-rank loci" originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher-rank Brill--Noether theory.
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The Strong Maximal Rank Conjecture and higher rank Brill–Noether theory
Ethan Cotterill, Adrián Alonso Gonzalo, and Naizhen Zhang
Instituto de Matemática, Universidade Federal Fluminense, Rua Prof Waldemar de Freitas, S/N, Campus do Gragoatá, CEP 24.210-201, Niterói, RJ, Brazil
Department of Mathematics, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Abstract.
In this paper, we compute the cohomology class of certain “special maximal-rank loci” originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher-rank Brill–Noether theory.
Naizhen Zhang is supported by the DFG Priority Programme 2026 “Geometry at infinity”. During much of the preparation of this work, he was supported by the Methusalem Project Pure Mathematics at KU Leuven.
Contents
-
3 Intersection theory on Grassmann bundles, and (shifted) Schur functions
-
3.1 The Chow ring of a Grassmann bundle and the Gysin morphism
-
3.3 Combinatorial properties of the Lagrange-Sylvester symmetrizer
-
3.4 The Littlewood–Richardson rule in terms of Schur functions
-
3.5 Characteristic classes of symmetric squares of vector bundles
-
5.3 Relating the SMRC degree to special values of shifted Schur functions
-
6 The Bertram-Feinberg-Mukai conjecture and its connection with the SMRC
1. Introduction
Ever since the inception of moduli of (complex) curves as an area of investigation in its own right, linear series have served as crucial tools for probing the intrinsic geometry of the moduli space via the extrinsic properties of a (variable) curve’s embeddings in projective spaces. Classically, a linear series is defined by a vector subspace of holomorphic sections of a line bundle ; in that case, the celebrated Brill–Noether theorem of Griffiths and Harris gives a complete description of the space of series on a curve that is general in moduli. There are many interesting variations on the basic Brill–Noether paradigm; in this paper, we will be concerned with two of these. The first concerns the (dimension of) the vector space of hypersurfaces of fixed degree containing the image of under a linear series, while the second involves replacing by a vector bundle of some higher rank , and produces a theory of rank- linear series.
For every integer , the dimension of the vector space of degree- hypersurfaces in containing the image of a curve under is determined by the rank of an -fold multiplication map . Multiplication maps for general linear series on general curves are the focus of the Maximal Rank Conjecture, or MRC, now a theorem thanks to the work of Eric Larson. A strong form of the MRC, or SMRC, addresses the dimensionality of spaces of special linear series whose multiplication maps fail to be of maximal rank. The SMRC is already very much open when and much of our effort in this paper will be devoted to certifying the positivity of (classes of) quadratic SMRC loci of (traditional) linear series on a general curve. These are degeneracy loci in moduli spaces of linear series, in which the quadratic multiplication map fails to have maximal rank. While we manage to realize our classes as alternating sums of meromorphic functions in the shifted Schur functions of Okounkov and Olshanski, certifying the positivity of these expressions is subtle in general.
Our main result establishes the non-emptiness of quadratic SMRC loci (modulo a couple of explicit exceptions) whenever the dimension of the target of is at most 6 less than that of the domain, by verifying the positivity of the corresponding SMRC classes. By an SMRC class, we mean the image of the cohomology class of an SMRC locus under the Gysin map; see Section 3.
Theorem 1.1** (= Propositions 5.12 and 5.13).**
Assume that the dimension of the domain of the multiplication map is more than that of the target, where . The SMRC class is strictly positive when except when either
- (i)
, and ; or
- (ii)
, and
and the SMRC class is unconditionally strictly positive whenever . Moreover, the SMRC class is always strictly positive whenever .
Here positivity is measured by an intersection with a complementary power of the theta divisor.
Quadratic SMRC loci are in fact intimately related to rank-two linear series; indeed, our original motivation for studying the SMRC is a celebrated conjecture of Bertram, Feinberg and Mukai (referred to hereafter as the BFM conjecture), which states the following:
Conjecture 1**.**
(Bertram-Feinberg-Mukai [BF98, Muk95]) Set . On a general curve of genus , the moduli space of stable rank two vector bundles with canonical determinant and sections is non-empty, and has expected dimension whenever is non-negative. When , the moduli space is empty.
We shall refer to the first and second items as the “existence” and “non-existence” portions of the conjecture, respectively. The non-existence portion is a theorem of Teixidor i Bigas [TiB08]. Meanwhile, the existence portion of the conjecture has been verified for many cases: for small genera, see [BF98] and [Muk93]; while for results asymptotic with respect to , see [TiB04], [LNP16] and [Zha16]. However, the general case remains very much open.
In [Zha16], the third author succeeded in verifying new instances of the BFM conjecture. In order to do so, he appealed to a theory of higher-rank limit linear series on reducible curves of compact type. In this paper, we instead study rank-two bundles with canonical determinant as extensions of line bundles by their Serre duals; the coboundary map in cohomology induced by the extension sequence is controlled by a quadratic multiplication map.
With Theorem 1.1 in hand, we are able to extend the list of genera for which the existence portion of the BFM conjecture is known to hold:
Theorem 1.2** (= Corollary 6.13).**
The existence portion of the BFM conjecture holds for .
In fact, we prove more; namely, that in each of these instances, the stable bundle that we construct is realized by an extension of a line bundle of the minimal degree allowed by classical Brill–Noether theory.
Roadmap
The material following this introduction is structured as follows. In section 2, we list some notation related to linear series that we will use systematically throughout the entire paper. In section 3, we assemble all of the technical tools we will subsequently apply in calculating the classes of SMRC loci. We start by reviewing intersection theory on the Grassmann bundle . An important fact is that the Gysin (pushforward) morphism is induced by the Lagrange–Sylvester symmetrizer on the Chern roots of the bundle ; this is recalled in Lemma 3.3. A key formula due to Laksov, Lascoux, and Thorup, which we recall in Theorem 3.23, realizes the Chern polynomial of the symmetric square of a vector bundle as a linear combination of the Segre classes of , whose coefficients are multiples of certain minors of an infinite matrix. The output of the LLT formula, in turn, turns out to be naturally related to the shifted Schur functions of Okounkov and Olshanski introduced in [OO97]. For our SMRC class calculation, we specifically need to understand evaluations of shifted Schur functions evaluated along staircase partitions and their polynomiality properties; these are the focus of Proposition 3.33.
Our exploration of the strong maximal rank conjecture starts in earnest in section 4, where we explicitly describe quadratic SMRC loci as degeneracy loci for maps of vector bundles over the Picard variety of a general curve . Subsection 4 introduces a dichotomy between the injective and surjective ranges, depending upon how relatively large the (dimensions of the) source and target of quadratic multiplication are. Insofar as BFM loci are concerned, injective cases correspond to cases where the genus is small, while the surjective range describes the “generic” case. Proposition 4.7 establishes that SMRC loci are always non-empty, and in fact contain excessively large components, in a particular regime of parameters . We refer to these as trivial instances as they arise from the failure of the associated linear series to be very ample.
In section 5, we write down explicit formulae for the Gysin pushforwards to the Picard variety of the intersections of quadratic SMRC loci with complementary powers of the theta divisor. The basic shape of these formulae depends on whether the associated triple belongs to the injective or surjective range. In Lemma 5.7 of Subsection 5.4, we compute explicitly (the degrees of) SMRC classes using structure theorems for shifted symmetric functions in tandem with a computer program. Proposition 5.12 establishes that for every fixed value of , our SMRC class formulae are positive for all greater than an explicit cutoff function in . The section culminates in Proposition 5.13.
