The Kodaira dimension of moduli spaces of curves with marked points, II

On the Kodaira dimension of the moduli space of hyperelliptic curves with marked points

It is known that the moduli space $\overline{\mathcal{H}}_{g,n}$ of genus $g$ stable hyperelliptic curves with $n$ marked points is uniruled for $n \leq 4g+5$. In this paper we consider the complementary case. We calculate the canonical divisor of $\overline{\mathcal{H}}_{g,n}$ and show that it is effective for $n=4g+6$ and big for $n\leq 4g+7$. This leads us to conjecture that $\overline{\mathcal{H}}_{g,n}$ has non-negative Kodaira dimension for $n = 4g+6$ and is of general type for $n \geq 4g+7$.

Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks

We extend some of the results obtained for subvarieties of the moduli stack of canonically polarized manifolds in "Base spaces of non-isotrivial families of smooth minimal models" (math.AG/0103122) to moduli of polarized minimal models of Kodaira dimension zero. To this aim we first show that for those manifolds there are no non isotrivial families of minimal models, which are birationally isotrivial. In the last chapter, we discuss the rigidity of curves in the moduli stack of canonically polarized manifolds and of polarized minimal models of Kodaira dimension zero.

Moduli of canonically polarized manifolds, higher order Kodaira-Spencer maps, and an analogy to Calabi-Yau manifolds

A Finsler metric, whose holomorphic curvature is bounded from above by a negative constant, is constructed on the moduli stack of canonically polarized manifolds including singularities. Demailly's version of Ahlfors' lemma yields hyperbolicity of the moduli stack.

The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, i.e. $U\oplus \langle -2k \rangle$-polarized K3 surfaces. Such moduli spaces are proved to be of general type for $k\geq 220$. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for $k < 11$ and for 19 other isolated values up to $k=64$.

超Kähler簇模空间的Kodaira维数

Kodaira dimension of moduli spaces of hyperkähler varieties

本文研究K3^[n]型与OG10型极化超Kähler簇模空间的Kodaira维数，将除子为1的已有结果推广到高维与大除子，并证明几乎所有2n维下次数充分大时模空间各分支均为一般型。

The birational type of the moduli space of even spin curves

We determine the Kodaira dimension of the moduli space of even spin curves for all genera, with one possible exception: The scheme S_g has negative Kodaira dimension for g<8 and it is of general type for g>8. The Kodaira dimension of S_8 is non-negative.

On the Kodaira dimension of moduli spaces of Abelian differentials

This paper lays the foundation for determining the Kodaira dimension of the projectivized strata of Abelian differentials with prescribed zero and pole orders in large genus. We work with the moduli space of multi-scale differentials constructed in [BCGGM2] which provides an orbifold compactification of these strata. We establish the projectivity of the moduli space of multi-scale differentials, describe the locus of canonical singularities, and compute a series of effective divisor classes. Moreover, we exhibit a perturbation of the canonical class which allows the corresponding pluri-canonic…

Kodaira 维数为一的椭圆面模空间

Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves

本文通过墙交叉技术和扭曲稳定映射技术，研究Kodaira 维数为一的椭圆面模空间，构造无限序列的不可约分量并描述其边界及稳定约简步骤。

Two-dimensional moduli spaces of vector bundles over Kodaira surfaces

We prove that any two-dimensional moduli space of stable 2-vector bundles, in the non-filtrable range, on a primary Kodaira surface is a primary Kodaira surface. If a universal bundle exists, then the two surfaces are homeomorphic up to unramified covers.

亏格 $3$ 带点曲线模空间的 Kodaira 分类

The Kodaira classification of the moduli space of pointed curves in genus $3$

本文通过证明 $n \geq 15$ 时带标记点曲线模空间 $\overline{\mathcal{M}}_{3,n}$ 为一般型，完成了亏格 $3$ 模空间的 Kodaira 分类，并分析了其奇点与典范类的性质。

The classification of universal Jacobians over the moduli space of curves

We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g<10, has Kodaira dimension zero for g=10 and Kodaira dimension 19 for g=11. For g>11, the variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by Iitaka's easy addition formula for fibre spaces. In particular, we disprove the expectation that P_g and M_g have the same Kodaira dimension for all genera.

