Rainer Weissauer
We construct a $\bar Q_l$-linear Tannakian category attached to a smooth projective curve C equivalent to the category of finite dimensional $\bar Q_l$-representations Rep(G), where G is $Sp(2g-2,\bar Q_l)$ or $Sl(2g-2,\bar Q_l)$ depending on whether C is hyperelliptic or not.
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Recent advances in Brill--Noether theory and the geometry of Brill--Noether curves
本文综述了Brill-Noether理论的最新进展，并介绍了推动这些进展的关键技术。
This paper gives an overview of the main results of Brill-Noether Theory for vector bundles on algebraic curves.
In this paper we determine the number of general points through which a Brill--Noether curve of fixed degree and genus in any projective space can be passed.
Final version, to appear in Mathematical Research Letters.
In this note we compute the number of general points through which a general Brill-Noether space curve passes.
A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Goettsche and Hirschowitz prove that on the projective plane every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if t…
Brill--Noether loci in genus $\leq 12$
本文通过分析 K3 曲面上不稳定 Lazarsfeld--Mukai 丛并结合经典结果与显式构造，确定了亏格不超过 12 时所有 Brill--Noether 轨迹的相对位置。
We provide an example of a trivalent, 3-connected graph G such that, for any choice of metric on G, the resulting metric graph is Brill-Noether special.
Trigonal curves provide an example of Brill-Noether special curves. Theorem 1.3 of [9] characterizes the Brill-Noether theory of general trigonal curves and the refined stratification by Brill-Noether splitting loci, which parametrize line bundles whose push forward to $\mathbb{P}^1$ has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill-Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected…
We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kahler manifold, proving that these strata have quadratic algebraic singularities.
The Brill-Noether Theory of the Moduli Spaces of Sheaves on Surfaces
本文综述了复射影曲面上高阶向量丛的Brill-Noether理论，聚焦有理面和K-平凡曲面上的弱定理及其应用。
Given the Prym variety of an étale double cover one can define analogues of the classical Brill-Noether loci on Jacobians of curves. Recent work by Lahoz and Naranjo shows that the Brill-Noether locus $V^2$ completely determines the covering. In this paper we describe the singular locus and the irreducible components of $V^2$.
We extend results on generic strange duality for K3 surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized K3s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized K3s.
We extend a previous result of Feyzbakhsh concerning the injectivity of a map of moduli spaces and we use this result to construct curves whose Brill-Noether loci have unexpected dimension.
Maximal Brill--Noether loci via the gonality stratification
本文通过将Brill–Noether轨迹限制到曲线模空间的gonality分层，给出ρ=-1轨迹支撑集不同的新证明，得到极大Brill–Noether轨迹猜想不包含关系的下界，并证明秩2情形在g≥28时极大及g=20时猜想成立。
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 $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.
Prym-Brill-Noether theory for ramified double covers
该研究开展分歧双复叠的Prym-Brill-Noether理论研究，改进并拓展了多位学者的相关经典结果，还证明一般Du Val曲线是Prym-Brill-Noether一般的。
New examples of twisted Brill-Noether loci I
本文在亏格$g\ge2$曲线上构造了多个预期维数为负的扭曲Brill-Noether轨迹新例子，并推导出Brill-Noether映射中支撑非空标准轨迹的新区域。
Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c_1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in sense of moduli is also generic in sense of Brill-Noether theory (here H is the positive generator…
This article classifies Knutsen K3 surfaces all of whose hyperplane sections are irreducible and reduced. As an application, this gives infinite families of K3 surfaces of Picard number two whose general hyperplane sections are Brill-Noether general curves.
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.
In this article, we classify the irreducible components of moduli stacks of torsion free sheaves of rank 2 on K3 surfaces of Picard number 1. For ruled surfaces, the components of moduli stacks of torsion free sheaves were classified by C.Walter. Moreover, by virtue of our result, we classify the irreducible components of Brill-Noether loci of Hilbert schemes of points on K3 surfaces.
For a projective nonsingular curve of genus $g$, the Brill-Noether locus $W^r_d(C)$ parametrizes line bundles of degree $d$ over $C$ with at least $r+1$ sections. When the curve is generic and the Brill-Noether number $ρ(g,r,d)$ equals $1$, one can then talk of the \emph{Brill-Noether curve}. In this paper, we explore the first two invariants of this curve, giving a new way of calculating the genus of this curve and computing its gonality when $C$ has genus 5.
A Laurent polynomial $f$ in two variables naturally describes a projective curve $C(f)$ on a toric surface. We show that if $C(f)$ is a smooth curve of genus at least 7, then $C(f)$ is not Brill-Noether general. To accomplish this, we classify all Newton polygons that admit such curves whose divisors all have nonnegative Brill-Noether number.
