Andrei Bud
We compute the dimension of the image of the forgetful map from $Z$ to the moduli space of curves, where $Z$ is a connected component of the stratum of $k$-differentials with an assigned partition $μ$, for the cases when $k=1$ with meromorphic partition and $k=2$ when the quadratic differentials have at worst simple poles.
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We construct explicit closed GL(2, R)-invariant loci in strata of meromorphic differentials of arbitrary large dimension with fully degenerate Lyapunov spectrum. This answers a question of Forni-Matheus-Zorich.
Complete subvarieties in the projectivized strata of meromorphic differentials
通过构造全局严格多次调和函数，证明具有指定零点和极点阶数的严格亚纯微分投影化分层不含正维完全子簇。
Connected components of generalized strata of meromorphic differentials with residue conditions
本文在最一般情形下分类了亚纯微分广义层级的连通分支，为理解多尺度紧化边界层级的不可约分支提供了关键基础。
Connected components of strata of residueless meromorphic differentials
本文分类了无留数亚纯微分层的连通分支，以理解多尺度紧化的边界复形，这些层是对极点留数施加最多条件数的层。
Teichmüller curves in hyperelliptic components of meromorphic strata
本文完整分类了亚纯微分层超椭圆分量中的Teichmüller曲线，通过非存在性判据排除了除特定Hurwitz覆盖外的所有候选者。
We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kahler manifold, proving that these strata have quadratic algebraic singularities.
We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic.
Virtual volumes of strata of meromorphic differentials with simple poles
本文提出了一种计算具有简单极点的微分方的层的虚拟体积的方法，并展示了其与可积系统的关系。
One-dimensional strata of residueless meromorphic differentials
本文研究椭圆曲线上无留数亚纯微分的一维分层，利用多尺度紧化给出其亏格和遗忘映射次数公式，并完成例外分层的连通分量分类。
In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarise…
In this paper we consider the following problem: For a given natural number $d$ and a prime $p$ determine all Newton polygons of polarized K3 surfaces of degree 2d over fields of characteristic $p$. This is an analogue of the Manin problem for Newton polygons of abelian varieties. This question is equivalent to determining the non-empty height strata of the moduli stack $\M_{2d}\otimes \F_p$ of K3 surfaces with a polarization of degree 2d over $\F_p$. We prove here that if $d$ is large enough and prime to $p$, then the height strata of $\M_{2d}\otimes \F_p$ are non-empty.
Moduli spaces for $Θ$-strata and non-reductive quotients
本文利用过滤栈和分级栈给出了关于分级幂零群作用几何商存在性的$\hat{U}$-定理的新证明，并将其推广至任意特征、族中作用及一般$Θ$-层。
The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most $n$ simple poles on genus $g$ complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic 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…
Tautological classes of strata of exact differentials
本文研究正合微分分层，证明其闭包类位于多尺度微分模空间的除子型示性环内，并借助有理函数关联给出了Hurwitz空间类示性的新证明与计算方法。
The meromorphic Hitchin fibration over stable pointed curves: moduli spaces
本文在稳定尖点曲线叠上构造了半稳定亚纯希格斯丛模空间的万有部分紧化，建立了容许节点谱覆盖下纤维与紧化雅可比簇的 BNR 对应类比，并研究了留数受幂零共轭类约束时的平坦固有希钦态射。
We study the induced Ekedahl-Oort stratification on the moduli space $\mathcal{M}_4$ of curves of genus $4$ in characteristic $3$. By constructing families of curves with given Ekedahl-Oort type and by computing the dimension of the intersection of the induced Ekedahl-Oort strata and the boundary divisor classes, we show that for certain induced Ekedahl-Oort strata in $\mathcal{M}_4$, they have the same codimension in $\mathcal{M}_4$ as the corresponding Ekedahl-Oort strata in moduli space $\mathcal{A}_4$ of principally polarized abelian varieties of dimension $4$.
