Virasoro constraints for moduli of weighted pointed stable curves

Log canonical models for the moduli space of pointed stable rational curves

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.

On some modular contractions of the moduli space of stable pointed curves

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.

Moduli stacks of permutation classes of pointed stable curves

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.

On the Picard group scheme of the moduli stack of stable pointed curves

The aim of the present paper is to study the (abstract) Picard group and the Picard group scheme of the moduli stack of stable pointed curves over an arbitrary scheme. As a byproduct, we compute the Picard groups of the moduli stack of stable or smooth pointed curves over a field of characteristic different from two.

Calculating intersection numbers on moduli spaces of pointed curves

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.

The rationality of the moduli space of one-pointed ineffective spin hyperelliptic curves via an almost del Pezzo threefold

Using the geometry of an almost del Pezzo threefold, we show that the moduli space of genus $g$ one-pointed ineffective spin hyperelliptic curves is rational for every $g\geq 2$.

Extremal effective divisors on the moduli space of n-pointed genus one curves

For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.

圆盘模空间的相交理论、开KdV与Virasoro代数

Intersection theory on moduli of disks, open KdV and Virasoro

该研究定义了稳定带点圆盘模空间的下降积分理论，证明其与开KdV层级τ函数、半Virasoro代数的关系，通过几何递推计算积分，得到适用于所有亏格的Witten猜想开版本。

The rationality of the moduli space of two-pointed ineffective spin hyperelliptic curves

By the geometry of the 3-fold quadric we show that the coarse moduli space of genus g ineffective spin hyperelliptic curves with two marked points is a rational variety for every $g \geq 2$.

Rational Picard group of moduli of pointed hyperelliptic curves

We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers.

On the first steps of the minimal model program for the moduli space of stable pointed curves

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation and we study their geometric properties. As a particular case, we recover the first few Hassett-Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

Some Moduli of $n$-pointed Fano Fourfolds

The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.

On Log Canonical Models of the Moduli Space of Stable Pointed Curves

We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence of a conjecture by Fulton regarding the ample cone of MBar_{0,n}, these log canonical models are equal to certain of Hassett's weighted pointed curve spaces.

低亏格带点代数曲线模空间的维数

The Dimension of the Moduli Space of Pointed Algebraic Curves of Low Genus

本文针对亏格不超过7的多种情形，显式计算了以给定数值半群为Weierstrass半群的带点代数曲线的模空间，并确定了所有亏格7半群的维数。

Quivers and moduli spaces of pointed curves of genus zero

We construct moduli spaces of representations of quivers over arbitrary schemes and show how moduli spaces of pointed curves of genus zero like the Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$ and the Losev-Manin moduli spaces $\overline{L}_n$ can be interpreted as inverse limits of moduli spaces of representations of certain bipartite quivers. We also investigate the case of more general Hassett moduli spaces $\overline{M}_{0,a}$ of weighted pointed stable curves of genus zero.

Moduli spaces of weighted pointed stable rational curves via GIT

We construct the Mumford-Knudsen space of n pointed stable rational curves by a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with respect to the symmetric linearization O(1,...,1). The intermediate blown-up spaces turn out to be the moduli spaces of weighted pointed stable curves for suitable ranges of weights. As an application, we provide a new unconditional proof of M. Simpson's Theorem about the log canonical models of the Mumford-Knudsen space. We also give a basis of the Picard group of the moduli spaces of weighted pointed stable curves.

Equivariant Cohomology of Certain Moduli of Weighted Pointed Rational Curves

We determine the action of the product of symmetric groups on the cohomology of certain moduli of weighted pointed rational curves. The moduli spaces that we study are of stable rational curves with m+n marked points where the first m marked points are distinct from all the others where as the last n may coincide among themselves. We give a recipe for calculating the equivariant Poincaré polynomials and list them for small m and n.

带有碰撞标记的单纯稳定曲线的Virasoro递推关系

Virasoro Recursions for Simplicially Stable Curves with Colliding Markings

本文针对带有碰撞标记的单纯稳定曲线，通过分解约化态射得到重构公式，推导修正后的Virasoro递推关系，并将有限支撑理论封装于Fock空间，拓展了Witten-Kontsevich势的相关研究。

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".

