The unirationality of the moduli space of curves of genus $\leq 14$

A note on the unirationality of a moduli space of double covers

In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.

光滑曲线上半稳定向量丛的万有模空间的单有理性

Unirationality of the universal moduli space of semistable bundles over smooth curves

本文通过构造从有理簇上的射影丛到模空间的显式支配有理态射，证明了秩、次数任意且亏格在2到9之间的半稳定向量丛万有模空间 $U_{r,d,g}$ 的单有理性，并将结论推广到 $U_{r,\mathcal{L},g}$。

Unirationality of the Hurwitz space H_{9,8}

In this paper we prove that the Hurwitz space H_{9,8}, which parameterizes 8-sheeted covers of P^1 by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.

Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.

$S_9^-$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 vector bundles from Chang and Ran's proof of the unirationality of $\mathcal{M}_g$, $g \leq 13$

We combine the idea of Chang and Ran [Invent. Math. 76 (1984), 41-54] of using monads of vector bundles on the projective 3-space to prove the unirationality of the moduli spaces of curves of low genus with our classification of globally generated vector bundles with small first Chern class $c_1$ on the projective 3-space to get an alternative argument for the unirationality of the moduli spaces of curves of degree at most 13 (based on the general framework of Chang and Ran).

Explicit constructions of K3 surfaces and unirational Noether-Lefschetz divisors

We provide methods to construct explicit examples of $K3$ surfaces. This leads to unirational constructions of Noether--Lefschetz divisors inside the moduli space of $K3$ surfaces of genus $g$. We implement Mukai's unirationality construction of the moduli spaces of $K3$ surfaces of genus $g\in\{6,\dots, 10, 12\}$, and we also present a new constructive proof of the unirationality of the moduli space of $K3$ surfaces of genus $11$. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of $K3$ surfaces of genus $g$.

典范曲线的 Hilbert 概型中双椭圆轨迹是单有理的

The bielliptic locus in the Hilbert scheme of canonical curves is unirational

本文证明了亏格 $g \geq 11$ 的典范曲线 Hilbert 概型中双椭圆曲线轨迹的单有理性，并以此给出了亏格 $g \geq 11$ 曲线模空间中双椭圆轨迹单有理性的另一证明。

Rational curves on $V_5$ and rational simple connectedness

In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many further unirationality results for special moduli spaces of rational curves on quadric hypersurfaces and del Pezzo surfaces are obtained via explicit birational methods.

Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds A_3(1,1,4)

We prove that the moduli space A_3(1,1,4) of polarized abelian threefolds with polarization of type (1,1,4) is unirational. By a result of Birkenhake and Lange this implies the unirationality of the isomorphic moduli space A_3(1,4,4). The result is based on the study the Hurwitz space H_{4,n}(Y) of quadruple coverings of an elliptic curve Y simply branched in n points. We prove the unirationality of its codimension one subvariety H^{0}_{4,A}(Y) which parametrizes quadruple coverings π:X --> Y with Tschirnhausen modules isomorphic to A^{-1}, where A\in Pic^{n/2}Y, and for which π^*:J(Y)--> J(X)…

Unirationality of Hurwitz spaces of coverings of degree <= 5

Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{*}O_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3, 4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.

Rank-3 stable bundles on rational ruled surfaces

In this paper, we compare the moduli spaces of rank-3 vector bundles stable with respect to different ample divisors over rational ruled surfaces. We also discuss the irreducibility, unirationality, and rationality of these moduli spaces.

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.

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.

Matrix factorizations and families of curves of genus 15

In this note, we explain how certain matrix factorizations on cubic threefolds lead to families of curves of genus 15 and degree 16 in P^4. We prove that the moduli space M={(C,L) | C a curve of genus 15, L a line bundle on C of degree 16 with 5 sections} is birational to a certain space of matrix factorizations of cubics, and that M is uniruled. Our attempt to prove the unirationality of this space failed with our methods. Instead one can interpret our findings as evidence for the conjecture that the basis of the maximal rational connected fibration of M has a three dimensional base.

Prym varieties and moduli of polarized Nikulin surfaces

We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled.

The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

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 $k=64$.

The rationality of the moduli spaces of trigonal curves of odd genus

The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.

Ekedahl-Oort strata on the moduli space of curves of genus four

We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic.

Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds

We prove that the moduli spaces A_3(D) of polarized abelian threefolds with polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space H_{3,A}(Y) which parameterizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^{-1}.

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.

Pryms of non-cyclic triple coverings and log canonical models of the spin moduli space of genus 2

We show that the moduli spaces of non-triple cyclic coverings of genus 2 curves and that of even spin curves of genus 2 are birationally isomorphic via the Prym map. We describe the log canonical models of the moduli space of even spin curves of genus 2 and use this to extend the above birational map.

The moduli of curves of genus 6 and K3 surfaces

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.

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 10.

Calabi-Yau Threefolds and Moduli of Abelian Surfaces I

We describe birational models and decide the rationality/unirationality of moduli spaces AA_{d} (and AA^{lev}_{d}) of (1,d)-polarized abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi-Yau threefolds with Euler characteristic zero.

关于亏格16曲线模空间的Kodaira维数

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。

Perfect points on genus one curves and consequences for supersingular K3 surfaces

We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of such fibrations without purely inseparable multisections. Finally, we discuss the consequences for the claimed proof of the Artin conjecture on unirationality of supersingular K3 surfaces.