We further analyze the moduli space of stable curves with level structure provided by Chiodo and Farkas in \cite{AA}. Their result builds upon Harris and Mumford analysis of the locus of singularities of the moduli space of curves and shows in particular that for levels 2, 3, 4, and 6 the locus of noncanonical singularities is completely analogous to the locus described by Harris and Mumford, it has codimension 2 and arises from the involution of elliptic tails carrying a trivial level structure. For the remaining levels (5, 7, and beyond), the picture also involves components of higher codime…
Bi-algebraic geometry of strata of differentials in genus zero
本文研究亏格零微分层中满足周期代数关系的代数子簇，证明了Ax-Schanuel与André-Oort型定理，并给出双代数簇的等价刻画及其叶状化性质，同时构造了无穷多新线性簇。
Hecke curves in Frobenius strata of moduli space of rank 2 vector bundles
本文研究了特征为2的代数闭域上光滑射影曲线X的模2向量丛模空间中的Frobenius分层，并证明其中存在一个由Hecke曲线覆盖的分层。
We describe the invariants of plane quartic curves -- nonhyperelliptic genus 3 curves in their canonical model -- as determined by Dixmier and Ohno, with application to the classification of curves with given structure. In particular, we determine modular equations for the strata in the moduli space $\Me_3$ of plane quartics which have at least seven hyperflexes, and obtain an computational characterization of curves in these strata.
This paper provides explicit closed formulas in terms of tautological classes for the cycle classes of the height and Artin invariant strata in families of K3 surfaces. The proof is uniform for all strata and uses a flag space as the computations in [arXiv:math/0412272] for the Ekedahl-Oort strata for families of abelian varieties, but employs a Pieri formula formula to determine the push down to the base space.
We prove that the moduli space of double covers ramified at two points $\mathcal{R}_{g,2}$ is uniruled for $3\leq g\leq 6$ and of general type for $g\geq 16$. Furthermore, we consider Prym-canonical divisorial strata in the moduli space $\overline{\mathcal{C}^n\mathcal{R}}_g$ parametrizing $n$-pointed Prym curves, and we compute their classes in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{C}^n\mathcal{R}}_g)$.
We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new irreducibility result for these strata.
We study the moduli space of Gieseker semi-stable sheaves on the complex projective plane supported on sextic curves and having Euler characteristic one. We determine locally free resolutions of length one for all such sheaves. We decompose the moduli space into strata which occur naturally as quotients modulo actions of certain algebraic groups. In some cases we give concrete geometric descriptions of the strata.
Double ramification loci, also known as strata of $0$-differentials, are algebraic subvarieties of the moduli space of smooth curves parametrizing Riemann surfaces such that there exists a rational function with prescribed ramification over $0$ and $\infty$. We describe the closure of double ramification loci inside the Deligne-Mumford compactification in geometric terms. To a rational function we associate its exact differential, which allows us to realize double ramification loci as linear subvarieties of strata of meromorphic differentials. We then obtain a geometric description of the clos…
For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-Möller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Rie…
We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.
Symplectic leaves of meromorphic Hitchin systems
本文重新研究了Bottacin和Markman独立发现的亚纯Higgs丛模空间的Poisson结构的辛叶，通过限制Hitchin映射的部分紧化构造了代数完全可积系统，并证明了该紧化由ξ-抛物Higgs丛模空间实现，同时提供了辛叶闭包正规化的辛解消。
The main goal of this article is to compute the class of the divisor of $\overline{\mathcal{M}}_3$ obtained by taking the closure of the image of $Ω\mathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to $\mathcal{M}_g$ and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge-Chen-Gendron-Gr…
Meromorphic differentials, twisted DR cycles and quantum integrable hierarchies
本文基于亚纯微分层交集理论定义了扭曲双重分歧层次，证明其可积性与tau对称性，并发现应用于平凡上同调场论时产生KdV层次，进而导出涉及多个特征类的交集数恒等式。
DR cycles and strata of differentials with spin parity
本研究聚焦曲线模空间周环中固定旋宇称的微分层类，证明其为可计算的 tautological 类，并建立了对应的精细化 DR 循环公式。
Moduli of sheaves on ribbons
本文研究了带状结构上无挠层模栈的几何性质，通过完全类型引入分层并分析其性质与闭包关系，揭示了Fano、Calabi-Yau和典范极化情形间的分叉以描述不可约分支，并计算了模栈在给定层处的切空间。
Our concern in this paper is to describe the p-rank statification on the Siegel moduli space with Iwahori level structure over fields of positive characteristic. We calculate the dimension of the strata and describe the closure of a given stratum in terms of p-rank strata. We also examine the relationship between the p-rank stratification and the Kottwitz-Rapoport stratification.
This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the study of the symmetries, a realization of the moduli spaces as quiver varieties is given and plays an essential role.
We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\ell) if g>=3 and \ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.
We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by cosets in a Weyl group. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.
