Degeneration of moduli spaces and generalized theta functions

K-模空间紧化的有限性

Finiteness of K-moduli compactifications under the boundedness of K-semistable degenerations

本文在K-半稳定退化对数有界性假设下证明了K-半（多）稳定性的有限半代数腔分解定理，该假设源于K-模空间紧化的有限性。

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.

On degeneration of surface in Fitting compactification of moduli of stable vector bundles

The new compactification of moduli scheme of Gieseker-stable vector bundles with the given Hilbert polynomial on a smooth projective polarized surface (S;H), over the field k = \bar k of zero characteristic, is constructed in previous papers of the author. Families of locally free sheaves on the surface S are completed by the locally free sheaves on the schemes which are certain modifications of S. We obtain the class of modified surfaces to appear in the construction. Keywords: moduli space, semistable coherent sheaves, blowup algebra, algebraic surface.

Cohomology of the moduli space of smooth plane quartic curves with an odd theta characteristic

We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.

Moduli of abelian surfaces, symmetric theta structures and theta characteristics

We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the parities of the $d_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d_1$ and $d_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geom…

纤维化曲面上层的退化

Degenerations of Sheaves on Fibered Surfaces

本文利用扩展退化为纤维化曲面构造稳定层模叠并得到枚举不变量，证明粘合情形的退化公式，并计算某些椭圆曲面上稳定层模空间的椭圆亏格。

Stable degenerations of symmetric squares of curves

The main result of this article is that the component of the Alexeev-Kollár-Shepherd-Barron moduli space of stable surfaces parameterizing stable degenerations of symmetric squares of curves is isomorphic to the moduli space of stable curves. The moduli space of stable curves is used to ensure that any degeneration can be replaced by a degeneration to the symmetric square of a stable curve. These degenerations are not stable, however, but are partially resolved by the relative Hilbert scheme of length two subschemes in the fibers of a family of stable curves. The article gives a geometric desc…

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants.

Theta characteristics and their moduli

We discuss various aspects of the geometry of theta characteristics including the birational geometry of the spin moduli space of curves, parametrization of moduli via special K3 surfaces, as well as the relation with classical theta function theory and string theory in the form of the superstring measure. A historical view of the development of the subject is also presented.

Degeneration of differentials and moduli of nodal curves on $K3$ surfaces

We consider, under suitable assumptions, the following situation: $\mathcal B$ is a component of the moduli space of polarized surfaces and $\mathcal V_{m,δ}$ is the universal Severi variety over $\mathcal B$ parametrizing pairs $(S,C)$, with $(S,H)\in \mathcal B$ and $C\in |mH|$ irreducible with exactly $δ$ nodes as singularities. The moduli map $\mathcal V\to \mathcal M_g$ of an irreducible component $\mathcal V$ of $\mathcal V_{m,δ}$ is generically of maximal rank if and only if certain cohomology vanishings hold. Assuming there are suitable semistable degenerations of the surfaces in $\mat…

A complete degeneration of the moduli of $G$-bundles on a curve

For a semi simple group G it is known the moduli stack of principal G-bundles over a fixed nodal curve is not complete. Finding a completion requires compactifying the group G. However it was shown in [34] that this is not sufficient to complete the moduli stack over a family of curves. In this paper I describe how to use an embedding of the loop group LG to provide a completion of the stack of G-bundles over a one dimensional family of curves degenerating to a nodal curve. The completion comes with a modular interpretation inspired by the work of Gieseker, Seshadri, Kausz and Thaddeus and Mar…

Stable degenerations of surfaces isogenous to a product of curves

We use the moduli space of stable curves to determine the stable (in the sense of Kollár-Shepherd-Barron) degenerations of surfaces isogenous to a product of stable curves. A recent family of examples of Catanese show that the moduli space of such smooth surfaces may have two components. The main result of this note is that these components do not meet in the stable compactification. To the best of my knowledge this gives the first example of a moduli space of stable surfaces with multiple connected components parameterizing diffeomorphic smooth surfaces. Comments are appreciated.

稳定曲线上的Higgs束模堆栈

Moduli stacks of Higgs bundles on stable curves

本文构造了稳定曲线上Higgs束导出模堆栈的平坦退化，基于Jun Li的扩展退化堆栈。研究了退化中的内在相对log-symplectic形式，并证明退化Hitchin映射的纤维完整且平滑，以及减去nilpotent锥的适当开集是拉格朗日的。

A tour of theta dualities on moduli spaces of sheaves

The purpose of this paper is twofold. First, we survey known results about theta dualities on moduli spaces of sheaves on curves and surfaces. Secondly, we establish new such dualities in the surface case. Among others, the case of elliptic K3 surfaces is studied in detail; we propose further conjectures which are shown to imply strange duality.

