Higher Noether-Lefschetz loci of elliptic surfaces

Deformation of pairs and Noether-Lefschetz loci in toric varieties

We continue our study of the Noether-Lefschetz loci in toric varieties and investigate deformation of pairs (V,X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a odd dimensional simplicial projective toric variety, with V\subset X. Under some assumptions, we prove that the cohomological class in H^{k,k}(X) associated to V remains of type (k,k) under an infinitesimal deformation if and only if V remains algebraic. Actually we prove that locally the Noether-Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 …

Components of maximal dimension in the Noether-Lefschetz locus for Beilinson-Hodge cycles on open surfaces

The main subject of study in this paper is the Noether-Lefschetz locus for Beilinson-Hodge cycles on the complement of the union of hyperplane sections in a surface in $\Bbb P^3$. We give an analogue of the result by M.Green and C.Voisin on the maximal components in the Noether-Lefschetz locus.

Density of Noether-Lefschetz loci of polarized irreducible holomorphic symplectic varieties and applications

In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…

Exceptional loci in Lefschetz theory

Let $ϕ:X\rightarrow \mathbb{P}^n$ be a morphism of varieties. Given a hyperplane $H$ in $\mathbb{P}^n$, there is a Gysin map from the compactly supported cohomology of $ϕ^{-1}(H)$ to that of $X$. We give conditions on the degree of the cohomology under which this map is an isomorphism for all but a low-dimensional set of hyperplanes, generalizing results due to Skorobogatov, Benoist, and Poonen-Slavov. Our argument is based on Beilinson's theory of singular supports for étale sheaves.

An application of wall-crossing to Noether-Lefschetz loci

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda (such as $\mathbb P^3$, the quintic threefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class then we show it has very negative square.

Noether-Lefschetz for K_1 of a Surface, Revisited

Let Z be a general surface in P^3 of degree at least 5. Using a Lefschetz pencil argument, we give an elementary new proof of the vanishing of a regulator on K_1(Z).

On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors

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.

Complex Multiplication and Noether-Lefschetz Loci of the Twistor Space of a K3 Surface

For an algebraic K3 surface with complex multiplication (CM), algebraic fibres of the associated twistor space away from the equator are again of CM type. In this paper, we show that algebraic fibres corresponding to points at the same altitude of the twistor base $S^2\simeq \mathbb{P}^1_\mathbb{C}$ share the same CM endomorphism field. Moreover, we determine all the admissible Picard numbers of the twistor fibres.

On the Brill-Noether loci of a curve embedded in a K3 surface

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.

通过gonality分层研究极大Brill–Noether轨迹

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。

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 $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.

Quasi-modular forms from mixed Noether-Lefschetz theory

The Gromov-Witten theory of threefolds admitting a smooth K3 fibration can be solved in terms of the Noether-Lefschetz intersection numbers of the fibration and the reduced invariants of a K3 surface. Toward a generalization of this result to families with singular fibers, we introduce completed Noether-Lefschetz numbers using toroidal compactifications of the period space of elliptic K3 surfaces. As an application, we prove quasi-modularity for some genus 0 partition functions of Weierstrass fibrations over ruled surfaces, and show that they satisfy a holomorphic anomaly equation.

Existence and density of general components of the Noether-Lefschetz locus on normal threefolds

We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed infinitely many of them. Moreover, we show that their union is dense in the natural topology.

Fano 与 Calabi–Yau 三维簇中曲面的 Noether–Lefschetz 轨迹的密度

Density of Noether-Lefschetz loci for surfaces in Fano and Calabi-Yau threefolds

该研究利用 Hodge 轨迹的一般定理，考察 Fano 与 Calabi–Yau 三维簇中充分丰富线性系统曲面族的 Noether–Lefschetz 轨迹，并比较一般分支与例外分支并集的不同密度行为。

Gromov-Witten theory and Noether-Lefschetz theory

Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the 3-fold total space. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds and Kudla-Millson for O(2,19) lattices to determine the Noether-Lefschetz degrees in classical families of K3 surfaces of degrees 2, 4, 6 and 8. For the quartic K3 surfaces, the Noether-Lefschetz degrees are proven to be the Fourier coeffi…

区分 Brill-Noether 轨迹

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。

Noether-Lefschetz theorem with base locus

We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty). Our method is an adaptation of Griffiths and Harris' degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups, which generalize several known results.