Boundary complexes of moduli spaces of curves in higher genus

Given a collection of boundary divisors in the moduli space of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in the moduli space of n-pointed curves of genus g.

Rational cohomology of the moduli space of trigonal curves of genus 5

We compute the rational cohomology of the moduli space of trigonal curves of genus 5. We do this by considering their natural embedding in the first Hirzebruch surface and by using Gorinov-Vassiliev's method.

On the rationality of the moduli of higher spin curves in low genus

The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for g >= 2 and r > 2. Using quite related geometric constructions, we almost complete the picture of the known results in genus g <= 4 showing the rationality of the moduli spaces of even and odd 4-spin curves of genus 3, of odd spin curves of genus 4 and of 3-spin curves of genus 4.

Equivariant cohomology of moduli spaces of genus three curves with level two structure

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked points and strata of the moduli space of Abelian differentials of genus three.

Quantum cohomology of moduli spaces of genus zero stable curves

We investigate the (small) quantum cohomology ring of the moduli spaces of stable n-pointed curves of genus 0. In particular, we determine an explicit presentation in the case n=5 and we outline a computational approach to the case n=6.

Cohomology of moduli spaces of curves of genus three via point counts

In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of…

16亏格曲线的模空间是单有理的

The moduli space of curves of genus 16 is uniruled

该研究运用非阿贝尔布里尔-诺特理论，证明16亏格曲线的模空间是单有理的，这是目前已知该类结果的最高亏格。

Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity

A formula for the generating function of the Weil-Petersson volumes of moduli spaces of pointed curves that is identical to the genus expansion of the free energy in two dimensional gravity is obtained. The contribution of arbitrary genus is expressed in terms of the Bessel function $J_0$.

低亏格拟极化 K3 曲面模空间的周环

Chow ring of moduli spaces of quasi-polarised K3 surfaces in lower genus

本文研究低亏格拟极化 K3 曲面模空间，证明亏格不超过 5 时其周环由重言类构成。

Moduli of Curves of Genus One with Twisted Fields

We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.

亏格3的曲线模空间中亏格105的一条完备曲线

A complete curve of genus 5 in the moduli space of curves of genus 3 and related questions

研究复数域上亏格3曲线模空间，通过特定方法证明其中存在一条亏格105的完备曲线。

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

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。

On the geometry of the moduli space of sheaves supported on curves of genus two in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 2, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers and we give a classification of the stable sheaves involving locally free resolutions.

The moduli space of étale double covers of genus 5 curves is unirational

We show that the coarse moduli space $\cR_5$ of étale double covers of curves of genus~5 over the complex numbers is unirational. We give two slightly different arguments, one purely geometric and the other more computational.

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。

The class of a Hurwitz divisor on the moduli of curves of even genus

We calculate the cycle class of the Hurwitz divisor $D_2$ on the moduli space of stable curves of genus $g=2k$ given by the degree $k+1$ covers of the projective line with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map of the Hurwitz space of admissible covers of degree $k+1$ and genus $2k$ to the moduli space of stable curves of genus $2k$.

Complete descriptions of the tautological rings of the moduli spaces of curves of genus lower or equal to 4 with marked points

We study here the tautological rings of the moduli spaces of compact Riemann surfaces of genus 1,2,3 and 4 with marked points. The paper presents the complete descriptions of these rings by describing the groups of all degrees.

The stringy E function of the moduli space of rank 2 bundles over a Riemann surface of genus 3

We compute the stringy E function of the moduli space of rank 2 bundles over a Riemann surface of genus 3. In doing so, we answer a question of Batyrev about the stringy E functions of the GIT quotients of linear representations.

On Deformations and Moduli of Genus 2 Fibrations

In this paper, we investigate the moduli of surfaces of general type admitting genus 2 fibrations with irregularity q = g_b + 1, where g_b >= 2 is the genus of the base. We prove that smooth fibrations are parametrized by a unique component in the moduli space. The same result applies to nonsmooth fibrations with special values of g_b. In the general case, we give a bound on the dimension of the corresponding connected components.

The rationality of the moduli space of curves of genus 3 after P. Katsylo

This article is a survey of P. Katsylo's proof that the moduli space of smooth projective complex curves of genus 3 is rational. We hope to make the argument more comprehensible and transparent by emphasizing the underlying geometry in the proof and its key structural features.

亏格六曲线的模空间与K-稳定性

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。

Moduli of sheaves supported on curves of genus two in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We give a classification of the stable sheaves involving locally free resolutions or extensions. We compute the Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs.

Heegner divisors in the moduli space of genus three curves

S. Kondō used periods of $K3$ surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by: hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational "splitting curves".

9 亏格 Prym 模空间的单有理性

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。

The moduli space of genus 4 even spin curves is rational

By the technique of 3-fold Mori theory, we prove that the moduli space whose general point parameterizes a couple of a smooth curve of genus 4 and a halfcanonical divisor with vanishing global section is rational.

Towards large genus asymtotics of intersection numbers on moduli spaces of curves

We explicitly compute the diverging factor in the large genus asymptotics of the Weil-Petersson volumes of the moduli spaces of $n$-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the Weil-Petersson volumes in the inverse powers of the genus with coefficients that are polynomials in $n$. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.

Picard group of moduli of curves of low genus in positive characteristic

We compute the Picard group of the moduli stack of smooth curves of genus $g$ for $3\leq g\leq 5$, using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on $\mathcal{M}_g$.