Finally, section 6 deals with the BFM conjecture and its relationship with the SMRC. In order to realize stable rank-two bundles with canonical determinant and prescribed numbers of sections as extensions of line bundles by their Serre duals, the crucial fact is that the map that sends an extension to its coboundary is dual to the quadratic multiplication map on sections of . The other main point is that we have more control over the slope-stability of vector bundles arising from extensions in which is a minimal quotient of the resulting vector bundle. Often minimality of the quotient implies that is in the kernel of the dual of the quadratic multiplication map. If the quadratic multiplication map is surjective, any such is trivial; so stable rank-two bundles arise from (extensions of) line bundles that belong to quadratic SMRC loci.
A nontrivial extension does not necessarily give rise to a stable bundle, however, so accordingly we develop additional tools for understanding when this happens. Proposition 6.2 gives a necessary geometric criterion: certain secant divisors to the image of under are obstructions to stability. On the other hand, Theorem 6.4, due to Nagata, gives (see Corollary 6.5) an upper bound on the degree of the minimal quotient line bundle of a rank-two vector bundle with canonical determinant. It allows us to identify a critical range of possible degrees for line bundles. Proposition 6.8 establishes that a nontrivial extension of by its Serre dual gives rise to a stable bundle when its degree is at least and is minimal among degrees of all potentially feasible line bundles; its Corollary 6.9 ultimately leads to new cases of the (existence portion of the) BFM conjecture. In subsection 6.2 we give a solution to the “BFM existence problem” in the injective range via extensions, which gives an alternative to earlier work of Bertram and Feinberg. Afterwards, we establish a regime of parameters in which earlier results yield the existence portion of the BFM conjecture; this culminates in Corollary 6.13. We also discuss a significant case at the numerological border of the surjective range, that of , in which our arguments are at present inconclusive. Our Claim 1 establishes that the BFM conjecture holds in the case provided that on a general curve i) the multiplication map associated with a complete is always surjective; and ii) there exist very ample complete for which fails to be surjective.
Acknowledgements
We are grateful to Wouter Castryck, Renzo Cavalieri, Marc Coppens, Joe Harris, Thomas Lam, Alex Massarenti, Brian Osserman, and Richard Stanley for useful comments and conversations. Special thanks are due to Peter Newstead and Montserrat Teixidor i Bigas for the detailed corrections and suggestions they provided after reading an earlier version of this paper. Finally, we are grateful for the CNPq postdoctoral scheme that allowed the first and third authors to meet; and to the anonymous referee, who flagged several errors and whose suggestions have helped improve the organization and quality of exposition.
2. Notation related to linear series
Notation 2.1**.**
Hereafter, will denote an irreducible smooth projective curve over an algebraically closed field .
Notation 2.2**.**
* will denote the moduli scheme of linear series over .*
Notation 2.3**.**
* will denote a Poincaré line bundle over .*
Fix an effective, reduced divisor of of degree , and let . The space is naturally a closed subscheme of . Indeed, it is the zero locus of the morphism
[TABLE]
where is the tautological subbundle, is the structure morphism and is the second projection.
Notation 2.4**.**
Let denote the corresponding closed immersion.
Notation 2.5**.**
Let denote the forgetful map , and let denote the second projection morphism in this case.
Notation 2.6**.**
Let denote the pull-back of to along .
Notation 2.7**.**
Let denote the universal family over , i.e. the pull-back of to along .
For the sake of convenience, we summarize the maps and spaces mentioned above in one commutative diagram:
[TABLE]
Notation 2.8**.**
In numerical examples, will always denote the genus of the underlying curve, the (projective) dimension of a rank-one linear series, the degree of a line bundle, the Euler characteristic of a line bundle, the dimension of a rank 2 linear series, and
[TABLE]
Notation 2.9**.**
Given non-negative integers , we let
[TABLE]
When , we also set
[TABLE]
Notation 2.10**.**
Fix a reference point on . Let denote the corresponding class of for , given by the image of the map . It is a codimension- class.
In order to describe the Chern classes of the Poincaré line bundle, we single out certain cohomology classes of .
Notation 2.11**.**
Let denote the class of the pull-back of the theta divisor, and let denote the pullback of the class of a point on .
3. Intersection theory on Grassmann bundles, and (shifted) Schur functions
In this section, we assemble all of the main ingredients required for our study of SMRC loci. We first review some well-known facts about intersection theory on Grassmann bundles.
3.1. The Chow ring of a Grassmann bundle and the Gysin morphism
Let be a locally-free sheaf of rank on a smooth projective variety , and let denote the natural morphism. The following structure theorem for the Chow ring of a Grassmann bundle is well-known.
Theorem 3.1** ([Ful98], 14.6.6).**
Let and denote the tautological subbundle and tautological quotient bundle of , respectively. The Chow ring is an algebra over generated by the tautological classes
[TABLE]
modulo the relations .
Let and denote the Chern roots of and , respectively. The Chern classes and are elementary symmetric functions and . Consequently, if we think of as a subring of , is a polynomial in which is symmetric in and in separately. The upshot is that we may express any intersection product involving , and as a product of symmetric functions in the Chern roots of and . We will put this observation to work in writing down the Gysin map . But first we recall another well-known fact, which we will also use.
Theorem 3.2** ([Gro58], Theorem 3.1).**
Let denote the complete flag bundle associated to a vector bundle of rank and be the Chern roots of . The Chow ring is an -algebra, generated by the elements modulo the relations
[TABLE]
Lemma 3.3** ([Pra88], Lemma 2.5).**
The Gysin morphism is induced by the map
[TABLE]
where acts on a polynomial by permuting the indices of the variables.
Remark 3.4*.*
The map is known as the Lagrange-Sylvester symmetrizer (see also [Tu17]). The attentive reader will notice that the the denominator on the right-hand side differs from that in the original formula of Pragacz by a power of . This disparity is explained by the fact that Pragacz stated the formula for the Gysin map with domain a Grassmannian of rank- quotient bundles, while our formula applies to the Gysin map whose domain is a Grassmannian of rank- sub-bundles. See also [FP06, §4.1] for further discussion.
3.2. Schur functions
Schur functions in variables are symmetric functions labeled by partitions of length at most . Two equivalent conventions for Schur functions appear in the literature and are convenient for different purposes. We introduce both of them here and comment on their equivalence.
Definition 3.5**.**
A partition of length is a finite sequence of non-negative integers arranged in non-increasing order. The conjugate of a partition is a partition whose corresponding Young diagram is obtained from the original diagram by interchanging rows and columns.
Definition 3.6**.**
Let be a partition of length . The Schur function is the symmetric polynomial
[TABLE]
Definition 3.7**.**
Let be a strictly increasing sequence of non-negative integers. Now set
[TABLE]
where is the -th coefficient in the formal expansion
[TABLE]
Definition 3.8**.**
For , we call the length of and denote . We also write .
Notation 3.9**.**
For any given partition of length , and any given integer , there is a unique strictly-increasing -term sequence defined by
[TABLE]
in which we set whenever . We denote this sequence by . Conversely, given a strictly-increasing sequence of non-negative integers, let
[TABLE]
denote the corresponding partition of length at most .
Inasmuch as there is a bijection between strictly-increasing sequences of positive integers and partitions, definitions 3.6 and 3.7 need to be reconciled. The Jacobi–Trudi lemma does the trick.
Lemma 3.10** (Lemma A.9.3 in [Ful98]).**
Let be a strictly-increasing sequence of non-negative integers. We have
[TABLE]
where are the parts of .
Remark 3.11*.*
Note that , so that .
Definition 3.12**.**
More generally, given any finite non-necessarily-increasing sequence of non-negative integers , we set
[TABLE]
The fact that agrees with Definition 3.7 whenever is a strictly increasing sequence follows from Lemma 3.10.
Notation 3.13**.**
Given a rank- vector bundle with Chern roots , we will follow the convention in [Ful98, §14.5] and refer to as the -th Segre class of , also written as . We set . Notice that the Jacobi-Trudi lemma expresses as a determinant in terms of the classes , as opposed to .
