Special scrolls whose base curve has general moduli

Some Loci of Rational Cubic Fourfolds

In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in $\mathbb P^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to $\mathcal C_{14}$ is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus…

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.

Classification of Incidence Scrolls(I)

The aim of this paper is to obtain a classification of scrolls of genus 0 and 1, which are defined by a one-dimensional family of lines meeting a certain set of linear spaces in ${\bf P}^n$. These ruled surfaces will be called incidence scrolls and such a set will be the base of the incidence scroll. Unless otherwise stated, we assume that the base spaces are in general position.

Another classification of Incidence Scrolls

The aim of this paper is the computation of the degree and genus of all incidence scrolls in Pn. For this, we fix the dimension of a linear space which have a base space of this fixed dimension. In this way, we can obtain all the incidence scrolls with a line as directrix curve, those whose base contains a plane, and so on.

Equations and Syzygies of K3 Carpets and Unions of Scrolls

We describe the equations and Gröbner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The ideals of these surfaces are nested in a simple way that allows us to analyze them inductively. We describe explicit Gröbner bases and syzygies for these objects over the integers and this lets us treat them in all characteristics simultaneously.

Projective normality of special scrolls II

We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. We relate it with the Clifford index of the base curve X. If d>=4g-2i-Cliff(X)+1, i>=3 and R is smooth, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.

On linear systems of curves on rational scrolls

In this paper we prove a conjecture on the dimension of linear systems, with base points of multiplicity 2 and 3, on an Hirzebruck surface.

三次四维流形的模空间与可约OADP曲面

Moduli of Cubic fourfolds and reducible OADP surfaces

本文研究Hassett除子$\mathcal C_8$与$\mathcal{C}_{12}$、$\mathcal{C}_{20}$的交集不可约分量，刻画一般元素的几何，并借助二次纤维化有理截面和可约OADP曲面探讨其有理性，最终用Macaulay计算给出显式方程。

三重固体与卷积

Triple solids and scrolls

研究光滑射影簇在度3有限态射下的卷积结构，证明维度限制并分类m=1和m=2情形，发现多种情况简化为Segre积。

Rolling Factors Deformations and Extensions of Canonical Curves

A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in `rolling factors' format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves.

Special Linear Series and Syzygies of Canonical Curves of Genus 9

In this thesis we give a complete description of the syzygies of irreducible, nonsingular, canoncial curves $C$ of genus 9. This includes a collection of all possible Betti tables for $C$. Moreover a direct correspondence between these Betti tables and the number and types of special linear series on $C$ is given. Especially for $\operatorname{Cliff}(C)=3$ the curve $C$ is contained in determinantal surface $Y$ on a 4-dimensional rational normal scroll $X\subset \mathbb{P}^8$ constructed from a base point free pencil of divisors of degree 5.

Stable Base Locus Decompositions of Kontsevich Moduli Spaces

In this paper, we determine the stable base locus decomposition of the Kontsevich moduli spaces of degree two and three stable maps to Grassmannians. This gives new examples of the decomposition for varieties with Picard rank three. We also discuss the birational models that correspond to the chambers in the decomposition.

Structure of the cycle map for Hilbert schemes of families of nodal curves

We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We determine the relevant cotangent sheaves and complexes. We determine the structure of certain projective bundles called node scrolls, which play an important role in the geometry of Hilbert schemes.

Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves

We obtain effective results for the global generation of pluritheta line bundles on moduli spaces of vector bundles on curves. The main ingredient is an independent result giving an upper bound on the dimension of the Hilbert scheme of coherent quotients of a given vector bundle.

Galois descent for the gonality of curves

We determine conditions for the invariance of the gonality under base extension, depending on the numeric invariants of the curve. More generally, we study the Galois descent of morphisms of curves to Brauer-Severi varieties, and also of rational normal scrolls.

栈曲线上向量丛的良模空间射影性

Projectivity of good moduli spaces of vector bundles on stacky curves

本文利用良模空间存在性判据证明栈曲线上半稳定向量丛模栈具有射影良模空间，并给出线丛无基点性的有效界，从而为抛物丛模空间提供了新的有效构造。

A GIT Construction of Moduli Spaces of Stable Maps in Positive Characteristic

In a previous paper, the author and David Swinarski constructed the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). That paper required the base field to be the complex numbers, a restriction which this paper removes: here the coarse moduli spaces of stable maps are constructed via GIT over a more general base.

Geometry and intersection theory on Hilbert schemes of families of nodal curves

We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We work out the action of the blowup or 'discriminant' polarization on some natural cycles in the Hilbert scheme, including generalized diagonals and cycles, called 'node scrolls', parametrizing schemes supported on singular points. We derive an intersection calculus for Chern classes of tautologic…

Symplectic structures on moduli spaces of sheaves via the Atiyah class

Several situations are known when a holomorphic 2-form on a moduli space of sheaves over some base S is induced by a holomorphic 2-form on S. Moreover, the closedness of the 2-form on the base implies the closedness on the moduli space, which provides a stock of symplectic structures on moduli spaces (Mukai, Kobayashi, O'Grady, Tyurin, Huybrechts--Lehn). A parallel theory was developed for bivector fields and Poisson structures (Bottacin). However, there exist symplectic moduli spaces of sheaves over bases that have no holomorphic forms at all. A well-known example (Beauville--Donagi) is the f…

On the moduli space of singular principal G-bundles over stable curves

In this paper we proof the existence of a linearization for singular principal G-bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ-(semi)stable singular principal G-bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over $\overline{M}_{g}$ to the construction of the universal moduli space of swamps.

