D. Arinkin
We study GL(2)-bundles with connections with a small parameter on a smooth projective curve. We describe an open subset in the moduli space of such bundles. The description degenerates into the Hitchin fibration as the parameter tends to zero.
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In this paper, we study the moduli spaces of parabolic connections with a quadratic differential. We endow these moduli spaces with symplectic structures by using the fundamental 2-forms on the moduli spaces of parabolic connections (which are phase spaces of isomonodromic deformation systems). Moreover, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.
Moduli spaces of framed logarithmic and parabolic connections on a Riemann surface
该研究构造了黎曼曲面上带标架的对数联络与抛物联络的模空间，证明其具有自然代数辛结构，并给出抛物联络模空间正则函数代数超越次数的上界。
In this paper, we treat moduli spaces of parabolic connections. We take étale coverings of the moduli spaces, and we construct a Hamiltonian structure of an algebraic vector field determined by the isomonodromic deformation for each étale morphism.
We define a moduli space of stable regular singular parabolic connections of spectral type on smooth projective curves and show the smoothness of the moduli space and give a relative symplectic structure on the moduli space. Moreover, we define the isomonodromic deformation on this moduli space and prove the geometric Painlevé property of the isomonodromic deformation.
A remark on the Moduli of parabolic connections
本文研究亏格至少为3的紧Riemann曲面上有理一般权重抛物联络的模空间，确定底层抛物丛稳定情形的周群，并讨论其有理性与有理连通性。
Recent work on the log minimal model program for the moduli space of curves, as well as past results of Caporaso, Pandharipande, and Simpson motivate an investigation of compactifications of the universal moduli space of slope semi-stable vector bundles over moduli spaces of curves arising in the Hassett-Keel program. Our main result is the construction of a universal moduli space of slope semi-stable sheaves which compactifies the moduli space of vector bundles over the moduli space of pseudo-stable curves.
Let $X$ be a compact Riemann surface of genus $g \geq 3$. We consider the moduli space of holomorphic connections over $X$ and the moduli space of logarithmic connections singular over a finite subset of $X$ with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational con…
We describe some results on moduli space of logarithmic connections equipped with framings on a $n$-pointed compact Riemann surface.
This paper studies the moduli space of stable surfaces of general type. The moduli space component containing the moduli point of a product of smooth curves of general type is proved to be the product of the moduli spaces of the curves, unless the curves have the same genus, in which case it is the symmetric product of the moduli space of curves. This result follows from a study of deformation theory of products and the singularities of products of nodal curves (namely that the product of nodal curves has semilogcanonical singularities).
This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.
We contribute to the foundations of tropical geometry with a view towards formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems---moduli of curves or otherwise---to be extended to logarithmic schemes…
We construct the generic component of the moduli space of the germs of Legendrian curves with generic plane projection topologicaly equivalent to a curve y^n = x^m, (n,m)=1.
The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.
We discuss the GIT moduli of semistable pairs consisting of a cubic curve and a line on the projective plane. We study in some detail this moduli and compare it with another moduli suggested by Alexeev. It is the moduli of pairs (with no specified semi-abelian action) consisting of a cubic curve with at worst nodal singularities and a line which does not pass through singular points of the cubic curve. Meanwhile, we make a comparison between Nakamura's compactification of the moduli of level three elliptic curves and these two moduli spaces.
The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.
We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d-closed via constructing an unfolding of the moduli space. Based on the Stokes data, we intro…
We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points.
We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic.
The new result is the unirationality of the moduli space of curves of genus 14. The method applies to lower genus.
The Determinant of Cohomology and Moduli of $λ$-Connections
本文研究了通过余子式余子式线丛的Atiyah藻酸盐对偶来理解稳定丛的Hodge-Deligne模空间。
In this paper, we investigate the apparent singularities and the dual parameters of rank 2 parabolic connections on $\mathbb{P}^1$ and rank 2 (parabolic) Higgs bundle on $\mathbb{P}^1$. Then we obtain explicit descriptions of Zariski open sets of the moduli space of the parabolic connections and the moduli space of the Higgs bundles. For $n=5$, we can give global descriptions of the moduli spaces in detail.
Connectedness of the moduli space of all reduced curves
通过等规范化曲线和Ishii的领土理论，证明了固定算术亏格的减少n点曲线模空间的连通性。
The space of smooth curves admits a beautiful compactification by the moduli space of Deligne-Mumford stable curves. In this paper, we undertake a systematic investigation of alternate modular compactifications of the space of smooth curves.
The slope of the moduli space of genus g curves is bounded from below by 60/(g+4) via a descendent calculation.
We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.
Birational description of moduli spaces of rank 2 logarithmic connections
本文通过分析抛物丛和表观奇点，证明射影空间与抛物丛模空间乘积的 Zariski 开子集可为秩2抛物对数联络模空间提供 Darboux 坐标，从而给出其双有理描述。
Moduli of curves on Enriques surfaces
本文计算Enriques曲面上光滑曲线模空间各不可约分支的模数，并刻画其到Prym曲线模空间映射的一般单射或支配性及例外情形。
We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve $C$ with elliptic quotient $π\colon C\to{\mathbb P}^1$, and the logarithmic connection $(E,\nabla)$ on ${\mathbb P}^1$, we may pullback the connection to the elliptic curve to obtain a new connection $(π^*E, π^*\nabla)$ on $C$. After suitable birational modifi…
For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.