Notation 3.14**.**
Fix a positive integer . Let be a set of elements in some commutative ring and let denote a partition of length at most . We set
[TABLE]
3.3. Combinatorial properties of the Lagrange-Sylvester symmetrizer
We next review some well-known properties of the Lagrange-Sylvester symmetrizer that we will use. The first is a combinatorial formula that describes the action of on Schur functions. For a reference, see [Las88].
Lemma 3.15**.**
Given two sequences of non-negative integers and , we have
[TABLE]
where denotes the concatenation of and .
Remark 3.16*.*
By definition, is clearly additive. Moreover, because
[TABLE]
and the Schur polynomials form a -basis for the ring of symmetric functions (see [Mac98, I.3.2]), the formula in Lemma 3.15 completely determines .
Corollary 3.17**.**
The map satisfies the following properties:
- (1)
* for every -invariant polynomial .* 2. (2)
* for every -tuple ; in particular, whenever .* 3. (3)
* is either a Schur polynomial in or zero for every -tuple .*
Proof.
Whenever is symmetric, we have and thus , which is claim (1). On the other hand, clearly . The first part of claim (2) follows now from Lemma 3.15. For the second part of claim (2), note that whenever , we have , where is the determinant of a matrix with two identical rows and hence must be zero.
From (2), we know that either or else is a strictly increasing sequence of non-negative integers. In the latter case, Lemma 3.10 establishes that is a Schur polynomial, which is claim (3). ∎
For our main application, will be the Picard variety of a smooth curve of genus and will be the pushforward of the twist of a Poincaré line bundle over , by the pullback of an effective divisor on of degree at least . Since ultimately we are interested in whether certain cohomology classes over are non-zero, we work up to numerical equivalence. We will apply the following well-known result of Mattuck.
Lemma 3.18** ([Mat65], Example 14.4.5 of [Ful98]).**
Suppose . The Segre class of the pushforward of a Poincaré line bundle is given by (see Notation 2.10). Moreover, is numerically equivalent to , where is the theta divisor class.
In our context, the lemma says that the total Segre class, , is equal to up to numerical equivalence and hence , i.e., that . Thus, the relations in are concisely expressed by the equations
[TABLE]
Moreover, it does no harm to assume that , so that is a rank- vector bundle over . Consequently, we have and . Representing elements in by elements in , we may think of the Gysin morphism as a map
[TABLE]
Here is the -th elementary symmetric function in ; while are arbitrary elements in ; and denotes an equivalence class.
Lemma 3.19**.**
For any strictly increasing sequence with , we have
[TABLE]
Proof.
Applying Corollary 3.17(2), we see that
[TABLE]
On the other hand, using the conventions established in Notation 3.13 it is clear that
[TABLE]
where . Substituting , we get
[TABLE]
It is well-known that the determinant
[TABLE]
is a multiple of the Vandermonde determinant and can be computed as ; cf. [ACGH85, Ch.7, proof of Thm 4.4]. The claim follows. ∎
3.4. The Littlewood–Richardson rule in terms of Schur functions
Schur functions provide a convenient way of writing down the product of Schubert classes on a Grassmann bundle.
Lemma 3.20**.**
Let be the ring of symmetric polynomials in variables. Fix with and suppose is any partition for which . We then have in .
Proof.
We begin by writing
[TABLE]
where denotes the set of all partitions of length at most and size , and where the coefficients are non-negative integers. Now suppose . Since is rectangular with rows, the length of is then necessarily exactly and is still a partition.
According to the Littlewood-Richardson rule, counts the number of Littlewood-Richardson tableaux. These are semi-standard Young tableaux of shape and weight whose characteristic property is that concatenating their reversed rows yields a word which is a lattice permutation; that is, in every initial part of the word, any number occurs at least as many times as . The characteristic property immediately forces all the entries in the first row of such a tableau to be 1, and no further ones can occur in this tableau, since entries in every column form a strictly increasing sequence. Thus, .
Now remove the first row; the result is a Littlewood-Richardson tableau of shape and weight on the alphabet . Applying the same argument as before, we get . By induction, we conclude that and in particular . ∎
Remark 3.21*.*
An alternative proof of the same result may be derived from Corollary 7.15.2 in [SF99], which establishes that is equal to the coefficient of in , where .
To wit, note that when , as the only semi-standard Young tableau on the alphabet is the one with every column equal to . It follows that
[TABLE]
Note also that is a strictly decreasing sequence. But is strictly decreasing if and only if the permutation is the identity. So if then necessarily , in which case .
Using the alternative convention for Schur polynomials as in Definition 3.7, Lemma 3.20 becomes the following statement.
Corollary 3.22**.**
Let be an arbitrary strictly increasing sequence of non-negative integers and (). We then have
[TABLE]
where .
3.5. Characteristic classes of symmetric squares of vector bundles
The symmetric square of the tautological subbundle on (the total space of) a Grassmann bundle plays a key role in the enumerative geometry of SMRC loci. Laksov, Lascoux and Thorup gave formulae (hereafter, the LLT formulae) that express the Chern and Segre classes of the symmetric square of a vector bundle in terms of its Segre classes.
Theorem 3.23** (Proposition 2.8.4, 2.8.6, [LLT89]).**
Let be a vector bundle of rank . We have
- (1)
; 2. (2)
.
Here both sums are taken over all strictly increasing sequences of non-negative integers. The coefficients are defined by
[TABLE]
and the coefficients are given by the following recursive relation:
- (1)
, ; 2. (2)
* if ;* 3. (3)
.111In (2) and (3), if , is interpreted as 0.
Remark 3.24*.*
Let denote the infinite matrix whose -th entry is , and let denote the submatrix of consisting of its -th columns and -th rows, where . We then have
[TABLE]
As pointed out in [LLT89], the convention that Laksov–Lascoux–Thorup use for the -th Segre class, differs from that of [Ful98] by a factor of . In other words, the Segre classes in [LLT89] are the Segre classes of in [Ful98]. Since we adopt the conventions of [Ful98] in this paper, we rewrote the original formula of Laksov-Lascoux-Thorup accordingly.
3.6. Shifted Schur functions
The LLT coefficients of the preceding subsection are closely related to the shifted Schur functions introduced by Okounkov and Olshanski in [OO97].
Notation 3.25** ([OO97], (5.2), (5.3)).**
Let be any Young diagram. Starting from the upper-left corner, identify the box in the -th row, -th column with the integer vector . The -th generalized raising factorial of is defined by
[TABLE]
Notation 3.26** ([OO97],(11.1)).**
Similarly, the -th generalized falling factorial of is defined by
[TABLE]
More generally, given any skew diagram , we set .
Example 3.27**.**
. In general, we have , where is the conjugate of .
Definition 3.28**.**
Let be a partition of length at most . The shifted Schur polynomial in variables with respect to is
[TABLE]
where (see Notation 3.9).
In [OO97], Okounkov and Olshanski introduced and studied the ring of shifted polynomials in variables to be the algebra consisting of all -variable polynomials that become symmetric after shifting variables according to
[TABLE]
The ring of shifted symmetric functions333Okounkov and Olshanski showed there is a natural map defined by setting equal to 0, and that a well-defined limit over all exists in the category of filtered algebras. carries structures similar to those of the ring of classical symmetric functions.
Theorem 3.29**.**
[Theorem 4.1, 4.2, [OO97]] There exists an involution satisfying the following properties:
- (1)
, for all . 2. (2)
.
Here denotes the conjugate of an arbitrary partition .
A key observation is that the coefficients can be written in terms of special values of shifted symmetric functions:
[TABLE]
in which by convention we interpret the empty product as 1 and is as given in Notation 3.9. Notice also that
[TABLE]
where denotes the Vandermonde polynomial in variables, and recall from Remark 3.11 that
[TABLE]
As a result, we are primarily interested in special evaluations of shifted Schur functions along staircase partitions.
Definition 3.30**.**
Given a nonnegative integer , the -th staircase partition is .