The weighted moduli spaces of sextics

We use the weighted moduli height as defined in \cite{sh-h} to investigate the distribution of fine moduli points in the moduli space of genus two curves. We show that for any genus two curve with equation $y^2=f(x)$, its weighted moduli height $\mathfrak h (\mathfrak{p}) \leq 2^3 \sqrt{3 \cdot 5 \cdot 7} \, \cdot H(f)$, where $H(f)$ is the minimal naive height of the curve as defined in \cite{height}. Based on the weighted moduli height $\mathfrak h$ we create a database of genus two curves defined over $\mathbb Q$ with small $\mathfrak h$ and show that for small such height ($\mathfrak h < 5…

Analytic moduli spaces of simple sheaves on families of integral curves

We prove the existence of fine moduli spaces of simple coherent sheaves on families of irreducible curves. Our proof is based on the existence of a universal upper bound of the Castelnuovo-Mumford regularity of such sheaves, which we provide.

The Moduli of Sections Has a Canonical Obstruction Theory

We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument we construct a dualizing sheaf and trace map, in the lisse-etale topology, for families of tame twisted curves, when the base stack is locally Noetherian.

Walls for Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases

Let $X$ be a complex projective manifold. Fix two ample line bundles $H_0$ and $H_1$ on $X$. It is the aim of this note to study the variation of the moduli spaces of Gieseker semistable sheaves for polarizations lying in the cone spanned by $H_0$ and $H_1$. We attempt a new definition of walls which naturally describes the behaviour of Gieseker semistability. By means of an example, we establish the possibility of non-rational walls which is a substantially new phenomenon compared to the surface case. Using the approach of Ellingsrud and Goettsche via parabolic sheaves, we were able to show t…

对数 Hodge 模空间的几何（与 Siqing Zhang 合著的附录）

Geometry of the logarithmic Hodge moduli space (with an Appendix joint with Siqing Zhang)

本文证明了对数 t-连接的 Hodge 模空间及其相关模空间在仿射直线上的光滑性与几何整性，并利用光滑性导出其平展上同调环的特殊化同构，证明了在正特征可分闭域情形下这些同构是反常层滤过的滤过同构。

Picard group of moduli of curves of low genus in positive characteristic

We compute the Picard group of the moduli stack of smooth curves of genus $g$ for $3\leq g\leq 5$, using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on $\mathcal{M}_g$.

On the restriction of the moduli part to a reduced divisor

Let $f \colon (X,Δ) \to Y$ be a fibration such that $K_X + Δ$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.

Tau function and moduli of spin curves

The goal of the paper is to give an analytic proof of the formula of G. Farkas for the divisor class of spinors with multiple zeros in the moduli space of odd spin curves. We make use of the technique developed by Korotkin and Zograf that is based on properties of the Bergman tau function. We also show how the Farkas formula for the {\it theta-null} in the rational Picard group of the moduli space of even spin curves can be derived from classical theory of theta functions.

Moduli of curves, Gröbner bases, and the Krichever map

We study moduli spaces of (possibly non-nodal) curves (C,p_1,\ldots,p_n) of arithmetic genus g with n smooth marked points, equipped with nonzero tangent vectors, such that ${\mathcal O}_C(p_1+\ldots+p_n)$ is ample and $H^1({\mathcal O}_C(a_1p_1+\ldots+a_np_n))=0$ for given weights ${\bf a}=(a_1,\ldots,a_n)$ such that $a_i\ge 0$ and $\sum a_i=g$. We show that each such moduli space $\widetilde{\mathcal U}^{ns}_{g,n}({\bf a})$ is an affine scheme of finite type, and the Krichever map identifies it with the quotient of an explicit locally closed subscheme of the Sato Grassmannian by the free act…

Base change behavior of the relative canonical sheaf related to higher dimensional moduli

We show that the compatibility of the relative canonical sheaf with base change fails generally in families of normal varieties. Furthermore, it always fails if the general fiber of a family of pure dimension n is Cohen-Macaulay and the special fiber contains a strictly S_{n-1} point. In particular, in moduli spaces with functorial relative canonical sheaves Cohen-Macaulay schemes can not degenerate to S_{n-1} schemes. Another, less immediate consequence is that the canonical sheaf of an S_{n-1}, G_2 scheme of pure dimension n is not S_3.

The kappa ring of the moduli of curves of compact type: I

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the kappa rings are computed. A universality property relating the higher genus kappa rings to the genus 0 rings is stated and proved in a sequel. The lambda_g formula for Hodge integrals arises as the simplest consequence.

