V. Gritsenko, K. Hulek, G. K. Sankaran
The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46,50,54,58,60.
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We sharpen our previous results on the g and n such that the moduli space of curves of genus g with n marked points is of general type.
The moduli space of (1,p)-polarized abelian surfaces is a quasi-projective variety. In the case when p is a prime, we study its Kodaira dimension. We show that it is of general type for p > 71 and some smaller values of p. This improves an earlier result of G. K. Sankaran.
We prove a strengthening of Kollár's Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also prove subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.
We describe explicitly the possible degenerations of a class of double Kodaira fibrations in the moduli space of stable surfaces. Using deformation theory we also show that under some assumptions we get a connected component of the moduli space of stable surfaces.
On the Kodaira dimension of the moduli space of curves of genus 16
本文证明了亏格16曲线模空间不是一般型，解决了该维数下的一个长期未决问题。
Let M be a moduli scheme of stable sheaves with fixed Chern classes on an Enriques surface or a hyper-elliptic surface. If its expected dimension is 7 or more, then M admits only canonical singularities. Moreover, if M is compact, then its Kodaira dimension is zero.
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.
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.
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 $n \geq 4g+7$.
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 $k=64$.
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。
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.
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…
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。
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.
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。
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.
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 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.
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.
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。
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…
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…
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.
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…
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.
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 …
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 …
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.
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…
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 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…
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 …
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.
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.
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.
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。
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.
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。
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…
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。
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.
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".
It is in general unknown which topological complex vector bundles on a non-algebraic surface admit holomorphic structures. We solve this problem for primary Kodaira surfaces by using results of Kani on curves of genus two with elliptic differentials. Some of the corresponding moduli spaces will be smooth compact and holomorphically symplectic.
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。
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。
Quasi-Projective Moduli for Polarized klt Good Minimal Models
本文利用Gabber定理证明局部稳定族的直接像弱正性，进而应用Viehweg放大准则，展示极化klt极小模型模空间的正则化是准射影的。
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.
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).
We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.
On the Kodaira dimension
该研究聚焦代数几何领域，探讨光滑态射情形下小平维数的行为表现，为相关维数性质研究提供参考。
We prove that the moduli space of polarized $K3$ surfaces of genus eleven with $n$ marked points is unirational when $n\leq 6$ and uniruled when $n\leq7$. As a consequence, we settle a long standing but not proved assertion about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked points has non-negative Kodaira dimension.
We prove the invariance of plurigenera under smooth projective deformations of varieties with nonnegative Kodaira dimensions.
Over any algebraically closed field of positive characteristic, we construct examples of fibrations violating subadditivity of Kodaira dimension.
On the behavior of Kodaira dimension under smooth morphisms
该研究证明了光滑射影簇的光滑态射下小平维数可加性的若干结果，推进了代数几何中关于小平维数性质的研究。
In this note, we apply the semi-ampleness criterion in Lemma 3.1 to prove many classical results in the study of abundance conjecture. As a corollary, we prove abundance for large Kodaira dimension depending only on [BCHM10].
We prove the existence of good log minimal models for dlt pairs of numerical log Kodaira dimension 0.
The non-abelian Brill-Noether divisor on $\overline{\mathcal{M}}_{13}$ and the Kodaira dimension of $\overline{\mathcal{R}}_{13}$
本文计算 $M_{13}$ 上非交换 Brill-Noether 除子的类，首次得到斜率低于 $6+10/g$ 的有效除子，并据此证明亏格13的Prym模空间为一般型及两个相关猜想。
Estimates on the Kodaira dimension for fibrations over abelian varieties
本文研究了以阿贝尔簇为纤维的纤维化的柯西-亚维斯维度估计，并加强了其子加性性质。
We prove that the variation in a smooth projective family of varieties admitting a good minimal model forms a lower bound for the Kodaira dimension of the base, if the dimension of the base is at most five and its Kodaira dimension is non-negative. This gives an affirmative answer to the conjecture of Kebekus and Kovacs for base spaces of dimension at most five.
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