Theorem 3.31** (Theorem 8.1, [OO97]).**
Let , be two partitions such that and . We have
[TABLE]
where denotes the number of SYT of (skew) shape .
Corollary 3.32**.**
Given any non-negative integer , we have .
Proof.
Note that is the size of . The corollary then follows from Theorem 3.31 together with the standard representation-theoretic fact that (see, e.g., [Mac98, Example I.7.3]):
[TABLE]
∎
In [OO97], Okounkov and Olshanski also compute a generating function for , which they use to deduce a Jacobi–Trudi type formula for shifted Schur functions. Their result leads to the following proposition.
Proposition 3.33**.**
Given any partition , let denote the function . We have for some . Futhermore, the polynomials are such that
- (1)
; 2. (2)
; and 3. (3)
.
Proof.
Our point of departure is the generating series for given in [OO97, Thm 12.1]:
[TABLE]
From (2), we deduce that
[TABLE]
for all . 444To clarify, we have when . Recall that an empty product is interpreted as 1.
Now
let . We then get , and the left-hand side of (3) can be written as
[TABLE]
Notice that . Accordingly, we have
[TABLE]
Meanwhile
, we have
[TABLE]
Let denote the latter -meromorphic function; note that
[TABLE]
It follows that . Applying formulae of Faulhaber for sums of (alternating) consecutive powers as in [Knu93] and [How96], we deduce that
[TABLE]
where is a polynomial of degree such that , and is the -th Euler polynomial defined by . At this stage, it is worth remarking that always holds. Indeed, when is odd, the change of variable preserves ; while if is even, it is easy to check that the exponential generating series and for Euler polynomials and (respectively) with even indices are equal.
On the other hand, it follows from the Faà di Bruno formula together with the fact that that can always be written as an integer linear combination of terms , where is a partition of and . An induction on now shows that there exist polynomials such that for all and . And using the fact that
[TABLE]
we conclude that the same polynomials are such that for and . Another induction on shows, moreover, that .
We now show that . To this end, note that when , [OO97, Thm 3.1] implies that . In particular, are roots of . Using , it follows immediately that for . On the other hand, since , we further get that . To determine the coefficient , we evaluate at and apply Theorem 3.31 to get
[TABLE]
Thus , which proves item (1).
To draw a similar conclusion for for an arbitrary partition , we apply the Jacobi–Trudi formula [OO97, Thm 13.1] for shifted Schur functions that relates arbitrary partitions to rectangular ones:
[TABLE]
Using the polynomiality of the , we conclude immediately from (4) that there exist for which , for and , which proves item (2).
Finally, note that the -th entry of the determinant in Equation 4 is a linear combination of ’s, among which is of highest degree in . Thus, when we expand the determinant, we find that the leading coeffcient is precisely , where . But
[TABLE]
which proves item (3). ∎
The final ingredients we shall need from [OO97] are two branching rules for shifted Schur functions:
Lemma 3.34**.**
[TABLE]
where runs through all partitions of the form , for some .
Lemma 3.35**.**
[TABLE]
where if and only if for every ), and is the generalized falling factorial (cf. Notation 3.26).
Remark 3.36*.*
Lemmas 3.34 and 3.35 are special cases of [OO97, Thm 9.1] and [OO97, Thm 11.1], respectively. We state them in the above forms for future reference as well as for the convenience of the reader.
4. The Strong Maximal Rank Conjecture for quadrics
4.1. The statement of the conjecture
The focus of this section is the Strong Maximal Rank Conjecture for quadrics, which we state as follows:
Conjecture 2** (Strong Maximal Rank Conjecture).**
Fix such that and . Let denote a (Brill–Noether and Petri) general curve of genus .
- (1)
Suppose . The determinantal locus
[TABLE]
is non-empty and every irreducible component is at least -dimensional. 2. (2)
When , for all in , the multiplication map has maximal rank.
Note that our formulation of the conjecture differs from the original version stated by Aprodu and Farkas in [AF11], where they impose the restriction . Aprodu and Farkas conjectured that with this extra assumption should be exactly -dimensional. However, we are mainly concerned with the non-emptiness of , and for our main application to rank two Brill–Noether theory it is necessary to remove the restriction .555A relaxed version of the conjecture was also stated in [FO12], removing the assumption that . However, we do not expect that in this generality the dimension of is exactly ; see Example 4.9.
Hereafter, we mainly focus on part (1) of the conjecture. First of all, we recall how to realize as the degeneracy locus of a vector bundle morphism over . To this end, fix an effective divisor of degree on and let and denote its pullbacks to and , respectively. Recall also that is the closed subscheme of that is the zero locus of the bundle map
[TABLE]
where is the tautological bundle of the relative Grassmannian. Let denote the pull-back of to . We collect the relevant morphisms in the following diagram:
[TABLE]
Set . As is a bundle of rank , is a bundle of rank over .
Now let be a Poincare line bundle on and let denote its pull-back to . We claim that is a rank vector bundle over . Since is Petri-general and , by Petri’s theorem we have , for every ; see Lemma 4.1 below. Thus , and it follows from Grauert’s theorem that is locally free of rank over and . From the long exact sequence in cohomology, it follows that the sequence
[TABLE]
is exact.
We now describe the morphism . To this end, let be the pull-back of to . Composing with the natural morphism
[TABLE]
we get a morphism , and hence . Composing the latter with the natural morphism yields .
From the definition of , is the zero map and thus is zero. Hence, the morphism factors through . The morphism is then induced via descent by the morphism of locally-free sheaves . As a consequence, is scheme-theoretically defined as the degeneracy locus of (cf. [Far09], [FO11]), which is closed in .
The following result is well-known; but for completeness and for lack of an adequate reference, we include its proof.
Lemma 4.1**.**
Let be a line bundle on a Petri-general curve for which ; then .
Proof.
By the Gieseker-Petri theorem, the multiplication map is injective. By Serre duality, we have . If , then and its dual are nonzero. Thus, there exists a nonzero and hence injective morphism from to , and the induced linear map is multiplication by some rational function . Since , there exist linearly independent sections of for which is a non-zero element in the kernel of , a contradiction. ∎
Definition 4.2**.**
We call the SMRC locus for quadrics.
The regime
Hereafter, we focus on the special case where . As we shall see later, this will be the case relevant to the Bertram–Feinberg–Mukai conjecture; it is also cohomologically simpler than the general case.
Lemma 4.3**.**
When , we have .
Proof.
This essentially follows from standard properties of cohomology and base change, as applied to the fibered square
[TABLE]
and the invertible sheaf on . Recall first . Note that implies for all . Consequently, cohomology and base change commute for in degree 0, i.e. . ∎
Remark 4.4*.*
The same argument shows that without making any assumption on relative to . However, without the hypothesis , we do not have and in general is not locally free.
Hereafter we adopt the following approach to studying the non-emptiness of the SMRC locus.
Strategy.
Let be the class of the degeneracy locus of in (in particular is the pullback of along ), and let be the class of the degeneracy locus of in . If , then ; in other words, is non-empty. Furthermore, if , then is non-empty.
We briefly outline how to calculate and . The former may be calculated using Porteous’ formula, for which we need to determine the Chern classes of and . To handle , we apply the LLT formulae of Theorem 3.23. To calculate the Chern classes of , on the other hand, we apply the Grothendieck–Riemann–Roch formula. Following Theorem 3.1, (the pullbacks of) these classes naturally belong to .
Meanwhile, is the zero locus of the bundle map
[TABLE]
so itself may be computed using Porteous’ formula. Recall that the Chern classes of are among the generators of over . On the other hand, is a direct sum of line bundles algebraically equivalent to the trivial bundle [ACGH85, Sec. VII.2]. So, modulo algebraic equivalence (and hence up to numerical equivalence), has trivial Chern classes. It follows that
[TABLE]
Assume . Porteous’ formula yields
[TABLE]
Example 4.5**.**
Applying Lemma 3.19 in tandem with (5), we recover the well-known expression for the class of inside :
[TABLE]
where . Equivalently,
[TABLE]
The injective and surjective ranges
Hereafter, we shall refer to cases for which as cases within the injective range and cases for which as cases within the surjective range. Since the surjective range will be the main focus of the remainder of the paper, we record the most salient inequalities operative in that range as follows:
Basic Inequality 4.6**.**
Let and suppose . We have
[TABLE]
4.2. Some known cases
Various cases of part (2) of Conjecture 2 have already been established.