Higher-rank Brill-Noether loci on nodal reducible curves

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.

Brill-Noether loci with fixed determinant in rank 2

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.

Geometry of Brill-Noether loci on Prym varieties

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 $V^2$.

Nonemptiness and smoothness of twisted Brill-Noether loci

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…

有限生成的诺特-利夫谢茨除子与立方四重面模空间的斜率

Finite generation of Noether-Lefschetz divisors and the slope of the moduli space of cubic fourfolds

本文研究了立方四重面模空间和K3表面的除子结构，通过引入斜率量和显式界限，揭示了有效锥的性质，并给出了K3模空间有理Picard群的生成方式。

亏格 $\leq 12$ 的 Brill--Noether 轨迹

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。

扭曲Brill-Noether轨迹的新例子 I

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。

Non-reduced components of the Noether-Lefschetz locus

For a fixed integer $d \ge 5$, the Noether-Lefschetz locus parametrizes smooth degree $d$ complex hypersurfaces in $\mathbb{P}^3$ with Picard number greater than $1$. There are infinitely many irreducible components of this locus. The aim of this article is to understand the scheme structure of these components. In particular, we study when is such a component non-reduced. We give a criterion for such a component to be non-reduced. Finally, we give examples of such components.

通过退化与二重覆盖研究极大 Brill-Noether 轨迹

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。

Noether-Lefschetz Theory and Néron-Severi Group

Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Y\subseteq \Ps^N$, with $dim Y=2r+1\geq 3$. We describe the Néron-Severi group $NS_r(X)$ of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.

Extensions of line bundles and Brill--Noether loci of rank-two vector bundles on a general curve

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.

Brill--Noether loci of stable rank--two vector bundles on a general curve

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.

超椭圆布里尔-诺特轨迹

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。

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 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.

A_{infinity}-structures, Brill-Noether loci and the Fourier-Mukai transform

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.

扭曲Brill-Noether定位的新例子II

New examples of twisted Brill-Noether loci II

本文构造了高 genus 曲线上的扭曲Brill-Noether定位的新例子，证明了Butler猜想并研究了相关定位的birational性、光滑性和不可约性。

Explicit constructions of K3 surfaces and unirational Noether-Lefschetz divisors

We provide methods to construct explicit examples of $K3$ surfaces. This leads to unirational constructions of Noether--Lefschetz divisors inside the moduli space of $K3$ surfaces of genus $g$. We implement Mukai's unirationality construction of the moduli spaces of $K3$ surfaces of genus $g\in\{6,\dots, 10, 12\}$, and we also present a new constructive proof of the unirationality of the moduli space of $K3$ surfaces of genus $11$. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of $K3$ surfaces of genus $g$.

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.

Noether-Lefschetz Theory with base locus

Let $Z$ be a closed subscheme of a smooth complex projective variety $Y\subseteq \Ps^N$, with $\dim\,Y=2r+1\geq 3$. We describe the intermediate Néron-Severi group (i.e. the image of the cycle map $A_r(X)\to H_{2r}(X;\mathbb{Z})$) of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.

Some examples of rank-2 Brill-Noether loci

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.

阿贝尔簇模空间上的诺特-利夫谢茨循环

Noether-Lefschetz cycles on the moduli space of abelian varieties

研究阿贝尔簇模空间中非单阿贝尔簇形成的区域的诺特-利夫谢茨循环性质，证明其在特定情况下非tautological，并探讨其与格罗莫夫-王理论的联系。

On the codimension of Noether-Lefshetz loci for toric threefolds

In this manuscript we sharpen the lower bound on the codimension of the irreducible components of the Noether-Lefschetz locus of surfaces in projective toric threefolds given in [BG17]. We also provide a simpler proof of Theorem 4.11 in [BG17], which allows one to avoid some technical assumptions.

The Noether-Lefschetz locus of surfaces in toric threefolds

The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $\mathbb P^3$ of degree $d \geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d \geq 4$, a very general surface in a simplicial toric threefold $\mathbb P_Σ$ (with orbifold singularities) has the same Picard number as $\mathbb P_Σ$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $\mathbb P_Σ$ in a linear system of a Cartier ample divisor with respect to a (-1)…

Brill-Noether loci with ramification at two points

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.