超椭圆曲线模空间的Kodaira分类

The Kodaira classification of the moduli of hyperelliptic curves

本文研究带标记点的超椭圆曲线模空间的双有理几何，完成了其Kodaira分类，并分析了奇点性质及伪有效锥的结构。

Moduli Spaces of Stably Irreducible Sheaves on Kodaira Surfaces

Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic symplectic manifolds by deformation type. This paper studies a natural Lagrangian fibration on these moduli spaces to determine that they are not Kähler or simply connected, ruling out most of the known deformation types of holomorphic symplectic manifolds.

The Kodaira dimension of Siegel modular varieties of genus 3 or higher

We consider the moduli space $A_{pol}(n)$ of (non-principally) polarised abelian varieties of genus $g\geq3$ with coprime polarisation and full level-$n$ structure. Based upon the analysis of the Tits building in math/0405321, we give an explicit lower bound on $n$ that is sufficient for the compactified moduli space to be of general type if one further explicit condition is satisfied.

关于形变广义Kummer簇模空间的小平维数

On the Kodaira Dimension of the Moduli of Deformation Generalised Kummer Varieties

本文证明与形变广义Kummer型紧致超Kähler流形模空间相关的正交模簇的一般型结果，确定特定条件下模空间为一般型的界，推论对应模空间也为一般型。

Real Kodaira surfaces

In this paper we give the topological classification of real primary Kodaira surfaces and we describe in detail the structure of the corresponding moduli space. One of the main tools is the orbifold fundamental group of a real variety. Our first result is that if $(S, σ)$ is a real primary Kodaira surface, then the differentiable type of the pair $(S,σ)$ is completely determined by the orbifold fundamental group exact sequence. This result allows us to determine all the possible topological types of $(S, σ)$. Finally, we show that once we fix the topological type of $(S, σ)$ corresponding to a…

The Kodaira dimension of the moduli space of Prym varieties

We study the enumerative geometry of the moduli space R_g of Prym varieties of dimension g-1 (also known as the space of admissible double covers). Our main result is that the compactification of R_g is of general type as soon as g>13. We achieve this by computing the class of two types of cycles on R_g: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym-Green conjecture on syzygies of Prym-canonical curves. In the appendix we show that even though R_g has non-canonic…

The Kodaira dimension of spaces of rational curves on low degree hypersurfaces

For a hypersurface in complex projective space $X\subset \PP^n$, we investigate the singularities and Kodaira dimension of the Kontsevich moduli spaces $\Kbm{0,0}{X,e}$ parametrizing rational curves of degree $e$ on $X$. If $d+e \leq n$ and $X$ is a general hypersurface of degree $d$, we prove that $\Kbm{0,0}{X,e}$ has only canonical singularities and we conjecture the same is true for the coarse moduli space $\kbm{0,0}{X,e}$.This investigation is motivated by the question of which Fano hypersurfaces are unirational.

The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces

Let $X$ be an elliptic surface over ${\bf P}^1$ with $κ(X)=1$, and $M=M(c_2)$ be the moduli scheme of rank-two stable sheaves $E$ on $X$ with $(c_1(E),c_2(E))=(0,c_2)$ in $\operatorname{Pic}(X)\times\mathbb{Z}$. We look into defining equations of $M$ at its singularity $E$, partly because if $M$ admits only canonical singularities, then the Kodaira dimension $κ(M)$ can be calculated. We show the following. (A) $E$ is at worst canonical singularity of $M$ if the restriction of $E_η$ to the generic fiber of $X$ has no rank-one subsheaf, and if the number of multiple fibers of $X$ is a few. (B) W…