- (1)
Aprodu and Farkas show in [AF11, Prop. 5.7] that when and , part (2) of Conjecture 2 holds; that is, the special maximal-rank locus is empty. In other words, the multiplication map is surjective for every degree line bundle . (Note that in this case, , and consequently every is a complete linear series.) In fact, the same is true for the -th multiplication map , for all . 2. (2)
Farkas and Ortega show in [FO11, Prop. 2.3] that when and , part (2) of Conjecture 2 holds; that is, is empty. In other words, is injective, for any -dimensional subspace of sections of a degree- line bundle . On the other hand, Teixidor i Bigas shows in [TiB03] that whenever , part (2) of Conjecture 2 holds for complete linear series. 3. (3)
More recently, two separate groups [JP18, LOTiBZ18] have shown (working independently, and using different methods) that for , and , the map is injective for every line bundle of degree on a general curve of genus . This means, in particular, that the respective loci where fails to be injective determine divisors in the space of linear series and in . This potentially has important implications for the birational geometry of the moduli space of curves in genus 22 and 23. Indeed, in [Far09, Far18]), Farkas computed the classes of the corresponding virtual divisors, and showed that their (virtual) slopes are strictly less than . To conclude, it remains to establish that the natural forgetful projections from to and are generically finite along SMRC divisors.
4.3. Excess components of the SMRC locus
We now turn to part (1) of Conjecture 2. We describe a family of cases within the surjective range for which the associated SMRC loci are always non-empty. In fact, it is easy to see that whenever they exist, non-very ample linear series contribute components of larger-than-expected dimension to SMRC loci.
Proposition 4.7**.**
Let be non-negative integers satisfying the following conditions:
- (1)
; 2. (2)
; and 3. (3)
.
For every Brill–Noether general curve , the SMRC locus has an excessively large component.
Proof.
The second condition, coupled with the fact that is Brill–Noether general, implies that there exist ’s (and hence ’s) on , and that every (and every ) is a complete linear series. Now let denote a on . Then for every point , is a on which is not base-point free, since the inclusion is an isomorphism.
On the other hand, condition (1) implies that , and hence must be base-point free. But the image of the multiplication map is contained in . The upshot is that is not surjective, and belongs to .
Finally, condition (3) implies that falls in the surjective range, and moreover, that
[TABLE]
But by construction, contains an isomorphic image of , of dimension . It follows, in particular, that contains an excessively large component. ∎
Remark 4.8*.*
Whenever satisfies the condition in Proposition 4.7, we say that is a trivial instance within the surjective range. This naturally raises the question of whether non-empty non-trivial SMRC loci exist.
Example 4.9**.**
It is easy to check that is a trivial instance in the surjective range. In this case , so there are finitely many ’s on . Each one of these generates a 1-dimensional family of linear series in along which fails to be surjective.
More generally, whenever contains a large component consisting of non-very ample linear series, may have excessively large components.
Lemma 4.10**.**
Suppose is a non-very ample on a general genus curve such that . Then is not surjective.
Proof.
In light of Proposition 4.7, it suffices to consider the case where is base-point free. As is not very ample, there is some pair of points (not necessarily distinct) for which
[TABLE]
Consequently, has no section which vanishes to order exactly 1 at and does not vanish at . But , so is very ample. It follows that is not surjective. ∎
Example 4.11**.**
Suppose . Every on a general genus 13 curve is a base-point free complete linear series with . It follows that every is of the form , where is an effective divisor of degree 6. Since a general curve of genus 13 has no , we have if and only if .
On the other hand, a fails to be very ample if and only if it contains a . In other words, fails to be very ample if and only there exist such that is a . Now let denote the incidence variety of degree 6 effective divisors contained in a . It is easy to see that the projection over the curve has one-dimensional fibers, so is irreducible and 2-dimensional. Furthermore, a 6-tuple of points can be contained in at most one , because there is no on a general curve of genus . It follows that has a component that is at least 2-dimensional. However, . So has an excessively large component in this case.
In light of Proposition 4.7 and Lemma 4.10, the following question is fundamental.
Question 4.12**.**
When , does contain a very ample ?
It is worth mentioning that in the original SMRC proposed by Aprodu and Farkas, the condition implies that every on a general curve is very ample. (Indeed, this follows from [Far08, Thm 0.1].) So answers to question 4.12 will naturally extend the work of Aprodu and Farkas. As we will see later, affirmative answers to question 4.12 will also lead to solutions to existence problems in higher-rank Brill-Noether theory.
5. Enumerative calculations for SMRC loci
5.1. Chern classes of and , and the degeneracy class of
Situation 5.1**.**
In this section, we make the running assumption that .
Our first goal is to determine the Chern classes of . Recall that is the pull-back of an effective divisor on to and is a Poincaré line bundle on . So it suffices to compute the Chern classes of . Notice also under our running assumption that for all , and hence for every . Grothendieck–Riemann–Roch now yields
[TABLE]
The Todd class of an abelian variety is trivial, so is the pull-back of . Accordingly, (7) reduces to
[TABLE]
We still need to compute , or equivalently . The latter is, however, given explicitly in [ACGH85, Ch. VIII]. The upshot is that up to numerical equivalence
[TABLE]
Equivalently, we have .
Applying the LLT formulae of Theorem 3.23 in our context, we obtain
[TABLE]
where the summations are over all degree- Schur functions in the Chern roots of , and the and are particular combinatorial coefficients defined by determinantal formulae; see Theorem 3.23 for their precise definition.
We now return to the class calculation initiated in section 4.1. By Porteous’ formula, the class in of the locus over which the vector bundle map fails to be of maximal rank is given by
[TABLE]
in which , denotes the set of variables
[TABLE]
and (see Notation 3.14).
5.2. Classes of SMRC loci
Combining our formulae from sections 4.1 and 5.1, we deduce that up to numerical equivalence the class of is given by
[TABLE]
where and . As we mentioned in section 4, there is now a basic dichotomy depending on the sign of .
5.2.1. The surjective range: .
This is the case of primary interest to us. In this case, and by applying [Ful98, Lemma 14.5.1] we may rewrite (8) as
[TABLE]
where and denotes the set of variables
[TABLE]
Simplifying, we find that the class of the SMRC locus is given by
[TABLE]
where . To go further, will explicitly rewrite the products using Lemma 3.20. Now assume . Let and . Applying Lemma 3.19 and Corollary 3.22 to the intersection product given by equation (9) as well as one of the LLT formulae, we get666Recall (Remark 3.11) that . In particular, for all .
[TABLE]
To prove the corresponding special maximal-rank locus is non-empty, for any smooth projective curve of genus , it suffices to show that
[TABLE]
Example 5.2**.**
An important case is that in which , , and . Here
[TABLE]
5.2.2. The injective range: .
In this case, and we get
[TABLE]
where and . Consequently,
[TABLE]
To show the corresponding SMRC locus is non-empty, it suffices to show that
[TABLE]
is nonzero. Although the non-emptiness of the SMRC locus in the injective range is not the focus of the current paper, we conclude this subsection by giving a concrete numerical example in this range.
Example 5.3**.**
Consider the case where , , and . Here
[TABLE]
and is numerically equivalent to a zero-cycle of degree 10.
5.3. Relating the SMRC degree to special values of shifted Schur functions
In order to certify the nonemptiness of SMRC loci, we need to show that the degree described by Equation 11 is non-zero. Interesting combinatorial questions arise as involves the combinatorial coefficients of Theorem 3.23(1).
