We take a sum $C_1+r C_2,\ r\in\mathbb Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside a smooth surface of degree $8$ and with $C_1\cap C_2=\emptyset$. We gather evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+ r C_2$ are distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a conjectural description of a counterexample to a conjecture of J. Harris. The methods used in this paper also produce in a rigorous way an infinite number of general components passing through the point representing the Fermat surface of degree $\leq 9$, and many non-reduced components for such degrees.
We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensi…
We prove that there exist smooth surfaces of degree d in projective 3-space such that the group of rational equivalence classes of decomposable 0-cycles has rank at least the integer part of (d-1)/3.
We study an analogue of the Noether-Lefschetz theorem in the context of Beilinson's Hodge conjecture. We estimate the dimension of the Noether-Lefschetz locus by the infinitesimal methods in Hodge theory.
We describe the Hodge theory of brilliant families of K3 surfaces. Their characteristic feature is a close link between the Hodge structures of any two fibres over points in the Noether-Lefschetz locus. Twistor deformations, the analytic Tate-Safarevic group, and one-dimensional Shimura special cycles are covered by the theory. In this setting, the Brauer group is viewed as the Noether-Lefschetz locus of the Brauer family or as the specialization of the Noether-Lefschetz loci in a family of approaching twistor spaces. Passing from one algebraic twistor fibre to another, which by construction i…
We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in P^3 whose function field has level 2 is dense in the set of those that have no real points.
We study the Noether-Lefschetz locus of a very ample line bundle L on an arbitrary smooth threefold Y. Building on results of Green, Voisin and Otwinowska, we give explicit bounds, depending only on the Castelnuovo-Mumford regularity properties of L, on the codimension of the components of the Noether-Lefschetz locus of |L|.
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 …
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.
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…
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.
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.
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).
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.
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.
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.
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.
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.
In this paper, we study maps between moduli spaces of lattice-polarized K3 surfaces induced by sublattices of prime index. We show that these maps can be used to determine if a rational point of the moduli space belongs to the Noether-Lefschetz locus. As an application, we prove that the Bombieri-Lang conjecture implies non-density statements for the rational points in the Noether-Lefschetz locus, as predicted by a conjecture of Shafarevich.
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.
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…
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.
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.
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 $V^2$.
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…
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.
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 $Z$.
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 prove a non abelian Torelli type result for smooth projective curves by working in the derived category of some associated polarized Quot schemes and defining Brill-Noether loci and Abel-Jacobi maps on them.
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 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.
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 $g$.
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.
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 $Z$.
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.
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 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)…
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.
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…
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.
Higher order isomonodromic deformation of Higgs bundles and a characterization of the non-abelian Noether-Lefschetz locus
Wuhan University（武汉大学）；Xiamen University（厦门大学）；University of Science and Technology of China（中国科学技术大学）；Institut für Mathematik, Universität Mainz（美因茨大学数学研究所）
We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles, i.e. sub-arithmetic manifolds of the same type. For compact manifolds this was proved in \cite{BMM11}, here we extend the results of \cite{BMM11} to non-compact manifolds. This allows us to apply our results to the moduli spaces of quasi-polarized K3 surfaces.
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.
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).
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.