We construct a new compactification of the moduli space H_g of smooth hyperelliptic curves of genus g. We compare our compactification with other well-known remarkable compactifications of H_g .
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.
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.
Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism …
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 compact moduli of orbifold projective curves
该研究提出稳定轨形射影曲线概念，证明其模栈与由栈点自同构群确定权重的Hassett加权带点稳定曲线模栈同构，构建了轨形射影曲线的紧致模空间。
We compute Chow groups of moduli spaces of rank 2 vector bundles on curves with determinant of odd degree in terms of generators and relations.
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.
Here we calculate the Chern classes of ${\bar {\mathcal M}}_{g,n}$, the moduli stack of stable n-pointed curves. In particular, we prove that such classes lie in the tautological ring.
Stable cohomology of moduli spaces of hyperelliptic curves on Hirzebruch surfaces
本文计算了 Hirzebruch 曲面上固定亏格超椭圆曲线模空间的稳定上同调，并用带标记点的非嵌入模空间进行描述。
The Segre-Verlinde correspondence for the moduli space of stable bundles on a curve
该研究建立光滑射影曲线上半稳定向量丛模空间的经典Verlinde数与模空间上自然泛复形Segre数的对应关系，为相关代数几何研究提供新视角。
The paper aims to give an account, both historical and geometric, on the diverse geography of rational parametrizations of moduli spaces related to curves. It is a contribution to the book Handbook of Moduli, editors G. Farkas and I. Morrison, to be published by International Press. Refereed version.
We prove that the coarse moduli space of curves of genus 6 is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces.
The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves. These new moduli spaces, which are modular compactifications of the moduli space of smooth pointed curves, are related with the minimal model program for the moduli space of stable pointed curves and have been introduced in a previous work of the authors. We interpret them as log canonical models of adjoints divisors and we then describe the Shokurov decomposition of a region of boundary divisors on the moduli space of stable pointed curves.
By associating to a curve C of genus g=2k and a pencil of degree d=k+1 the so-called trace curve (resp. the reduced trace curve) we define a rational map from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to a moduli space of stable curves. We study the induced map between the divisor class groups of these moduli spaces of curves.
We survey recent developments and open problems about extremal effective divisors and higher codimension cycles in moduli spaces of curves.
These are notes for a series of five lectures on "Moduli and degenerations of algebraic curves via tropical geometry" delivered at the CIMPA-CIMAT-ICTP School on Moduli of Curves, February 29-March 4, 2016 in Guanajuato, Mexico.
We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine $H^k(\sgbar, \Q)$ for $g \ge 2$ and $k \le 3$, where $\sgbar$ denotes the moduli space of spin curves of genus $g$.
Line bundles on the moduli space of Lie algebroid connections over a curve
本研究探究紧黎曼曲面上全纯李代数胚联络模空间的代数几何性质，构造其光滑紧化并给出边界除子数值有效判据，计算皮卡群、分析相关阿蒂亚丛，证明部分联络空间上正则函数为常数，还探讨了模空间的有理连通性。
We run Mori's program for the moduli space of pointed stable rational curves with divisor $K +\sum a_{i}ψ_{i}$. We prove that, without assuming the F-conjecture, the birational model for the pair is the Hassett's moduli space of weighted pointed stable rational curves, without any modification of weight coefficients.
We consider the stack of stable curves of genus g with a given dual graph and we give an explicit desingularization of its closure in the moduli stack of stable curves. We study in particular the one-dimensional substack of curves with at least 3g-4 nodes.
We study the tropicalization of the moduli space of algebraic spin curves, exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves, prove that it is isomorphic to the skeleton of the Berkovich analytification of the moduli space of algebraic spin curves, and give a geometric interpretation of the retraction map in terms of a tropicalization map.
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.
Moduli of Genus Six Curves and K-stability
本文研究亏格六曲线的K-模空间，通过显式描述五次对数Fano对的穿墙结构对其层进行分类，并揭示K-模空间在GIT模空间与K3曲面模空间之间的插值关系。
A geometry of the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one
本文利用希钦映射和椭圆曲线希格斯丛模空间的平坦退化，刻画了算术亏格为一的不可约节点曲线上希格斯对模空间的几何结构。
We study the moduli space of logarithmic connections of rank 2 on the Riemann sphere minus n points with fixed spectral data. There are two natural Lagrangian maps: one towards apparent singularities of the associated fuchsian scalar equation, and another one towards moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.
The notion of $m/Γ$-pointed stable curves is introduced. It should be viewed as a generalization of the notion of m-pointed stable curves of a given genus, where the labels of the marked points are only determined up to the action of a group of permutations $Γ$. The classical moduli spaces and moduli stacks are generalized to this wider setting. Finally, an explicit construction of the new moduli stack of $m/Γ$-pointed stable curves as a quotient stack is given.
In this short note, we show that any rational curve passing through the generic point in a moduli space of stable bundles with rank $r$ and fixed determinant on a smooth projective curve of genus $g\ge 4$ has degree (with respect to the anti-canonical line bundle of the moduli space) at least $2r$. It has degree $2r$ if and only if it is a Hecke curve.
We discuss analogs of Faber's conjecture for two nested sequences of partial compactifications of the moduli space of smooth curves. We show that their tautological rings are one-dimensional in top degree but do not satisfy Poincare duality.
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