Thomae's formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves

Determinantal formulae for Jacobian theta functions that go back to Klein are elaborated, via an idea due to Matone and Volpato. Also, the natural square roots of theta constants on the moduli space of curves whose existence was shown by Tsuyumine are proved to have a spinorial structure.

Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture

In this paper we study the degeneration behavior of the norm of the Riemann $θ$-function in a family of principally polarized abelian varieties over the punctured complex unit disc in terms of the associated polarized real torus. As an application, we obtain the degeneration behavior of the Zhang--Kawazumi invariant $φ(M_t)$ of a family of Riemann surfaces $M_t$ in terms of Zhang's invariant $φ(Γ)$ of the associated metrized reduction graph $Γ$. This allows us to deduce a uniform lower bound for the essential minimum of the Néron-Tate height on the tautological cycles of any Jacobian variety o…

A Gieseker type degeneration of moduli stacks of vector bundles on curves

Given a generically smooth stable curve over a discrete valuation ring such that its special fibre is irreducible with one double point, we construct a moduli stack over that descrete valuation ring which is a model for the moduli stack of vector bundles over the generic fibre of the curve. The model has the following nice properties: 1. It is regular. 2. Its special fibre is a normal crossing divisor. 3. The normalization of its special fibre is a locally trivial KGl-bundle over the moduli stack of vector bundles over the normalization of the special fibre of the curve. Here KGl is a canonica…

The Moduli Space of Abelian Varieties and the Singularities of the Theta Divisor

In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.

Theta characteristics on tropical curves

We give an explicit description of theta-characteristics on tropical curves and characterize the effective ones. We construct the moduli space for tropical theta-characteristics of genus g as a generalized cone complex. We describe the fibers of the specialization map from the classical moduli space of theta-characteristics on algebraic curves to our moduli space of tropical theta-characteristics.

Stable Degenerations of Surfaces Isogenous to a Product II

In this note, we describe the possible singularities on a stable surface which is in the boundary of the moduli space of surfaces isogenous to a product. Then we use the $\mathbb Q$-Gorenstein deformation theory to get some connected components of the moduli space of stable surfaces.

层的平移辛模叠的退化

Degenerations of shifted symplectic moduli stacks of sheaves

本文通过推广 Brav--Dyckerhoff 的构造，证明了 Calabi--Yau 族上凝聚层的导出模叠具有相对平移辛结构，并识别其为伪完美对象模的子叠，为枚举几何提供了潜在应用。

Generalized theta linear series on moduli spaces of vector bundles on curves

This article is based on lecture notes prepared for the August 2006 Cologne Summer School. The first part contains background material and references for beginners. The second (and main) part is a survey of the current status in the theory of pluri-theta linear series and generalized theta divisors on moduli spaces of vector bundles on curves. It emphasizes relatively new techniques employed in the analysis of linear series on these moduli spaces, namely the use of moduli spaces of stable maps for understanding Quot schemes, and the Fourier-Mukai functor in the study of coherent sheaves on abe…

Vector bundles with Theta divisors I: Bundles on Castelnuovo curves

In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.

曲线上向量丛模空间半稳定退化的几何

Geometry of a semistable degeneration of moduli of vector bundles on curves

研究具有一个节点的不可约曲线上稳定秩2度1向量丛的Gieseker模空间，通过两步爆破和自然对合粘合描述其正规化，计算了Picard群、虚拟Poincaré多项式及交集上同调的闭式公式。

四次椭圆曲线退化与法诺三维簇的K-模空间

Degenerations of elliptic quartics and K-moduli of Fano threefolds

本文研究经两次爆破构造的法诺三维簇的K-模空间，将其同构于带CM线丛线性化的VGIT商空间，分类了该形变族的K-(半/多)稳定成员，相关证明用到希尔伯特概型几何等新要素。

Degeneration of intermediate Jacobians and the Torelli theorem

Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed odd degree determinant. In this article we prove the analogous result in the case $X$ is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated Néron model of a family of such moduli spaces.

The universal theta divisor over the moduli space of curves

We carry out a complete birational classification of the universal theta divisor Th_g over the moduli space of curves of genus g, and show that Th_g enjoys good rationality properties for g<12, and is a variety of general type for g\geq 12. The key ingredient is an intersection-theoretic study of the universal antiramification locus of the Gauss map. We also present a complete classification of the universal symmetric product of degree g-2 over M_g.

On properness of K-moduli spaces and optimal degenerations of Fano varieties

We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Theta-stratification on the moduli stack of Fano varieties.