On the Kodaira dimension of the moduli space of nodal curves

We show that the compactification of the moduli space of $n-$nodal curves of genus g, i.e. $\mathcal{N}_{g,n}:= \mathcal{M}_{g,2n} /G$, with $G:=(\mathbb{Z}_2)^n \rtimes S_n$, is of general type for $g \geq 24$, for all $n \in \mathbb{N}$. While this is a fairly easy result, it requires completely different techniques to extend it to low genus $5 \leq g \leq 23$. Here we need that the number of nodes varies in a band $n_{\mathrm{min}}(g) \leq n \leq n_{\mathrm{max}}(g)$, where $n_{\mathrm{max}}(g)$ is the largest integer smaller than (or in some cases equal to) $\frac{7}{2}(g-1)-3$. The lower …

On the moduli part of the Kawamata-Kodaira canonical bundle formula

It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration $(X,B)\to Z$ is semi-ample. In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking comes from a section of $K_X+B$. We apply the above result to settle a conjecture of Fujino and Gongyo: if $f\colon X\to Z$ is a smooth surjective morphism of smooth projective varieties with $-K_X$ semi-ample, then $-K_Z$ is also semi-ample. We list several counter-examples to show …

The Geometry of the Moduli Space of Curves of Genus 23

We prove that the Kodaira dimension of the moduli space M_{23} of curves of genus 23 is at least 2. We also present some evidence for the hypothesis that the Kodaira dimension of the moduli space is actually equal to 2. Note that for g > 23 the moduli space is of general type, while for g\leq 22, Harris and Morrison conjectured that M_g is uniruled. The result on M_{23} is obtained by investigating the relative position of three explicit multicanonical divisors which are of Brill-Noether type.

Pencils on surfaces with normal crossings and the Kodaira dimension of $\overline{\mathcal{M}}_{g,n}$

We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline{\mathcal{M}}_{g,n}$ is not pseudo-effective in some range, implying that $\overline{\mathcal{M}}_{12,6},\overline{\mathcal{M}}_{12,7},\overline{\mathcal{M}}_{13,4}$ and $\overline{\mathcal{M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline{\mathcal{M}}_{12,8}$ and $\overline{\mathcal{M}}_{16}$. We also show that the moduli of $(4g+5)$-pointed hyperelliptic curves $\mathcal{H}_{g,4g+5}$ is uniruled. Together with a re…

Kodaira dimension of universal holomorphic symplectic varieties

We prove that the Kodaira dimension of the n-fold universal family of lattice-polarized holomorphic symplectic varieties with dominant and generically finite period map stabilizes to the moduli number when n is sufficiently large. Then we study the transition of Kodaira dimension explicitly, from negative to nonnegative, for known explicit families of polarized symplectic varieties. In particular, we determine the exact transition point in the cases of Beauville-Donagi and Debarre-Voisin, where the Borcherds Phi_{12} form plays a crucial role.

Kodaira fibrations and beyond: methods for moduli theory

Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. The paper contains some new results but is mostly a survey paper, dealing with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variatio…

Kodaira dimension of moduli of special cubic fourfolds

A special cubic fourfold is a smooth hypersurface of degree three and dimension four that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether-Lefschetz divisors C_d in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the "low-weight cusp form trick" of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of C_d. For example, if d = 6n + 2, then we show that C_d is of general type …

Frobenius Structures and Generalized Deformation of Kodaira Manifolds

It is known that generalized deformation in the sense of Hitchin-Gaultieri is a geometric realization of the degree-2 component of Kontsevich-Barannikov's homological approach to extended deformation. Through extended deformation, one associates a Frobenius structure to the extended moduli space. In this notes, we prove that on primary Kodaira manifolds the restriction of the Frobenius structure on the degree-2 component of the extended moduli space is trivial. It generalizes the author's past observation on Kodaira surface.

Monodromy of Kodaira Fibrations of Genus $3$

A Kodaira fibration is a non-isotrivial fibration $f\colon S\rightarrow B$ from a smooth algebraic surface $S$ to a smooth algebraic curve $B$ such that all fibers are smooth algebraic curves of genus $g$. Such fibrations arise as complete curves inside the moduli space $\mathcal{M}_g$ of genus $g$ algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case $g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of $\mathcal{M}_3$ parametrizing curves whose…

Moduli of algebraic varieties

We develop a moduli theory of algebraic varieties and pairs of non-negative Kodaira dimension. We define stable minimal models and construct their projective coarse moduli spaces under certain natural conditions. This can be applied to a wide range of moduli problems in algebraic geometry.