Recall from Equation 1 that each is a special value of a shifted Schur function. In particular, setting , we may rewrite the formula for the SMRC degree in Equation 11 as follows:
[TABLE]
where , is the staircase partition of Definition 3.30, and is the dimension of the irreducible representation of indexed by (see [FH13, 4.11]).
Next, we rewrite in terms of evaluations of shifted Schur functions along staircases. Indeed, by [OO97, Thm 3.1], we have unless . It follows that
[TABLE]
Remark 5.4*.*
Notice that the coefficient is the coefficient of the cohomology class of (see Example 4.5).
Hereafter, let be the “reduced” version of defined by . We shall introduce one last auxiliary notion in order to simplify the computation of in the next subsection.
Definition 5.5**.**
Given positive integers and , let
[TABLE]
where and is the generalized raising factorial (see Notation 3.25).
Clearly is nonzero if and only if is nonzero. Note that the class calculation carried out for in Example 4.5 implies that the class of equation (10) is precisely
[TABLE]
Remark 5.6*.*
Using the standard (Hall) scalar product on the ring of symmetric functions , we may rewrite in a more compact form. Indeed, the Schur functions determine an orthonormal basis for with respect to , while the Murnaghan–Nakayama rule [SF99, Thm 7.17.4] implies that , where denotes the -th power sum symmetric function. It follows that
[TABLE]
5.4. for small values of
We now apply the technical results of the final part of Subsection 5.3 to compute whenever .
Lemma 5.7**.**
Set and . We have
- (1)
; 2. (2)
; 3. (3)
; 4. (4)
; 5. (5)
; 6. (6)
F_{g,r,d}(5)=\frac{2^{2N-5}}{(A+4)_{9}(N-5)!5!}\cdot\big{[}B(B-1)(B-3)(72+A^{4}(B-4)(B-2)-20A^{2}(B-4)(B-1)+6B(13B-48))\big{]}; 7. (7)
F_{g,r,d}(6)=\frac{2^{2N-6}}{(A+5)_{11}(A+1)_{3}(N-6)!6!}\cdot\big{[}B(B-1)(B-3)((B^{3}-11B^{2}+38B-40)A^{8}-(41B^{3}-411B^{2}+1198B-840)A^{6}+2(229B^{3}-2009B^{2}+4636B-1680)A^{4}-2(629B^{3}-4629B^{2}+8256B-1280)A^{2}+240B^{3}-1440B^{2}+4800B)\big{]}; and 8. (8)
F_{g,r,d}(7)=\frac{2^{2N-7}}{(A+6)_{13}(A^{2}-1)(N-7)!7!}\cdot\big{[}B(B-1)(B-3)(B-6)((B^{3}-11B^{2}+38B-40)A^{8}-(71B^{3}-711B^{2}+2068B-1440)A^{6}+14(112B^{3}-977B^{2}+2233B-840)A^{4}-2(5699B^{3}-40149B^{2}+67266B-9680)A^{2}+15780B^{3}-73200B^{2}+105300B-9000)\big{]}.
Proof.
When , the results follow directly from the definition of . Now say . Theorem 3.29 and Corollary 3.32 imply that , and it follows that
[TABLE]
The determination of for is similar, but more involved. We apply Proposition 3.33, Lemma 3.34 and Lemma 3.35 to compute all of the functions with . In doing so, we use MATLAB to explicitly solve a number of linear systems of equations. ∎
In Lemma 5.7 we computed the functions explicitly for . Since is an alternating sum of all with , we thus obtain explicit formulas for whenever . When is small, we can explicitly check the positivity of (and hence the positivity of ). A key point is that whenever , all are expresed in terms of and ; it turns out to be useful to estimate the ratio .
Lemma 5.8**.**
Let . Under the numerical assumptions of Basic Inequality 4.6, we have
- (1)
; and if , then . 2. (2)
.
Proof.
The fact that for every in the surjective range is clear. Moreover, we have
[TABLE]
Whenever , we have , and therefore , so that . On the other hand, when , Basic Inequality 4.6 implies that
[TABLE]
and hence that . If , the only possibility is that , in which case , i.e. ; while if , no triple satisfies our numerical conditions.
The only other situation in which may not be strictly positive is when and . When this happens, will be strictly positive as soon as and , as it then follows that . It therefore remains to check those cases in which or .
Accordingly, suppose that and . If vanishes, then necessarily and . In that case, the fact that implies that and . On the other hand, if , then and together force and . So item (1) is proved.
To verify item (2), note first that Basic Inequality 4.6 also implies that
[TABLE]
or equivalently, that
[TABLE]
It follows that , i.e., that . So we deduce that . ∎
Example 5.9**.**
Let as above. When , we have
[TABLE]
Applying Lemma 5.8(1), we deduce that is non-negative, and strictly positive except when .
Similarly, when , Basic Inequality 4.6 implies that , while
[TABLE]
Applying Lemma 5.8(2), we see that whenever ,
[TABLE]
in which the rightmost quantity is the larger root of the polynomial . It follows that in this case.
We are left to deal with those cases in which . When , from Basic Inequality 4.6 we have and thus . Lemma 5.8(2) now yields . Applying the argument of the preceding paragraph, we deduce that .
When , the unique triple that fulfills all of our numerical constraints is , in which case . Finally, , there is a unique possibility, namely ; in this case, as well.
Calculating explicitly becomes difficult as soon as . However, by applying a single branching rule (Lemma 3.34) for shifted Schur functions, we may conclude that is a positive class when is relatively large with respect to . To this end, we introduce the following rational function on partitions.
Notation 5.10**.**
Given any partition of length at most , let
[TABLE]
Notice that the rational function is precisely the reciprocal of the product of all the shifted contents , for . The following result is straightforward to verify.
Lemma 5.11**.**
We have
[TABLE]
for all partitions and related by .
Proposition 5.12**.**
Fix a choice of . The SMRC degree is a positive rational number whenever .
Proof.
Let , where , whenever . Whenever , we further set
[TABLE]
For any fixed value of , is a quadratic polynomial in with highest-degree coefficient equal to .
Given
a partition , let . According to Lemma 3.34, we have
[TABLE]
where . Equivalently, we have
[TABLE]
where .
Now say is odd. Define , for . then decomposes as , and equation (17) implies that
[TABLE]
Here , while the quotients are positive rational numbers such that , by the usual branching rule. More importantly, we have
[TABLE]
since and . Therefore, provided holds for all , we get . Since , it further suffices to establish that . Now is a quadratic polynomial in , with positive highest-degree coefficient, it is easy to see that when is sufficiently large with respect to .
Similarly, when is even, we decompose as , where and for . An argument analogous to that used in the case of odd allows us to conclude that when is sufficiently large with respect to .
On the other hand, is equivalent to the statement that the quadratic polynomial is positive. It is not hard to see that the largest root of in is less than . ∎
Finally, we classify all admissible triples according to the positivity of their SMRC loci whenever .
Proposition 5.13**.**
The class of equation (10) is strictly positive when except when either
- (i)
, and ; or
- (ii)
, and
and is unconditionally strictly positive whenever .
Proof.
Given Example 5.9, it only remains to show that for between 3 and 7.
Applying Proposition 5.12, we conclude that whenever . Thus we are left over with finitely many cases of feasible triples . We then determine a lower bound for for each value of by applying Basic Inequality 4.6:
- (1)
When , ; 2. (2)
when , ; 3. (3)
when , .
Finally, we check the positivity for the finitely many remaining possible triples using MATLAB; see the ancillary file:
https://drive.google.com/file/d/1Az5WOZyoa_UzQvktT7KuilTpbddL4ANn/view.
In this file, we list those values of all for which and lies between the given lower bound and . ∎
Remark 5.14*.*
The fact that is consistent with Mumford’s theorem [Mum10, Thm 6], which establishes that the quadratic multiplication map associated with (the complete series determined by) a line bundle with is surjective.
