Brill Noether Loci Research Articles

Components of Brill–Noether Loci for Curves with Fixed Gonality

We describe a conjectural stratification of the Brill–Noether variety for general curves of fixed genus and gonality. As evidence for this conjecture, we show that this Brill–Noether variety has at least as many irreducible components as predicted by the conjecture and that each of these components has the expected dimension. Our proof uses combinatorial and tropical techniques. Specifically, we analyze containment relations between the various strata of tropical Brill–Noether loci identified by Pflueger in his classification of special divisors on chains of loops.

Prym–Brill–Noether Loci of Special Curves

Abstract We use Young tableaux to compute the dimension of $V^r$, the Prym–Brill–Noether locus of a folded chain of loops of any gonality. This tropical result yields a new upper bound on the dimensions of algebraic Prym–Brill–Noether loci. Moreover, we prove that $V^r$ is pure dimensional and connected in codimension $1$ when $\dim V^r \geq 1$. We then compute the 1st Betti number of this locus for even gonality when the dimension is exactly $1$ and compute the cardinality when the locus is finite and the edge lengths are generic.

A Riemann–Kempf singularity theorem for higher rank Brill–Noether loci

Given a vector bundle $V$ over a curve $X$, we define and study a surjective rational map $\mathrm{Hilb}^d (\mathbb{P} V ) - \mathrm{Quot}^{0, d} ( V^* )$ generalising the natural map $\mathrm{Sym}^d X \to \mathrm{Quot}^{0, d} ({\mathcal O}_X)$. We then give a generalisation of the geometric Riemann--Roch theorem to vector bundles of higher rank over $X$. We use this to give a geometric description of the tangent cone to the Brill--Noether locus $B^r_{r, d}$ at a suitable bundle $E$ with $h^0 (E) = r+n$. This gives a generalisation of the Riemann--Kempf singularity theorem. As a corollary, we show that the $n$th secant variety of the rank one locus of $\mathbb{P} \mathrm{End} E$ is contained in the tangent cone.

Linear series on a stable curve of compact type and relations among Brill-Noether loci

Let C be a stable curve of compact type such that two smooth curves with general moduli are bridged by a chain of n-elliptic curves, which will be called a TCBE curve. In the present paper, we investigate the existence of a smoothable limit linear series on C whose Brill-Noether number ρ satisfies −n≤ρ<0. Through this work, we find some relationships between a family of TCBE curves of genus g and a Brill-Noether locus in the moduli space of stable curves of genus g. Further, these results show some relations among Brill-Noether loci.

Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing

AbstractLetCbe a curve of genusg=11{g=11}org≥13{g\geq 13}on a K3 surface whose Picard group is generated by the curve class[C]{[C]}. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai’s program to this situation: we show how to reconstruct the K3 surface containing the curveCas a Fourier–Mukai transform of a Brill–Noether locus of vector bundles onC.

Bounds on the dimension of the Brill–Noether schemes of rank two bundles

The aim of this note is to find upper bounds on the dimension of Brill-Noether locus' inside the moduli space of rank two vector bundles on a smooth algebraic curve. We deduce some consequences of these bounds.

A stratification of $$B^4(2,K_C)$$B4(2,KC) over a general curve

Let C be a smooth curve of genus $$g\ge 10$$ with general moduli. We show that the Brill–Noether locus $$B^4(2,K_C)$$ contains irreducible subvarieties $${\mathcal {B}}_3\supset {\mathcal {B}}_4\supset \cdots \supset {\mathcal {B}}_n$$, where each $${\mathcal {B}}_k$$ has dimension $$3g-10-k$$ and $${\mathcal {B}}_3$$ is an irreducible component of the expected dimension the Brill–Noether number $$\rho =3g-13$$.

Moduli spaces of bundles and Hilbert schemes of scrolls over $$\nu $$ ν -gonal curves

The aim of this paper is twofold. We first strongly improve our previous main result Choi et al. (Proc Am Math Soc 146(8):3233–3248, 2018, Theorem 3.1), concerning classification of irreducible components of the Brill–Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees d, with at least $$d-2g+4$$ independent global sections, on a general $$\nu $$ -gonal curve C of genus g. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable.

Extending the double ramification cycle using Jacobians

We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with one of those defined by the last-two named authors (using an extended Brill–Noether locus on a suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber–Vakil using a virtual fundamental class on a space of rubber maps.

Non-emptiness of Brill–Noether loci over very general quintic hypersurface

In this article we study Brill–Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill–Noether loci over very general quintic hypersurface in P3, and use the Petri map to produce components of expected dimension.

Brill–Noether loci of rank 2 vector bundles on a general $\nu $-gonal curve

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its irreducible components having at least expected dimension. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable minimality properties, providing information on the birational geometry of such components as well as on the very-ampleness of F.

Upper bounds for some Brill–Noether loci over a finite field

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics

Abstract A parametrization of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican, provide a common parameter space for these loci, and show that the Simpson moduli space M = M 4m ± 1(ℙ2) is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. Two different proofs of this statement are given.

Higher rank BN-theory for curves of genus 6

Higher rank Brill–Noether theory for genus 6 is especially interesting as, even in the general case, some unexpected phenomena arise which are absent in lower genus. Moreover, it is the first case for which there exist curves of Clifford dimension greater than 1 (smooth plane quintics). In all cases, we obtain new upper bounds for non-emptiness of Brill–Noether loci and construct many examples which approach these upper bounds more closely than those that are well known. Some of our examples of non-empty Brill–Noether loci have negative Brill–Noether numbers.

Connectedness of Brill–Noether Loci via Degenerations

Abstract We show that limit linear series spaces for chains of curves are reduced. Using recent advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification at up to two points. We find that in general, these spaces may not be connected even when they have positive dimension, but we prove a criterion for connectedness which generalizes the theorem previously proved by Fulton and Lazarsfeld in the case without imposed ramification.

Some examples of rank-2 Brill\u2013Noether loci

In this paper, we construct some examples of rank-2 Brill–Noether loci with “unexpected” properties on general curves. The key examples are in genus 6, but we also have interesting examples in genus 5 and in higher genus. We relate some of our results to the recent proof of Mercat’s conjecture in rank 2 by Bakker and Farkas.

Higher Brill–Noether loci and Bogomolov–Gieseker type inequality

We discuss the Brill–Noether loci of the moduli of \(\mu \)-stable sheaves on a smooth projective variety, and obtain some necessary conditions for these loci to be non-empty. As an application of our result, we prove Bogomolov–Gieseker type inequalities concerning the third Chern character of the \(\mu \)-stable sheaves on Calabi–Yau threefolds.

Expected dimensions of higher-rank Brill-Noether loci

In this paper, we prove a new expected dimension formula for certain rank two Brill-Noether loci with fixed special determinant. This answers a question asked by Osserman and also leads to a new and much simpler proof of a theorem in his 2015 work. Our result generalizes the well-known result by Bertram, Feinberg and independently Mukai on expected dimension of rank two Brill-Noether loci with canonical determinant and partially verifies a conjecture (in rank two) of Grzegorczyk and Newstead on coherent systems.

Special divisors on marked chains of cycles

We completely describe all Brill–Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill–Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality.

Spinors, Lagrangians and rank 2 Higgs bundles

The paper considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W. Each spinor in the null-space generates through the moment map a Higgs bundle, and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of SL(2) we obtain Lagrangian submanifolds of the rank 2 moduli space which link up with m-period points on the Prym variety of the spectral curve as well as Brill-Noether loci on the moduli space of semistable bundles. The case of genus 2 is investigated in some detail.