We discuss in this short note how basic facts about divisors on moduli spaces of pointed curves give a solution to the Brill-Noether problem of nonexistence of linear series with prescribed ramification at unassigned points.
Distinguishing Brill-Noether loci
本文通过构造携带特定特殊线性系的曲线，证明曲线模空间中 Brill-Noether 轨迹间的非包含关系，并完全证明 Maximal Brill-Noether Loci 猜想。
In this paper we deal with Brill-Noether theory for higher-rank sheaves on a polarized nodal reducible curve $(C,\underline{w})$ following the ideas of [arXiv:alg-geom/9511003v1]. We study the Brill-Noether loci of $\underline{w}$-stable depth one sheaves on $C$ having rank $r$ on all irreducible components and having small slope. In analogy with what happens in the smooth case, we prove that these loci are closely related to BGN extensions. Moreover, we produce irreducible components of the expected dimension for these Brill-Noether loci.
In the 1990's, Bertram, Feinberg and Mukai examined Brill-Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their results to treat a much broader range of fixed-determinant Brill-Noether loci. The main technique is a careful study of symplectic Grassmannians and related concepts.
We develop a theory of Brill-Noether divisors on the moduli space of stable spin curves of genus g, and compute the classes of these loci. A spin Brill-Noether cycle is defined in terms of the relative position of the spin structure with respect to certain Raynaud theta-divisors in the Jacobian of the curve.
Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study twisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ and degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill-Noether loci are nonempty, and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and "nontwisted" Brill--Noether loci. We describe the tangent cones to the twisted Bril…
We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety $G^{r}_{d}(C)$ that parametrizing linear series of degree $d$ and dimension $r$ on a general smooth projective curve $C$ is indecomposable when $d\leq g(C)-1$.
Maximal Brill-Noether loci via degenerations and double covers
本文利用曲线链上的极限线性系与非分歧二重覆盖的源曲线轨迹，证明了 Brill-Noether 轨迹的若干新不包含关系，并重新证明期望余维数满足 $-ρ\leq \lceil g/2\rceil$ 时 Brill-Noether 轨迹具有期望维数的分支。
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.
Let $M_{\mathbb{P}^2}(v)$ be a moduli space of semistable sheaves on $\mathbb{P}^2$, and let $B^k(v) \subseteq M_{\mathbb{P}^2}(v)$ be the \textit{Brill-Noether locus} of sheaves $E$ with $h^0(\mathbb{P}^2, E) \geq k$. In this paper we develop the foundational properties of Brill-Noether loci on $\mathbb{P}^2$. Set $r = r(E)$ to be the rank and $c_1, c_2$ the Chern classes. The Brill-Noether loci have natural determinantal scheme structures and expected dimensions $dim B^k(v) = dim M_{\mathbb{P}^2}(v) - k(k - χ(E))$. When $c_1 > 0$, we show that the Brill-Noether locus $B^r(v)$ is nonempty. Wh…
We compute the rational cohomology groups of the smooth Brill-Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci $W^r_d(C)$, parametrizing complete linear series on $C$ of degree $d$ and dimension at least $r$.
In this paper we study Brill-Noether loci for rank-two vector bundles and describe the general member of some components as suitable extensions of line bundles.
In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to Hilbert scheme of scrolls.
We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.
Hyperelliptic Brill-Noether loci
该研究证明超椭圆θ除子及布里尔-诺特轨迹是局部 catalecticant 行列式簇，为其结构研究提供重要推论。
We prove that a certain Brill-Noether locus over a non-hyperelliptic curve $C$ of genus 4, is isomorphic to the \textit{Donagi-Izadi cubic threefold} in the case when the pencils of the two trigonal line bundles of $C$ coincide.
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 of Osserman. 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.
We show that certain divisors of Brill-Noether and Gieseker-Petri type span extremal rays of the effective cone in the moduli space of stable genus one curves with $n$ ordered marked points. In particular, they are different from the infinitely many extremal rays found in \cite{ChenCoskun}.
Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions
本文通过椭圆K3曲面上的Bridgeland稳定性条件，发展Hurwitz-Brill-Noether理论的新方法，证明椭圆K3曲面上的曲线是首个普遍的光滑k-gonal曲线例子，并给出主要存在性与不存在性结果的新证明，以及数域上普遍曲线的显式构造。
A Brill-Noether locus is a subvariety of M_g consisting of curves having certain linear series g^r_d. We study the relative position of Brill-Noether loci with respect to the gonality stratification of M_g. We construct smooth curves in P^r of given degree and genus and having the `expected' gonality. As an application we give a new proof of our result about the Kodaira dimension of the moduli space of curves of genus 23.