Tautological module and intersection theory on Hilbert schemes of nodal curves

We study intersection theory on the relative Hilbert scheme of a family of nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an additive group called 'discriminant module', generated by diagonal loci, node scrolls, and twists thereof, and determine the action of the discriminant or big diagonal divisor on this group by intersection. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry. The latest version inc…

Moduli of vector bundles on higher-dimensional base manifolds - Construction and Variation

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker-Maruyama moduli spaces with respect to two different chosen polarisations are related via Thaddeus-flips through other "multi-Gieseker"-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding D…

机器学习用于亏格二曲线模空间及其在同源密码学中的应用

Machine learning for moduli space of genus two curves and an application to isogeny based cryptography

本研究利用机器学习检测亏格二曲线的分裂雅可比簇，证明少量低高度曲线具有该性质，并展示Igusa不变量可高精度预测，为同源密码学提供新工具。

The moduli space of the tropicalizations of Riemann surfaces

In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by a graph defined by its (hyperbolic) pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratificat…

关于秩2自由代数的模空间的一个注记

A note on the moduli spaces of free algebras of rank 2

本文用函子化方法刻画一般基环上秩2代数的模问题，并将其模空间构造为仿射概形在群概形作用下的预层商。

A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

The Chow ring of the moduli space of marked rational curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an algorithm for computing the intersection degree of tuples of Keel's divisor classes. This computation is a concrete but complicated algorithmic question in the field. Also, we give a simple complexity argument for the algorithm. Additionally, we introduce three identities on multinomial coefficients, as well as proofs for them.

Moduli of Riemann Surfaces, Transcendental Aspects

This is an informal set of lecture notes on moduli spaces of curves based on a set of lectures given at the ICTP last summer. It begins at an elementary level and discusses the genus 1 case in detail. The notes then give an informal treatement of Teichmuller space and higher genus moduli spaces. The point of view is generally topological and analytical.

Picard theorems for moduli spaces of polarized varieties

As a result of our study of the hyperbolicity of the moduli space of polarized manifold, we give a general big Picard theorem for a holomorphic curve on a log-smooth pair $(X,D)$ such that $W=X\setminus D$ admits a Finsler pseudometric that is strongly negatively curved when pulled back to the curve. We show, by some refinements of the classical Viehweg-Zuo construction, that this latter condition holds for the base space $W$, if nonsingular, of any algebraic family of polarized complex projective manifolds with semi-ample canonical bundles whose induced moduli map $ϕ$ to the moduli space of s…

Moduli Spaces for D-branes at the Tip of a Cone

For physicists: We show that the quiver gauge theory derived from a Calabi-Yau cone via an exceptional collection of line bundles on the base has the original cone as a component of its classical moduli space. For mathematicians: We use data from the derived category of sheaves on a Fano surface to construct a quiver, and show that its moduli space of representations has a component which is isomorphic to the anticanonical cone over the surface.

Big Picard theorem for moduli spaces of polarized manifolds

Consider a smooth projective family of complex polarized manifolds with semi-ample canonical sheaf over a quasi-projective manifold $V$. When the associated moduli map $V\to P_h$ from the base to coarse moduli space is quasi-finite, we prove that the generalized big Picard theorem holds for the base manifold $V$: for any projective compactification $Y$ of $V$, any holomorphic map $f:Δ-\{0\}\to V$ from the punctured unit disk to $V$ extends to a holomorphic map of the unit disk $Δ$ into $Y$. This result generalizes our previous work on the Brody hyperbolicity of $V$ (i.e. there are no entire cu…

Double coset construction of moduli space of holomorphic bundles and Hitchin systems

We present a description of the moduli space of holomorphic vector bundles over Riemann curves as a double coset space which is differ from the standard loop group construction. Our approach is based on equivalent definitions of holomorphic bundles, based on the transition maps or on the first order differential operators. Using this approach we present two independent derivations of the Hitchin integrable systems. We define a "superfree" upstairs systems from which Hitchin systems are obtained by three step hamiltonian reductions. A special attention is being given on the Schottky parameteriz…

Automorphic functions on moduli spaces of bundles on curves over local fields: a survey

This paper is the written version of D.Kazhdan's plenary talk at ICM 2022. It is dedicated to an exposition of recent results and (mostly) conjectures attempting to construct an analog of the theory of automorphic functions on moduli spaces of bundles on curves over local fields (both archimedian and non-archimedian). The talk is based on joint works of D.Kazhdan with A.Braverman, P.Etingof, E.Frenkel and A.Polishchuk.

箭袋丛的模叠及其在 Higgs 丛中的应用

Moduli stacks of quiver bundles with applications to Higgs bundles

本文提出由单纯集索引的箭袋丛模叠构造方法，给出 Artin 叠的基条件，并用于恢复 Nakajima 箭袋簇、另构 Higgs 丛模叠及证明其代数性。

quiver connections 的模空间与非阿贝尔霍奇理论

Moduli stacks of quiver connections and non-Abelian Hodge theory

本文构建了参数为λ的连接丛图的模空间，并展示了在特定条件下这些模空间的代数性质。