Gromov-Witten theory and Noether-Lefschetz theory for holomorphic-symplectic varieties

We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov-Witten techniques we also determine the generating series of Noether-Lefschetz numbers of a general pencil of Debarre-Voisin varieties. This reproves and extends a result of Debarre, Han, O…

通过 K3 曲面研究极大 Brill-Noether 轨迹

Maximal Brill-Noether loci via K3 surfaces

本文提出通过研究曲线上线性系统在极化 K3 曲面上的提升来区分曲线模空间中的 Brill-Noether 轨迹，并基于对 Lazarsfeld-Mukai 丛稳定性的分析，证明了亏格为 9-19、22 和 23 时的极大 Brill-Noether 轨迹猜想。

Brill-Noether Loci and the Gonality Stratification of $M_g$

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.

A Noether-Lefschetz Theorem for Spectral Varieties with Applications

We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field $k$ of odd characteristics or characteristic 0. We follow the strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an application, we calculate the generic fibers of Hitchin systems over a smooth quintic surface of Picard number 1.

Geometry of the Lagrangian Grassmannian Sp(3)/U(3) with applications to Brill-Noether loci

The geometry of Sp(3)/U(3) as a subvariety of Gr(3,6) is explored to explain several examples given by Mukai of non-abelian Brill-Noether loci, and to give some new examples. These examples identify Brill-Noether loci of vector bundles on linear sections of the Lagrangian Grassmannian Sp(3)/U(3) with orthogonal linear sections of the dual variety and vice versa. A main technical result of independent interest is the fact that any nodal hyperplane section of the Lagrangian Grassmannian projected from the node is a linear section of the Grassmannian Gr(2,6).

The Osculating cone to special Brill-Noether Loci

In this paper, we describe the osculating cone to Brill-Noether loci $W^{0}_{d}(C)$ at smooth isolated points of $W^{1}_{d}(C)$ for a smooth canonically embedded curve $C$ of even genus $g=2(d-1)$. In particular, we show that the canonical curve $C$ is a component of the osculating cone. The proof is based on techniques introduced by George Kempf in 1986.

Brill-Noether 轨迹与微分分层

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。

On the cohomology of Brill-Noether loci over Quot schemes

Let C be a smooth projective curve over the field of the complex numbers. We consider Brill-Noether loci over the moduli of maps from C to the Grassmannian G(m,n) and the corresponding Quot schemes of quotients of a trivial vector bundle on C compactifying the spaces of morphisms. We study in detail the case in which m=2, n=4. We prove results on the irreducibility and dimension of these Brill-Noether loci and we address explicit formulas for their cohomology classes. We study the existence problem of these spaces which is closely related with the problem of classification of vector bundles ov…

Injectivity of the Petri map for twisted Brill-Noether loci

Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map P:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})--> H^0(C, K) is injective.

某些可约和不可约Brill-Noether定位集

Some reducible and irreducible Brill-Noether loci

本文基于链上椭圆曲线极限线束，证明Farkas猜想，发现可约Brill-Noether定位集的实例，并展示当r≥3时的可约Hilbert方案，同时给出r=2时Brill-Noether定位集不可约的最优界。

Noether-Lefschetz 轨迹例外分支的非稠密性

Non-density of the exceptional components of the Noether-Lefschetz locus

本文证明 $\mathbb{P}^3$ 中高次光滑曲面 Noether-Lefschetz 轨迹的例外分支非 Zariski 稠密，并推广到适当三维簇，回答了 Voisin 的问题。

A Noether-Lefschetz theorem for vector bundles

In this note we use the monodromy argument to prove a Noether-Lefschetz theorem for vector bundles.

Noether-Lefschetz locus and a special case of the variational Hodge conjecture

For a fixed integer $d$, we study here the locus of degree $d$ hypersurfaces $X$ in $\mathbb{P}^{2n+1}$ such that $H^{2n}(X,\mathbb{Q}) \cap H^{n,n}(X,\mathbb{C}) \not= \mathbb{Q}$. We call this locus \textit{the Noether-Lefschetz locus}. Any irreducible component of this locus is locally a Hodge locus. So, we see that the study of this locus is very closely related to the variational Hodge conjecture. In this article we show that the Hodge locus corresponding to the cohomology class of a complete intersection subscheme in $\mathbb{P}^{2n+1}$ of codimension $n+1$ and degree less than $d$ satis…