$S_9^-$的合理性与标定点旋曲线的模空间

The unirationality of $S_9^-$ and moduli spaces of pointed spin curves

本文研究了 genus 9 的奇旋曲线模空间的合理性，并展示了通过局部平凡射影包将其与标定点的奇稳定旋曲线模空间联系起来，从而得到最高 genus 的合理性结果，同时探讨了标定点旋曲线模空间的 Kodaira 维数。

On the effective cone of the moduli space of pointed rational curves

We compute the effective cone for the moduli space of stable rational curves with at most six marked points.

On the birational geometry of moduli spaces of pointed curves

We prove that the moduli space ${\Cal M}_{g,n}$ of smooth curves of genus $g$ with $n$ marked points is rational for $g=6$ and $1 \le n \le 8$, and it is unirational for $g=8$ and $1 \le n \le 11$, $g=10$ and $1 \le n \le 3$, $g=12$ and $n = 1$.

Mori's program for the moduli space of pointed stable rational curves

We prove that, assuming the F-conjecture, the log canonical model of the pair $(\bar{M}_{0,n}, \sum a_i ψ_i)$ is the Hassett's moduli space of weighted pointed stable rational curves without any modification of weight coefficients. For the boundary weight cases, we prove that the birational model is the GIT quotient of the product of the projective lines. This is a generalization of Simpson's theorem for symmetric weight cases.

Moduli spaces of weighted pointed stable curves

A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sum of the corresponding weights is no greater than one. We construct moduli spaces for these objects using methods of the log minimal model program, and describe the induced birational morphisms between moduli spaces as the weights are varied. In the genus zero case, we explain the connection to Geometric Invariant Theory quotients of points in the projective line, and to compactifications of moduli spaces stu…

Ample divisors on moduli spaces of weighted pointed rational curves, with applications to log MMP for $\bar{M}_{0,n}$

We introduce a new technique for proving positivity of certain divisor classes on $\bar{M}_{0,n}$ and its weighted variants. Our methods give an unconditional description of the spaces of symmetric weighted pointed rational curves as log canonical models of $\bar{M}_{0,n}$.

稳定性标点曲线模栈的非交换刚性

Noncommutative rigidity of the moduli stack of stable pointed curves

本文证明了稳定性标点曲线模栈的第二Hochschild上同调群在大部分(g,n)情况下为零，揭示了其非交换刚性特性。

On the dimension of the stratum of the moduli of pointed curves by Weierstrass gaps

The dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the market point is computed for three families of symmetric semigroups of multiplicity six. We also collect the dimensions of such moduli spaces for all semigroups of genus not greater than seven. A question related to an improvement of a Deligne--Pinkham's bound is also formulated, suggesting that the positive graded part of the first module of the cotangent complex associated to a semigroup algebra is a missing invariant. The answer for this question is positive for all these moduli that we k…

Moduli of real pointed quartic curves

We describe a natural open stratum in the moduli space of smooth real pointed quartic curves in the projective plane. This stratum consists of real isomorphism classes of pairs $(C, p)$ with $p$ a real point on the curve $C$ such that the tangent line at $p$ intersects the curve in two distinct points besides $p$. We will prove that this stratum consists of $20$ connected components. Each of these components has a real toric structure defined by an involution in the Weyl group of type $E_7$.

带标点曲线模空间的导出范畴 -- II

Derived category of moduli of pointed curves -- II

研究带n个标点的稳定有理曲线模空间，证明其具有在对称群S_n作用下不变的完全例外集合，且其整系数K群是排列S_n-模。

Rational cohomology of the moduli spaces of pointed genus 1 curves

We give a combinatorial description of the rational cohomology of the moduli spaces of pointed genus 1 curves with $n$ marked points and level $N$ structures. More precisely, we explicitly describe the $E_2$ term of the Leray spectral sequence of the forgetful mapping $\EuScript{M}_{1,n}(N)\to\EuScript{M}_{1,1}(N)$ and show that the result is isomorphic to the rational cohomology of $\EuScript{M}_{1,n}(N)$ as a rational mixed Hodge structure equipped with an action of the symmetric group $\mathfrak{S}_n$. The classical moduli space $\EuScript{M}_{1,n}$ is the particular case N=1.