Kodaira dimension and zeros of holomorphic one-forms, revisited

We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on Simpson's results on the relation between rank one Higgs bundles and local systems of one-dimensional complex vectors spaces, and the structure of the cohomology jump loci in their moduli spaces.

次数为2的特殊$K3^{[2]}$-四形模空间的小平维数

Kodaira dimension of moduli of special $K3^{[2]}$-fourfolds of degree 2

该研究针对次数为2的$K3^{[2]}$-四形模空间的Noether-Lefschetz轨迹，计算了除有限个判别式外对应特殊除子的Kodaira维数，证明多数情形下该空间为一般型代数簇。

Relations among divisors on the moduli space of curves with marked points

We prove relations among the classes of certain divisors on the moduli spaces of curves with marked points, generalizing the Brill-Noether Ray Theorem of Eisenbud and Harris.

The moduli space of bilevel-6 abelian surfaces

The moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension. By contrast, Mukai has shown that the moduli space of bilevel-t abelian sufaces is rational for t=2,3,4,5.

The moduli space of marked supersingular Enriques surfaces

We construct a moduli space of adequately marked Enriques surfaces that have a supersingular K3 cover over fields of characteristic $p \geq 3$. We show that this moduli space exists as a scheme locally of finite type over $\mathbb{F}_p$. Moreover, there exists a period map from this moduli space to a period scheme and we obtain a Torelli theorem for supersingular Enriques surfaces.

四维Koszul Artin-Schelter正则代数的模空间的若干分支

Some components of the moduli space of Koszul Artin-Schelter regular algebras of dimension four

通过计算已知四维Koszul Artin-Schelter正则代数族的霍奇上同调和柯达依-斯宾塞映射，结合映射满射、双射的性质分析，确定了该类代数模栈的若干分支。

Jumps, folds, and singularities of Kodaira moduli spaces

For any integer $k$ we construct an explicit example of a twistor space which contains a one--parameter family of jumping rational curves, where the normal bundle changes from $O(1)+O(1)$ to $O(k)+O(2-k)$. For $k>3$ the resulting anti--self--dual Ricci-flat manifold is a Zariski cone in the space of holomorphic sections of $O(k)$. In the case $k=2$ we recover the canonical example of Hitchin's folded hyper-Kahler manifold, where the jumping lines form a three--parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fol…

论 Hurwitz 空间的 Kodaira 维数

On the Kodaira dimension of Hurwitz spaces

本文研究 Hurwitz 空间的 Kodaira 维数，证明相应 Hurwitz 栈的典范丛为大，并得出所有亏格大于一的三角曲线粗模空间均为一般型。

Complete descriptions of the tautological rings of the moduli spaces of curves of genus lower or equal to 4 with marked points

We study here the tautological rings of the moduli spaces of compact Riemann surfaces of genus 1,2,3 and 4 with marked points. The paper presents the complete descriptions of these rings by describing the groups of all degrees.

The Chow group of the moduli space of marked cubic surfaces

Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space.As an application we relate a recent result of Freitag on the Hilbert polynomial of a certain ring of modular forms to the Riemann-Roch theorem for the moduli space.

标记立方平面曲线的紧致模空间

Compact Moduli Spaces of Marked Cubic Plane Curves

本文通过几何不变理论研究标记立方平面曲线的紧致模空间，分析奇点和点位置以理解模论中的墙交叉现象。

The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surface II

The relation between the uniformizing equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces and an Appell-Lauricella hypergeometric system in nine variables is clarified.

Moduli Spaces of Germs of Semiquasihomogeneous Legendrian Curves

We construct a moduli space for Legendrian curves singularities which are contactomorphic-equivalent and equisingular through a contact analogue of the Kodaira-Spencer map for curve singularities. We focus on the specific case of Legendrian curves which are the conormal of a plane curve with one Puiseux pair.