Connectedness of the boundaries of the strata of differentials
本文研究微分射影化层边界的连通性，利用亚纯微分层的仿射性证明任意完备代数紧化中边界连通，并借助 Teichmüller 曲线分析全纯微分情形。
Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat structure defined by the meromorphic differentials. We provide formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve $X_{1}(N)$. This defines a family of graphs with…
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. And then we obtain an algorithm expressing intersection numbers $<τ_{n,m}\prod_{i=0}^{r-1}τ_{0,i}^{k_i}>_g$ via correlation functions of primaries.
In the geometric version of the Langlands correspondence, irregular singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study irregular singular connections on the projective line. Fundamental strata were originally used to classify cuspidal representations of the general linear group over a local field. In the geometric setting, fundamental strata play the role of the leading term of a connection. We introduce the concept of a regular stratum, which allows us to generalize the conditio…
We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space $\overline{\mathcal M}_{g,n}$ of stable genus $g$ curves with $n$ ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in $\overline{\mathcal M}_{3,1}$ and the locus of hyperelliptic curves in $\overline{\mathcal M}_4$ are extremal cycles. In addition, we exhibit infinitely many extremal…
We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincaré-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relati…
Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic
本文在任意特征下，为约化群概型构造了配备有界极点 $t$-连接的 $G$-丛的亚纯 Hodge 模空间，证明了半稳定轨迹的光滑性，并定义了正特征下常态的 Hodge-Hitchin 态射。
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not su…
Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n at least 3, and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by O_L of signature of (1,n-1). We show that each Newton stratum of M, other than the supersingular stratum, is irreducible.
One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to CP^2 and a projection of the image curve from an appropriate point p in CP^2 to the pencil of lines through p. We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata.
We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to strata. For non-generic strata similar results can be shown by a case-by-case inspection. We also propose to study a notion of 'linear manifold' that comprises Teichmueller curves, Hilbert modular s…
The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of th…
Spin refinement of moduli spaces of residueless meromorphic differentials and the BKP hierarchy
本文通过按自旋结构奇偶性加权无留数亚纯微分层的上同调类构造无穷秩部分CohFT，应用DR层级构造并约化后证明所得演化偏微分方程组与BKP层级在坐标变换下一致，从而给出BKP层级的自旋细化。
Cohomology ring of the strata of holomorphic differentials of genus 3
该研究证明了3次全纯微分层多尺度紧化连通分支的有理Chow环由边界层类生成，推导得有理闭链类映射为同构且紧化无奇次有理上同调。
Summary of main work 1999-2012
The boundary of the multi-scale differential compactification of strata of abelian differentials admits an explicit combinatorial description. However, even for low-dimensional strata, the complexity of the boundary requires use of a computer. We give a description of the algorithms implemented in the SageMath package diffstrata to enumerate the boundary components and perform intersection theory on this space. In particular, the package can compute the Euler characteristic of strata using the methods developed by the same authors in [CMZ20].
Moduli spaces of residueless meromorphic differentials and the KP hierarchy
本研究证明无留数亚纯微分模空间的上同调类构成无限秩部分CohFT，通过双分歧层级构造得到哈密顿系统，且其两零点多极点微分约化情形在变量替换下与KP层级一致。
Tau function and moduli of meromorphic forms on algebraic curves
本研究探讨复代数曲线上固定非零留数单极点亚纯1-形式的模空间，将Bergman tau函数视为线丛截面分析其渐近行为，还得到轨迹到曲线模空间投影的有理Picard群显式线性分解。
Finiteness of open substacks with proper moduli spaces
针对有限型代数stack，通过构造atlas的有限分层，证明了仅存在有限多个具有真充足模空间的开子stack。
We give a generalization to higher genera of the famous formula $12 λ=δ$ for genus 1. We also compute the classes of certain strata in the Satake compactification as elements of the push down of the tautological ring.
Ends of the strata of differentials
本文利用多尺度微分模空间与广义层连通分量分类的退化技术，枚举了黎曼曲面上亚纯1-形式的层的端，并推导出全纯1-形式的定理。
Euler characteristics of strata of $k$-differentials
本文计算了光滑尖点曲线模空间上 $k$-微分层的轨道欧拉示性数，给出双曲函数封闭公式、相交理论表达式及递归关系，并确定大亏格渐近行为。
In this paper we consider the stratification on the moduli space of principally polarized abelian surfaces in characteristic $p>0$ defined by the height of the formal group associated to $H^2(X,O_X)$. We compute the cycle classes of the strata and compare these with those of the $p$-rank stratification.
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