关于稳定带标曲线模空间之间忘却映射的分解定理

A Murphy's law for supports of forgetful maps between moduli spaces of stable pointed curves

本文研究稳定带标曲线模空间间忘却映射的分解定理，证明当m趋于无穷时，$\bar{\textit{M}}_{g,n}$的每个层闭包为分解定理的直和项支集，证明用到Hassett的模空间及跨壁论证。

Higher Weil-Petersson Volumes of Moduli Spaces of Stable $n$-pointed Curves

Moduli spaces of compact stable $n$-pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the Weil-Petersson volume forms and constitute a version of Mumford classes. We give various new formulas for the integrals of these forms and their generating functions. We also discuss their relation to the Kuenneth formula in quantum cohomology.

Geometry of the moduli space of $n$-pointed K3 surfaces of genus 11

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.

The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces

We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz' bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods, and their degenera…

On moduli of pointed real curves of genus zero

We introduce the moduli space $R \bar{M}_{2k,l}$ of pointed real curves of genus zero and give its natural stratification. The strata of $R \bar{M}_{2k,l}$ correspond to real curves of genus zero with different degeneration types and are encoded by their dual trees with certain decorations. By using this stratification, we calculate the first Stiefel-Whitney class of $R \bar{M}_{2k,l}$ and construct the orientation double cover $R \widetilde{M}_{2k,l}$$ of $R \bar{M}_{2k,l}$.

The Mori cones of moduli spaces of pointed curves of small genus

We compute the Mori cone of curves of the moduli space \M_{g,n} of stable n-pointed curves of genus g in the case when g and n are relatively small. For instance, we show that for g<14 every curve in \M_g is numerically equivalent to an effective sum of 1-strata (loci of curves with 3g-4 nodes). We also prove that the nef cone of \M_{0,6} is composed of 11 natural subcones all contained in the convex hull of boundary classes. We apply this result to classify the fibrations of the moduli space of rational curves with n<7 marked points.

Complete intersections and movable curves on the moduli space of six-pointed rational curves

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of th…

On the unirationality of moduli spaces of pointed curves

We show that $\mathcal{M}_{g,n}$, the moduli space of smooth curves of genus $g$ together with $n$ marked points, is unirational for $g=12$ and $2 \leq n\leq 4$ and for $g=13$ and $1 \leq n \leq 3$, by constructing suitable dominant families of projective curves in $\mathbb{P}^1 \times \mathbb{P}^2$ and $\mathbb{P}^3$ respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genus $g$ together with $n$ unordered points, establishing their unirationality for $g=11, n=7$ and $g=12, n =5,6$.

Intersection numbers on Deligne-Mumford moduli spaces and quantum Airy curve

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.

Derived category of moduli of pointed curves -- I

This is the first paper in the sequence devoted to derived category of moduli spaces of curves of genus $0$ with marked points. We develop several approaches to describe it equivariantly with respect to the action of the symmetric group permuting marked points. We construct an equivariant full exceptional collection on the Losev-Manin space which categorifies derangements.

光滑带点曲线的模空间上的非上同调环

Non-tautological cycles on moduli spaces of smooth pointed curves

本研究推广了已有方法，证明除有限多组亏格$g$和点数$n$外，光滑带点曲线模空间$\bM_{g,n}$上均存在非上同调代数上同调类，覆盖了绝大多数此前未解决的情形。

Moduli of log twisted $\mathcal{N} =1$ SUSY curves

The goal of the present paper is to construct a smooth compactification of the moduli superstack classifying pointed $\mathcal{N} =1$ SUSY (= $\text{SUSY}_1$) curves. This construction is based on the Abramovich-Jarvis-Chiodo compactification of the moduli stack classifying spin curves. First, we give a general framework of a theory of log superschemes (or more generally, log superstacks). Then, we introduce the notion of a pointed (stable) log twisted $\text{SUSY}_1$ curve; it may be thought of as a logarithmic and twisted generalization of the classical notion of a pointed $\text{SUSY}_1$ cu…

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves

For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves.