Brill--Noether Generality of Curves and K3 Surfaces
本文证明了在Brill-Noether一般的准极化K3曲面中，任何光滑曲线在对应线性系统中具有Brill-Noether普遍性。
We prove that the pullbacks of the virtual fundamental classes of the Brill-Noether loci under any Abel-Jacobi section lie in the tautological ring of the moduli space of stable curves. This resolves a conjecture of Pagani, Ricolfi and van Zelm, and is part of a broader program to understand the logarithmic intersection theory of $\Mgn$.
We use A_{infinity}-formalism to study variation of cohomology spaces under formal deformations of coherent sheaves on projective varieties. As an application we describe formal neighborhoods of twisted Brill-Noether loci at some points. Another application is a computation of the Fourier-Mukai transform of some natural line bundles on symmetric powers of a curve.
Prym-Brill-Noether Theory for General Covers
通过热带曲线组合与Coxeter群拓扑，界定了k-椭圆曲线平展双覆盖的Prym-Brill-Noether簇维数，否定了Creech等人的猜想并给出新猜想。
In this paper, we compute the number of general points through which a general Brill-Noether curve in $\mathbb{P}^4$ passes. We also prove an analogous theorem when some points are constrained to lie in a transverse hyperplane. As explained in arXiv:1809.05980, these results play an essential role in the first author's proof of the Maximal Rank Conjecture.
In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with hyperplane sections isomorphic to C. Most of our results are based on degeneration techniques.
We explicitly construct Brill--Noether general $K3$ surfaces of genus $4,6$ and $8$ having the maximal number of elliptic pencils of degrees $3, 4$ and $5$, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill--Noether general $K3$ surfaces of genus $4$ and $6$ without stable Lazarsfeld--Mukai bundles of minimal $c_2$.
Let C be a smooth projective complex curve of genus $g\geq2$. We investigate the Brill-Noether locus consisting of stable bundles of rank 2 and canonical determinant having at least $k$ independent sections. Using the Hecke correpondence we construct a fundamental class, which determines the non-emptiness of this locus at least when $C$ is a Petri curve. We prove that in many expected cases the Brill-Noether locus is non-empty. For some values of $k$ the result is best possible.
In this paper, we construct some examples of rank-2 Brill-Noether loci with "unexpected" properties on general curves. The key example is in genus 6, but we also have interesting examples in rank 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.
Let $X$ be a non-singular algebraic curve of genus $g$. We prove that the Brill-Noether locus $\bns $ is non-empty if $d= nd' +d'' $ with $0< d'' <2n$, $1\le s\le g$, $d'\geq (s-1)(s+g)/s $, $n\leq d''+(n-k)g$, $(d'',k)\ne(n,n)$. These results hold for an arbitrary curve of genus $\ge 2$, and allow us to construct a region in the associated ``Brill-Noether $\pa$-map'' of points for which the Brill-Noether loci are non-empty. Even for the generic case, the region so constructed extends beyond that defined by the so-called ``Teixidor parallelograms.'' For hyperelliptic curves, the same methods g…
We use results of M. Aprodu and E. Sernesi to extend a result by Fulton--Harris--Lazarsfeld in Brill--Noether theory of line bundles %and, as well, a result by Aprod-Sernesi in theory of Secant Loci, to Brill--Noether loci of stable bundles inside the moduli space of rank two stable vector bundles on a smooth projective algebraic curve. As a consequence; if $B^k_{2, d}$ is of expected dimension, then we prove that its smoothness is equivalent with the smoothness of $B^{k}_{2, d+1}$.
Brill-Noether loci and strata of differentials
本文证明微分射影化分层几乎不包含于带点 Brill-Noether 除子，并利用该结果研究 Auel-Haburcak 猜想，得到极大 Brill-Noether 轨迹间的新非包含关系，进而证明亏格 6 的二次分层是有理连通的。
We prove the injectivity of the Petri map for linear series on a general curve with given ramification at two generic points. We also describe the components of such a set of linear series on a chain of elliptic curves.
Weak Brill-Noether bundles on nodal Enriques surfaces
利用节点型Enriques曲面作为Grassmannian中Reye同余的实现，研究向量丛的弱Brill-Noether问题，考虑不可约齐次丛的限制并分析其弱Brill-Noether性质和斜率稳定性。
In this paper we consider the Brill-Noether locus $W_{\underline d}(C)$ of line bundles of multidegree $\underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $g\geq2$. We give an explicit description of the irreducible components of $W_{\underline d}(C)$ for a semistable multidegre $\underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.
We generalize the Embedding Theorem of Eisenbud-Harris from classical Brill-Noether theory to the setting of Hurwitz-Brill-Noether theory. More precisely, in classical Brill-Noether theory, the embedding theorem states that a general linear series of degree d and rank r on a general curve of genus g is an embedding if r is at least 3. If \(f \colon C \to \mathbb{P}^1\) is a general cover of degree k, and L is a line bundle on C, recent work of the authors shows that the splitting type of \(f_* L\) provides the appropriate generalization of the pair (r, d) in classical Brill--Noether theory. In…
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