Wei-Ping Li, Zhenbo Qin
We determine the topological Euler number of certain moduli space of 1-dimensional closed subschemes in a smooth projective variety which admits a Zariski-locally trivial fibration with 1-dimensional fibers. The main approach is to use virtual Hodge polynomials and torus actions. The results might shed some light on the corresponding Donaldson-Thomas invariants.
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We calculate the Euler characteristic of associated vector bundles over the moduli spaces of stable parabolic bundles on smooth curves. Our method is based on a wall-crossing technique from Geometric Invariant Theory, certain iterated residue calculus and the tautological Hecke correspondence. Our work was motivated by the results of Teleman and Woodward on the index of K-theory classes on moduli stacks.
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.
The generating function for $S_n$-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g>1 is calculated. This answer generalizes the known ones for genera 2 and 3 and answers obtained by J. Bergstrom for any genus and n<8 points.
For a partition $lambda=\{lambda_1 \geq λ_2 \geq λ_3 \}$ of non-negative integers, we calculate the Euler characteristic of the local system $V_λ$ on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some $λ$ of low degree, we make a guess for the motivic Euler characteristic of $V_λ$ using counting of curves over finite fields.
We study moduli spaces of vector bundles on a two-dimensional neighbourhood $Z_k$ of an irreducible curve $\ell = CP^1$ with $\ell^2 = -k$ and give an explicit construction of these moduli as stratified spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on $Z_k$ and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/ .
On the Virtual Euler Characteristic of the Moduli Space of Stable Pairs on Surfaces
本文研究了环面上稳定对理论，确定了固定点轨迹上的虚拟切空间，提出计算虚拟欧拉示性数的程序并以射影平面为例说明，通过特例验证支持了有理性与对称性猜想。
Given a smooth toric variety $X$, the action of the torus $T$ lifts to the moduli space $\mathcal{M}$ of stable sheaves on $X$. Using the pioneering work of Klyacho, a fairly explicit combinatorial description of the fixed point locus $\mathcal{M}^T$ can be given (as shown by earlier work of the author). In this paper, we apply this description to the case of torsion free sheaves on a smooth toric surface $S$. A general expression for the generating function of the Euler characteristics of such moduli spaces is obtained. The generating function is expressed in terms of Euler characteristics of…
Moduli Spaces of Twisted Endomorphisms and Euler Characteristics
本文为（拟）射影簇上的层定义了扭曲自同态的模空间，在自同构有限阶时构造非交换代数层并建立等价描述，进而计算其希尔伯特方案的欧拉示性数，推广了单位自同构的结果。
Motivic Euler characteristics of moduli spaces of curves
本文计算了曲线模空间的上同调欧拉特征，补充了9对未知情形，并首次计算了亏格4时n=7的非多项式情形。
Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.
We give a formula for the S_n - equivariant Euler characteristics of the moduli spaces of genus g curves with n marked points
We show how to calculate the Euler characteristic of a local system associated to an irreducible representation of the symplectic group of genus 3 on the moduli space of curves of genus 3 and the moduli space of principally polarized abelian varieties of dimension 3.
Chen--Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles
本文研究曲线上稳定抛物 Higgs 丛模空间在 Jacobian 的 $r$ 挠群作用下的不动点分支，并计算抛物丛商轨形的 Euler 示性数及 Chen–Ruan 上同调结构。
Let ${mathcal M}_g^n$ be the moduli space of n-pointed Riemann surfaces of genus g. Denote by ${\bar {\mathcal M}}_g^n$ the Deligne-Mumford compactification of ${mathcal M}_g^n$. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of ${\bar {\mathcal M}}_g^n$ for any g and n such that n>2-2g.
For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most basic intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification. The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of Abelian differentials and computational tools in the Chow ring, such as normal bundles to boundary divisors.
Euler characteristics of moduli of twisted sheaves on Enriques surfaces
本文研究 Enriques 曲面上稳定扭曲层的模空间的虚拟欧拉特征数，证明其仅依赖于虚拟维数 $N$，奇数时为零，偶数时为 $2\cdot e(Y^{[N/2]})$。
We show that a partition function of topological twisted N=4 Yang-Mills theory is given by Seiberg-Witten invariants on a Riemannian four manifolds under the condition that the sum of Euler number and signature of the four manifolds vanish. The partition function is the sum of Euler number of instanton moduli space when it is possible to apply the vanishing theorem. And we get a relation of Euler number labeled by the instanton number $k$ with Seiberg-Witten invariants, too. All calculation in this paper is done without assuming duality.
We define a one-dimensional family of "Euler" stability conditions on $\mathbb{P}^n$ which are conjectured to converge to Gieseker stability for coherent sheaves. Here, we focus on ${\mathbb P}^3$, first identifying Euler stability conditions with double-tilt stability conditions, and then we consider moduli of one-dimensional sheaves, proving some asymptotic results, boundedness for walls, and then explicitly computing walls and wall-crossings for sheaves supported on rational curves of degrees $3$ and $4$.
The weight two compactly supported Euler characteristic of moduli spaces of curves
本文通过图复形与操作子方法推导出曲线模空间权二紧支撑等变欧拉示性数的生成函数公式。
We show that the virtual Euler characteristics of the moduli spaces of $s$-pointed algebraic curves of genus $g$ can be determined from a polynomial in $1/γ$ where $γ$ permits specialization, through $γ=1,$ to the complex case treated by Harer and Zagier and, through $γ=1/2$, to the real case. This polynomial appears to have geometric significance, and may be the virtual Euler characteristic of some moduli space, as yet unidentified. This is related to a conjecture that the indeterminate $b=γ^1-1$ is associated with a combinatorial invariant of cell-decompositions through matrix models and the…
We show that each of the irreducible components of moduli of rank 2 torsion-free sheaves with odd Euler characteristic over a reducible nodal curve is rational.
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.
Recently, Minahan, Nemeschansky, Vafa and Warner computed partition functions for N=4 topological Yang-Mills theory on rational elliptic surfaces. In particular they computed generating functions of Euler characteristics of SU(2)-instanton moduli spaces. In mathematics, they are expected to coincide with those of Gieseker compactifications. In this paper, we compute Euler characteristics of these spaces and show that our results coincide with theirs. We also check the modular property of Z_{SU(2)} and Z_{SO(3)} conjectured by Vafa and Witten.
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 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.
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。
Nonnegativity of signed Euler characteristics of moduli of curves and abelian varieties
本文证明了在主极化abelian varieties和光滑曲线的模堆上perverse sheaf的欧拉特征非负性，并探讨了其在正特征下的失效情况。
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.
We investigate the moduli space of sheaves supported on space curves of degree 4 and having Euler characteristic 1. We give an elementary proof of the fact that this moduli space consists of three irreducible components.
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。
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。
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 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 calculate the Euler characteristics of the local systems S^k(V) \otimes S^l\Wedge^2(V) on the moduli space M_2 of curves of genus 2, where V is the rank 4 local system R^1π_*C.
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.
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