Theta-stratifications, Theta-reductive stacks, and applications

These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the "beyond geometric invariant theory" program for analyzing moduli problems in algebraic geometry. We discuss methods for analyzing stability in general moduli problems, focusing on the moduli of coherent sheaves on a smooth projective scheme as a toy example. We describe several applications: a general structure theorem for the derived category of coherent sheaves on an algebraic stack; some results on the topology of moduli stacks; and a "v…

O'Grady spaces and symplectic resolution of moduli spaces of Higgs bundles

We describe here a degeneration of the symplectic desingularization of the moduli spaces of topologically trivial $GL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$-Higgs bundles over a hyperelliptic curve, into O'Grady's ten and six dimensional exceptional examples of irreducible holomorphic symplectic manifolds. Most of this note is a survey of work on these degenerations by Donagi-Ein-Lazarsfeld, de Cataldo-Maulik-Shen and Felissetti-Mauri, although in certain cases we provide details that were missing the previous articles.

A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles

In this paper I present a new geometric approach to the factorization rule for generalised theta functions. Let $X$ be an irreducible projective nodal curve with one singularity and let $Y$ be its normalization. Recently I have constructed the moduli stack $GVB(X)$ of rank $n$ Gieseker vector bundles on $X$ and have shown that its normalization is a locally trivial fibration over the moduli stack $VB(Y)$ of vector bundles on $Y$, where the fibre is a canonical compactification of $Gl_n$. In this paper I prove a canonical direct sum decomposition of the space of global sections of a power of th…

Degenerations of triple coverings and Thomae's formula

In this paper, we prove Thomae's formula for a triple covering of $\bold P^1$ with arbitrary index. This formula gives a relation between theta constants, determinants of period integrals and the difference products of branch points. To specify a symplectic basis of the curve, we use the combinatorics of binary trees on $\bold P^1$. This symplectic basis behaves so well for degenerations that we obtain the absolute constant in this formula and reduce it to a special case treated in [Bershadsky-Radul], [Nakayashiki].

Some intersections in the Poincare bundle, and the universal theta divisor on the moduli space of (semi)abelian varieties

We compute all the top intersection numbers of divisors on the total space of the Poincare bundle restricted to the product of a curve and the abelian variety. We use these computations to find the class of the universal theta divisor and $m$-theta divisor inside the universal corank 1 semiabelian variety -- the boundary of the partial toroidal compactification of the moduli space of abelian varieties. We give two computational examples: we compute the boundary coefficient of the Andreotti-Mayer divisor (computed by Mumford but in a much harder and ad hoc way), and the analog of this for the u…

Degenerations of Godeaux surfaces and exceptional vector bundles

A recent construction of Hacking relates the classification of stable vector bundles on a surface of general type with $p_g = 0$ and the boundary of the moduli space of deformations of the surface. In the present paper we analyze this relation for Godeaux surfaces. We provide a description of certain boundary components of the moduli space of Godeaux surfaces. Also we explicitly construct certain exceptional vector bundles of rank 2 on Godeaux surfaces, stable with respect to the canonical class, and examine the correspondence between the boundary components and such exceptional vector bundles…

来自 Theta 特征的曲线模空间上的上同调类

Cohomology classes on moduli of curves from Theta Characteristics

本文利用 theta 特征的 Galois–Stiefel–Whitney 类，证明全实域上曲线与主极化阿贝尔簇模栈在指定次数存在非平凡上同调类，并计算亏格 3 情形的 Brauer 群拉回。

Parametrization of Sing(Theta) for a Fano 3-fold of genus 7 by moduli of vector bundles

According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve G, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of G. It is proven that, for a generic X, the Abel-Jacobi map of the family of elliptic sextics on X factors through the moduli space of rank-2 vector bundles with c_1=-K_X and deg c_2=6 and that the latter is birational to the singular locus of the theta divisor of J(X).

Irreducible degenerations of primary Kodaira surfaces

We classify irreducible d-semistable degenerations of primary Kodaira surfaces. As an application we construct a canonical completion for the moduli space of primary Kodaira surfaces.

Theta functions on the moduli space of parabolic bundles

Let X be a smooth projective connected curve of genus $g \ge 2$ and let I be a finite set of points of X. Fix a parabolic structure on I for rank r vector bundles on X. Let $M^{par}$ denote the moduli space of parabolic semistable bundles and let $L^{par}$ denote the parabolic determinant bundle. In this paper we show that the n-th tensor power line bundle ${L^{par}}^n$ on the moduli space $M^{par}$ is globally generated, as soon as the integer n is such that $n \ge [\frac{r^2}{4}]$. In order to get this bound, we construct a parabolic analogue of the Quot scheme and extend the result of Popa …

The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the jacobian

We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2Θ$. The embedded tangent space at a semi-stable non-stable bundle $ξ\oplusξ^{-1}$, where $ξ$ is a degree zero line bundle, is shown to consist of those divisors in $|2Θ|$ which contain $Sing(Θ_ξ)$ where $Θ_ξ$ is the translate of $Θ$ by $ξ$. We also obtain geometrical results on the structure of this tangent space.