6. The Bertram-Feinberg-Mukai conjecture and its connection with the SMRC
In this section, we explain the connection between the BFM conjecture (Conjecture 1) and the Strong Maximal Rank Conjecture for quadrics.
Our point of departure is the fact that every stable rank-two vector bundle with canonical determinant fits into a short exact sequence of the form
[TABLE]
Every extension (18) naturally determines an element . Given any such extension , we have , where is the linear map in the cohomological long exact sequence induced by (18).
In order to verify (the existence portion of) Conjecture 1, our aim is to produce extensions (18) of line bundles whose associated rank-two vector bundles are stable and satisfy , whenever . In doing so, we try to simultaneously ensure that is as small as possible relative to and that has small rank. Heuristically speaking, we specify a stable vector bundle by identifying one of its minimal quotient line bundles, to the extent that this is possible given the cohomological condition .
The impulse to consider extensions of (relatively) small degree line bundles comes from two sources. First of all, maximal sub-line bundles (and hence minimal quotient bundles) of a vector bundle are in some sense canonical. For example, it is known that the space of maximal subbundles of any rank-two vector bundle is at most one-dimensional [LN83, Cor. 4.6], and is further known to be finite or even a singleton [LN83, Cor. 3.2 and Prop. 3.3] for general rank-two vector bundles with certain prescribed degrees and Segre invariants. Second, it is easier to certify that rank-two bundles determined by extensions of line bundles of relatively small degrees are stable. In fact, we shall see that in many such cases stability is automatic.
On the other hand, since we focus on extensions of line bundles in which has relatively large cokernel dimension, quite often the loci of the line bundles being extended are related to the SMRC loci for quadrics we defined in Section 4. To see this, we analyze the two respective conditions:
- (1)
; and 2. (2)
.
The locus of line bundles in with is easy to describe: it is precisely the Brill-Noether locus .
We now turn to the dimension of . The assignment that takes an extension to the corresponding coboundary map describes a linear map from to , which is dual to the multiplication map . It is easy to see that the condition may be reformulated as the statement that some -dimensional subspace is such that , where is the restriction of to , and is viewed as a linear function on . (See [CF15, Remark 5.7] for a more general statement.)
The upshot is that in order to show existence of rank two linear series with canonical determinant and many sections, it is useful to study loci in of specified rank.
Definition 6.1**.**
Let denote the locus of in where .
It is not hard to see that is a determinantal scheme of expected codimension . Indeed, it is precisely the -th degeneracy locus of the pull-back of the universal linear map on . Accordingly, we have a natural filtration
[TABLE]
Note that if and only if the multiplication map is surjective.
For our application to higher-rank Brill–Noether theory, we will mainly consider non-general extensions of (possibly) non-general linear series by their Serre duals. Whether or not invertible subsheaves of lift to subsheaves of (stable) rank two bundles is related to the existence of on that fail to impose independent conditions on global sections of .
Proposition 6.2**.**
Suppose is nonzero, where with . Let denote the rank-two vector bundle obtained from , and suppose further that . Suppose is an effective divisor on .
- (1)
Whenever there exists a morphism for which the composition is the identity and realizes as a sub-bundle of , we have
[TABLE] 2. (2)
If, moreover, and is surjective, then is not stable.
Proof.
The assumption in item one implies there is a short exact sequence
[TABLE]
and hence . On the other hand, Riemann–Roch together with the long exact sequence in cohomology induced by the inclusion of in imply that
[TABLE]
and the claim in item one follows immediately.
Now suppose and is surjective. The multiplication map factors as
[TABLE]
The fact that implies that is in the kernel of ; since contains , vanishes on . It follows that is in the kernel of the dual of the inclusion map , namely
[TABLE]
This means, in turn, that the top row of the following diagram splits:
[TABLE]
where . Thus is an invertible sub-sheaf of (and hence of ). Since , we have , and consequently is not stable. ∎
Remark 6.3*.*
Geometrically, Proposition 6.2 relates the liftability of invertible subsheaves of to the existence of divisors whose images under span secant spaces of dimensions prescribed by the rank of the evaluation map . This, in turn, explains our terminological use of “secant divisors”.
Finally, a classical result of Nagata gives a lower bound on the degree of a maximal subbundle of a rank-two vector bundle:
Theorem 6.4** ([Nag70]).**
Any maximal-degree line subbundle of a rank-two vector bundle over a non-singular projective curve of genus over an algebraically closed field satisfies
[TABLE]
Applying this theorem to the case where is a rank-two vector bundle with canonical determinant yields the following useful statement.
Corollary 6.5**.**
The degree of the minimal quotient line bundle of a rank-two vector bundle with canonical determinant is at most .
So by replacing a line bundle with its Serre dual if necessary, it suffices to consider for which when constructing stable rank-two vector bundles with canonical determinant via extensions as in (18).
6.1. The search for minimal quotient line bundles
In order to prove the existence portion of the BFM conjecture it suffices to show that for every and for the maximal integer for which , there exists a stable rank-two vector bundle over a general genus curve for which and .
Recall that we are interested in extensions of the form (18) over a curve that is (Brill-Noether and Petri-) general. We would also like to minimize the degree of in such extensions. In light of this, Corollary 6.5 motivates the following definition.
Definition 6.6**.**
The minimal BN-compatible degree with respect to is
[TABLE]
Now suppose . The Brill-Noether theorem and Serre duality together imply that
[TABLE]
where , and .
Definition 6.7**.**
Fix positive integers such that . Suppose and that satisfies the constraints in (19); we then say that is BN-compatible with respect to .
It is not always the case that a stable rank-two vector bundle for which and has a quotient line bundle of degree . For example, if for all line bundles of degree , the multiplication map is surjective and , then every extension of the form
[TABLE]
splits (and hence is not stable). However, on the positive side we have the following result.
Proposition 6.8**.**
Suppose is a non-trivial extension of line bundles such that and , then is stable and is a minimal quotient line bundle of .
Proof.
Let be a sub-line bundle of . The composition of morphisms of line bundles is either zero or injective. If it is zero, the morphism must factor through the kernel of , which is . Then .
If it is injective, let . Since by construction, we must have either or . However, the second case is impossible as this would force and thereby violate our assumption that is a non-trivial extension.
It follows that admits no sub-line bundle of degree or greater. Hence is stable. The fact that is minimal follows from the definition of and the stability of . ∎
Corollary 6.9**.**
The existence portion of the BFM conjecture holds under either of the following two circumstances.
- (1)
There exists a BN-compatible pair such that , and falls within the injective range. 2. (2)
There exists a BN-compatible pair such that , , falls within the surjective range, and .
Proof.
In both cases, the multiplication map fails to be surjective and hence its dual fails to be injective. Case (1) follows from the classical Maximal Rank Conjecture, which is now a theorem of Eric Larson; see [Lar17]. Larson’s theorem implies that there exists a nonzero extension in with , in which case and hence . In both cases, the stability of follows from Proposition 6.8. ∎
Remark 6.10*.*
Note that we ignore the possibility that for a practical reason. Namely, when and is relatively large with respect to , the condition will always hold as there is no strictly semi-stable rank-two vector bundle on a Brill-Noether general curve with independent sections. See [Zha16, Lemma 3.12] for a precise statement.
Example 6.11**.**
The following are two examples of those cases covered by Corollary 6.9.
- (1)
Set . In this case, , is a BN-compatible pair that falls within the surjective range, and . It follows from our calculation of the SMRC class in Proposition 5.13 that is nonempty. Consequently, the existence portion of the BFM conjecture holds for . 2. (2)
Set . In this case, , and is a BN-compatible pair that falls within the injective range. Thus, the existence portion of the BFM conjecture also holds for .