Chow groups of the moduli spaces of weighted pointed stable curves of genus zero

The moduli space $\bar{M}_A$ of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of the moduli space $\bar{M}_A$ are generated by the strata of $\bar{M}_A$ and give all additive relations between them. We also observe that the Chow groups $A_i \bar{M}_A$ and the homology groups $H_{2i} \bar{M}_A$ are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data $A$.

Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves

We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for $\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results ar…

Tautological relations in moduli spaces of weighted pointed curves

Pandharipande-Pixton have used the geometry of the moduli space of stable quotients to produce relations between tautological Chow classes on the moduli space $M_g$ of smooth genus g curves. We study a natural extension of their methods to the boundary and more generally to Hassett's moduli spaces $\overline M_{g, w}$ of stable nodal curves with weighted marked points. Algebraic manipulation of these relations brings them into a Faber-Zagier type form. We show that they give Pixton's generalized FZ relations when all weights are one. As a special case, we give a formulation of FZ relations for…

Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter …

到极化代数簇模空间的带基点映射的有限性

Finiteness of pointed maps to moduli spaces of polarized varieties

该研究证明了到具有极大模变异的光滑射影族底空间的带基点映射的有限性，通过建立非紧代数簇带基点映射的刚性，结合已解决的Viehweg双曲性猜想，得到了曲线到该底空间的Hom概形的最优维数界等应用。

一些带标记点的自旋曲线模空间的有理性

Uniruledness of some moduli spaces of pointed spin curves

本文证明了九个带自旋结构的标记点曲线模空间是有理的，通过构造有理曲线覆盖。

Effective cones of quotients of moduli spaces of stable n-pointed curves of genus zero

Let X_n := \bar M_{0,n}, the moduli space of n-pointed stable genus zero curves, and let X_{n,m} be the quotient of X_n by the action of the symmetric group S_{n-m} on the last n-m marked points. The cones of effective divisors of X_{n,m}, m = 0,1,2, are calculated. Using this, upper bounds for the cones Mov(X_{n,m}) generated by divisors with moving linear systems are calculated, m = 0,1, along with the induced bounds on the cones of ample divisors of \bar M_g and \bar M_{g,1}. As an application, the cone of effective divisors of \bar M_{2,1} is analyzed in detail.

稳定尖点曲线上亚纯希钦纤维化的模空间

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。

Moduli spaces of nonspecial pointed curves of arithmetic genus 1

In this paper we study the moduli stack ${\mathcal U}_{1,n}^{ns}$ of curves of arithmetic genus 1 with n marked points, forming a nonspecial divisor. In arXiv:1511.03797 this stack was realized as the quotient of an explicit scheme $\widetilde{\mathcal U}_{1,n}^{ns}$, affine of finite type over ${\Bbb P}^{n-1}$, by the action of ${\Bbb G}_m^n$ . Our main result is an explicit description of the corresponding GIT semistable loci in $\widetilde{\mathcal U}_{1,n}^{ns}$. This allows us to identify some of the GIT quotients with some of the modular compactifications of ${\mathcal M}_{1,n}$ defined …

Permutohedral spaces and the Cox ring of the moduli space of stable pointed rational curves

We study the Cox ring of the moduli space of stable pointed rational curves, \M_{0,n}, via the closely related permutohedral (or Losev-Manin) spaces. Our main result establishes \binom{n}{2} polynomial subrings of the Cox ring, thus giving collections of boundary variables that intersect the ideal of relations trivially. As applications, we give a combinatorial way to partially solve the Riemann-Roch problem for \M_{0,n}, and we show that all relations in degrees of Cox(\M_{0,6}) arising from certain pull-backs from projective spaces are generated by the Plucker relations.

Graph homology of moduli space of pointed real curves of genus zero

The moduli space $\bar{M}_S^σ(R)$ parameterizes the isomorphism classes of $S$-pointed stable real curves of genus zero which are invariant under relabeling by the involution $σ$. This moduli space is stratified according to the degeneration types of $σ$-invariant curves. The degeneration types of $σ$-invariant curves are encoded by their dual trees with additional decorations. We construct a combinatorial graph complex generated by the fundamental classes of strata of $\bar{M}_S^σ(R)$. We show that the homology of $\bar{M}_S^σ(R)$ is isomorphic to the homology of our graph complex. We also gi…