Verlinde bundles and generalized theta linear series

In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties. We study what are called the Verlinde bundles in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.

Theta Functions and Szegö Kernels

We study relations between two fundamental constructions associated to vector bundles on a smooth complex projective curve: the theta function (a section of a line bundle on the moduli space of vector bundles) and the Szegö kernel (a section of a vector bundle on the square of the curve). Two types of relations are demonstrated. First, we establish a higher--rank version of the prime form, describing the pullback of determinant line bundles by difference maps, and show the theta function pulls back to the determinant of the Szegö kernel. Next, we prove that the expansion of the Szegö kernel at…

MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence

Motivated by string-theoretic arguments Manschot, Pioline and Sen discovered a new remarkable formula for the Poincare polynomial of a smooth compact moduli space of stable quiver representations which effectively reduces to the abelian case (i.e. thin dimension vectors). We first prove a motivic generalization of this formula, valid for arbitrary quivers, dimension vectors and stabilities. In the case of complete bipartite quivers we use the refined GW/Kronecker correspondence between Euler characteristics of quiver moduli and Gromov-Witten invariants to identify the MPS formula for Euler cha…

Weighted compactifications of configuration spaces and relative stable degenerations

We study a compactification of the configuration space of n distinct labeled points on an arbitrary nonsingular variety. Our construction provides a generalization of the original Fulton-MacPherson compactification that is parallel to the generalization of the moduli space of n-pointed stable curves carried out by Hassett.

Degeneration and curves on K3 surfac

This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces contain infinite number of irreducible rational curves. It is known that all K3 surfaces, except those contained in the countable union of hypersurfaces in the moduli space of K3 surfaces satisfy this property. In this paper we present a new approach for constructing curves on varieties which admit nice degenerations. We apply this technique to the above problem …

阿贝尔簇模空间上theta除子的纯扩张

Pure extension of the theta divisor over the moduli space of abelian varieties

研究阿贝尔簇模空间上theta除子的扩张问题，通过两种方式进行扩张，证明两种扩张选择的差异及相关公式，并讨论算术应用。

Degenerations of (1,7)-polarized abelian surfaces

The moduli space of (1,7)-polarized abelian surfaces with a level structure was shown by Manolache and Schreyer to be rational with compactification a Fano 3-fold of genus 12 which is also a compactification of the variety of powersum presentations of the Klein quartic curve. In this paper the possible degenerations of the abelian surfaces are classified in terms of degenerations of powersum presentations. In particular, the surface degenerations into fourteen planes are recovered. These were independently discovered by Alfio Marini using degenerations of twisted cubic curves.

Involutions on moduli spaces of vector bundles and GIT quotients

Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//\operatorname{PGL(2)}$. Then, we use recent results of Kumar to identify the restriction of the theta map to these GIT quotients with some explicit osculating projection. As a corollary of this construction, we obtain a birational equivalence between the ramification locus of th…

Relative VGIT and an application to degenerations of Hilbert schemes

We generalize the classical semi-continuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism from X to an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes, and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed by Gulbrandsen-Halle-Hulek.

轨道曲面的退化作为非交换代数簇

Degenerations of orbifold curves as noncommutative varieties

本文从非交换代数几何角度研究轨道投影曲线在堆叠点碰撞下的退化特性。

Logarithmic Gromov-Witten theory with expansions

We construct relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we show that there exist proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$, equipped with virtual classes. Each point in such a moduli space parameterizes a map from a nodal curve to an expanded degeneration of $X$ that is dimensionally transverse to the strata. In the context of maps to a simple normal crossings degeneration, the virtual fundamen…

Maps between moduli spaces of vector bundles and the base locus of the theta divisor

Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting…

Extensions of the universal theta divisor

The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently be…

Compactifications of moduli of points and lines in the projective plane

Projective duality identifies the moduli spaces $\mathbf{B}_n$ and $\mathbf{X}(3,n)$ parametrizing linearly general configurations of $n$ points in $\mathbb{P}^2$ and $n$ lines in the dual $\mathbb{P}^2$, respectively. The space $\mathbf{X}(3,n)$ admits Kapranov's Chow quotient compactification $\overline{\mathbf{X}}(3,n)$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $\mathbb{P}^2$ with $n$ "broken lines". Gerritzen and Piwek proposed a dual perspectiv…

连续线性级数

Continuous Linear Series

本文引入连续线性级数这一新对象，用精细模空间参数化线性级数到两光滑分支横截相交奇异曲线的退化，该模空间作为Hilbert商构成Osserman及Eisenbud-Harris极限线性级数模空间的新紧化。