6.2. Existence in small genera
The existence portion of the BFM conjecture for small genera () was first established by Bertram and Feinberg in [BF98]. Here, we show how these cases of the conjecture can be easily recovered from our MRC-based viewpoint.
The -values listed here are maximal with respect to the given -values such that . As we shall see, the case is exotic in the sense that , but there is no stable bundle of rank two with two sections on a genus 2 curve. We nevertheless describe the situation in this case, since the approach is the same as for other low genera cases.
We will show that every semi-stable rank-two vector bundle with canonical determinant and two sections is strictly semi-stable. The minimal quotient line bundle of a rank two vector bundle with canonical determinant on a genus 2 curve has degree at most . For to hold, must fit into an extension of the form
[TABLE]
since and together imply that .
Now consider as a point in . By [LN83, Prop. 1.1], lies in whenever is not stable, where is the image of the morphism . In particular, a general extension will contribute a stable bundle . On the other hand, it is easy to check directly that is an isomorphism, since . It follows that the dual is also an isomorphism. Consequently, every point in corresponds to a symmetric bilinear map on . We claim that the locus where the bilinear map has rank one is precisely . Indeed, we have if and only if , where is some 1-dimensional subspace of , in which case we have for some . But then ; that is, is the image of on . It follows that is a sub-bundle of , and that is semi-stable.
The same argument also yields the existence of stable rank two bundles with canonical determinant and one global section.
In this case, . We conclude by applying Corollary 6.9(1) to a general on a general curve of genus 3.
In this case, . A general extension of a general on a genus 4 curve by its Serre dual yields a vector bundle with three sections; and since , the bundle is at least semi-stable. If we further assume that the , say , is base-point free, the bundle is in fact stable. Indeed, if is BN-compatible then necessarily . On the other hand, if is base-point free, then its associated complete series has no sub-, and hence has no destabilizing subbundle of degree 3, which means that is stable.
It is easy to see that a base-point free exists on a general genus 4 curve. Namely, consider the natural (addition) map , where is the Brill-Noether locus of ’s inside . By the Brill-Noether theorem, the domain is 1-dimensional, while the target is 2-dimensional.
In this case, , and the smallest for which some pair is BN-compatible is with . Now let be a general , and let be a general extension of by its Serre dual. By arguing much as in the case, we conclude there is some non-trivial extension that produces a semi-stable vector bundle with 4 sections. In fact, since is 2-dimensional, the locus in of such extensions is 1-dimensional. By [LN83, Prop. 1.1], to conclude that the resulting vector bundle is stable, we need to certify that , where is the image of under the morphism .
We now claim that is not contained in . To see this, note first that for to lie on a secant line of spanned by and means that . (In this case is very ample, so we may identify points on with points on .) Moreover, . By the Brill-Noether theorem, a on a general genus 5 curve is base-point free, so , for any (not necessarily distinct). It thus suffices to show that there exists a codimension-1 subspace of containing but not of the form . And indeed, the closed immersion identifies with (further identified with a line in the dual vector space), while the locus of codimension-1 sub-spaces of containing is a line in . Since a positive genus curve cannot contain a line, there is some codimension-1 subspace of containing but not of the form ; the claim follows, and we conclude.
In this case, , and we conclude by applying Corollary 6.9(1) to a general on a general curve of genus 6.
In this case, , and we conclude by applying Corollary 6.9(1) to a general on a general curve of genus 7.
In this case, , and we conclude by applying Corollary 6.9(1) to a general on a general curve of genus 8. This is essentially the example produced by Mukai in [Muk93].
In this case, , and the smallest for which some pair is BN-compatible is with . Similar to the case, we consider the bundle obtained by a general extension of some on a general genus-9 curve by its Serre dual. To conclude that is stable, it suffices to produce a that admits no sub-. A well-known theorem of Farkas on inclusion of linear series with base points implies that such ’s exist; see [Far08, Thm 0.1].
Much as in the case, here we apply [Far08, Thm 0.1] to conclude the existence of a general on a genus 10 curve that admits no sub-, and then proceed with the same argument as before.
In this case, , and we conclude by applying Corollary 6.9(1) to a general on a general curve of genus 11.
In this case, and we conclude by applying Corollary 6.9(1) to a general on a general curve of genus 12 curve.
6.3. New non-emptiness certificates for BFM loci
We begin by giving a list of previously unknown cases (of the existence portion) of the BFM conjecture that follow directly from our non-emptiness result for special maximal rank loci. The running numerical assumptions are that (i) , ; and (ii) . It is easy to see that for any considered here, there is always some positive integer for which . We deduce the following result.
Theorem 6.12**.**
The moduli space of stable rank two vector bundles with canonical determinant and sections over a general curve of genus is non-empty whenever there exists an integer solution to one of the two following systems of inequalities:
[TABLE]
Proof.
In the system on the left, the first inequality implies that ; the second inequality implies that falls within the surjective range; and the third inequality implies that
[TABLE]
In particular, . Similarly, in the system on the right, we have and falls within the injective range.
It then follows from Corollary 6.9 and Proposition 5.13 that the corresponding moduli spaces of rank two vector bundles are non-empty. ∎
Using Theorem 6.12, we obtain some sharp existence results for the BFM conjecture, some of which were previously unknown.
Corollary 6.13**.**
The existence portion of the BFM conjecture holds for . Moreover, there exists a vector bundle verifying the existence portion of the conjecture with a minimal quotient line bundle of degree as defined in Definition 6.6.
Remark 6.14*.*
Those cases in which cases were previously settled in [LNP16]. In fact, by combining Corollary 6.13 with results in [BF98], [Zha16] and [LNP16], we conclude that whenever , the existence portion of the BFM conjecture holds except when .
We now turn our attention to the case , which is of minimal genus among the remaining open cases of the BFM conjecture. While we do not manage to definitively settle this case, we uncover some interesting geometry in the process.
6.4. The case of ,
For the BFM conjecture to hold in this case, we must produce an extension of the form
[TABLE]
such that is stable and over a Brill–Noether–Petri general curve. By the theorem of Nagata, it suffices to search within the degree range . Imposing reduces the possibilities to and .
When , the Brill–Noether theorem forces and . In this case, any stable is associated with a nontrivial extension belonging to the kernel of . In particular, if is stable, the multiplication map cannot be surjective. However, in this case
[TABLE]
so the SMRC predicts that is always surjective. Note that the image of any whose quadratic multiplication map fails to be surjective lies on a Fano threefold of type . Explicitly, is obtained from the blow-up of along a line via a morphism that contracts the proper transforms of those lines on that intersect . Deciding whether the SMRC holds in this case amounts to a statement about how linear series transform under the birational isomorphism ; we intend to pursue this line further in a subsequent paper.
When , the Brill-Noether theorem forces and . By the same argument as in the case , if a stable bundle with sufficiently many sections exists, the corresponding multiplication map cannot be surjective. However, we expect the SMRC locus to have dimension
[TABLE]
so the BFM and SMRC conjectures are compatible in this case.
Notice that in this case the Brill–Noether theorem also implies that (the complete linear series determined by) any destabilizing subbundle of must be a . On the other hand, from Proposition 6.2 it follows that no giving rise to a stable vector bundle contains a . Meanwhile, it follows from Proposition 5.13 that Conjecture 2 (SMRC) holds for . We deduce the following.
Claim 1**.**
*Assume that Conjecture 2 (SMRC) holds for . Then the existence portion of the BFM conjecture holds for on a general curve if and only if there exists some in which is very ample. *
By [Far08, Thm 0.1], the locus of containing a sub- is at most 2-dimensional.777Indeed, the incidence variety of degree 6 effective divisors contained in a is 2-dimensional, and the image of the map is a 2-dimensional locus inside consisting of containing a . So if we can show that the locus in where the multiplication map fails to have maximal rank is non-empty and is not contained inside the locus of admitting a sub-, we will verify the existence of a rank two linear series with sections and canonical determinant.
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