Birational Geometry of moduli spaces of pointed curves

This is the authors doctoral thesis written at the Humboldt-University Berlin. It contains material from the three separate papers: "On the Kodaira dimension of the moduli space of nodal curves", "On quotients of $\overline{\mathcal{M}}_{g,n}$ by certain subgroups of $S_n$" and "The moduli space of hyperelliptic curves with marked points".

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of n-pointed stable curves of genus g, for g greater than 2. These stacks are smooth, irreducible and have dimension 4g-3+n, yielding a geometrically meaningful compactification of the degree d universal Picard stack over the moduli stack of smooth curves with marked points.

标记非交换三次曲面的紧模空间

Compact moduli of marked noncommutative cubic surfaces

本文通过箭图关系的GIT模概形构造标记非交换三次曲面的紧模空间，证明其为包含六点构型空间的环面簇，并双有理参数化一类AS-正则Z-代数。

Toric Construction and Chow Ring of Moduli Space of Quasi Maps from $\mathbb{P}^1$ with Two Marked Points to $\mathbb{P}^1 \times \mathbb{P}^1$

In this paper, we present explicit toric construction of moduli space of quasi maps from $\mathbb{P}^1$ with two marked points to $\mathbb{P}^1 \times \mathbb{P}^1$, which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.

$\overline{\mathcal{M}}_{22}$ 与 $\overline{\mathcal{M}}_{23}$ 的小平维度

The Kodaira dimensions of $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$

本文通过计算与秩 6 线性系相关的小斜率虚拟除子类，并引入新的热带方法证明其有效性，从而证明了亏格 22 和 23 的曲线模空间是一般型的。

具有五个标记点的 $\mathbb{P}^1$ 上抛物丛的模空间

Moduli spaces of parabolic bundles over $\mathbb{P}^1$ with five marked points

本文研究五标记点 $\mathbb{P}^1$ 上秩2、次数1的抛物对数平坦丛与Higgs丛的模空间/模叠，分析其叶状结构与分层，并在非特殊权重系统下证实Simpson猜想。

具有标记线的三次曲面的KSBA模空间

KSBA moduli spaces of cubic surfaces with a marked line

本文研究了通过KSBA稳定对方法构造具有标记线的三次曲面模空间的紧凑化，提出非均匀权重的稳定对参数化，并给出了新的KSBA粗模空间。

Moduli Space of Quasi-Maps from P^{1} with Two Marked Points to P(1,1,1,3) and j-invariant

In this paper, we construct toric data of moduli space of quasi maps of degree $d$ from P^{1} with two marked points to weighted projective space P(1.1,1,3). With this result, we prove that the moduli space is a compact toric orbifold. We also determine its Chow ring. Moreover, we give a proof of the conjecture proposed by Jinzenji that a series of intersection numbers of the moduli spaces coincides with expansion coefficients of inverse function of -log(j(tau)).

A linear system on Naruki's moduli space of marked cubic surfaces

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Naruki's toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional project…

9 亏格 Prym 模空间的单有理性

The uniruledness of the Prym moduli space of genus 9

本文证明亏格 9 的 Prym 曲线模空间 R_9 单有理性，并指出这是 R_g 具有负 Kodaira 维数的最大亏格。

The global geometry of the moduli space of curves

This is a survey written for the Proceedings of the AMS Summer Institute in Algebraic Geometry held in Seattle in 2005. Topics discussed in the survey include the ample and the effective cone of the moduli space of curves, Kodaira dimension, Slope Conjecture, log canonical models etc.

Moduli of Curves and Spin Structures via Algebraic Geometry

Here we investigate some birational properties of two collections of moduli spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin curves. In particular, we focus on vanishings of Hodge numbers of type (p,0) and on computations of the Kodaira dimension. Our methods are purely algebraic geometric and rely on an induction argument on the number of marked points and the genus of the curves (cf. mathAG/9803001).