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

The trigonal construction in the ramified case

To every double cover ramified in two points of a general trigonal curve of genus g, one can associate an étale double cover of a tetragonal curve of genus g+1. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties.

Trigonal deformations of rank one and Jacobians

In this paper we study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure (IVHS) is of rank 1. We show that if the genus g is greater or equal to 8 or g=6,7 and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete a result of Naranjo and Pirola. We show in fact that if the genus g is greater or equal to 6, the hyperelliptic locus is the only 2g-1-dimensional sub-locus Y of the moduli space of curves of genus g, such that for the general element [C] in Y, its Ja…

三角规范纤维化曲面

Trigonal Canonically Fibered Surfaces

本文修复了相对亏格5的规范纤维化曲面的Xiao猜想证明中的漏洞，简化了原证明并改进了曲面几何亏格的界，同时将该方法推广至相对亏格3和4的曲面，获得了相应的Noether型不等式。

Irreducibility and stable rationality of the loci of Weierstrass points on curves of genus at most six

We show that for any numerical semigroup H of genus g at most 6, the locus of Weierstrass points on curves of genus g with Weierstrass semigroup H is irreducible and that for all but possibly two semigroups it is stably rational.

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.

论三角曲线的亚历山大不变量

On the Alexander Invariants of Trigonal Curves

本文证明辫群 B3 中大多数亏格零子群对亚历山大不变量无贡献，并引入“本原”亏格零子群来完全决定不变量，进而分类特殊子类并重新得到不可约三角曲线二面体覆盖的已知分类。

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.

The rationality of certain moduli spaces of curves of genus 3

We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, η) where C has genus 3 and ηis a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs (C, <η>) as above and where <η> is the cyclic subgroup of order 3 in Pic_0(C) generated by η.

Superspecial trigonal curves of genus $5$

This paper provides an algorithm enumerating superspecial trigonal curves of genus $5$ over finite fields. Executing the algorithm over a computer algebra system Magma, we enumerate them over finite fields $\mathbb{F}_{p^a}$ for any natural number $a$ if $p \leq 7$ and for odd $a$ if $p \leq 13$.

关于某些 Fano 三维簇的有理性

On the rationality of certain Fano threefolds

本文研究具典范奇点的 Gorenstein Fano 三维簇有理性，证明三角、含平面、高亏格及含普通直线等四类情形在相应阈值下均为有理簇。

Rationality of the universal K3 surface of genus 8

The aim of the present paper is to prove the rationality of the universal family of polarized $ K3 $ surfaces of degree 14. This is achieved by identifying it with the moduli space of cubic fourfolds plus the data of a quartic scroll. The last moduli space is finally proved to be rational since it has a natural structure of $\mathbb P^n$-bundle over a $ k $-stably rational variety with $k \leq n$.

无效 spin 亏格 4 thetanull 轨迹的有理性

The rationality of ineffective spin genus-4 thetanull loci

本文证明了由亏格4且thetanull消失的曲线C与无效thetacharacteristic θ构成的对[C,θ]所给出的除子是有理簇，并得出Prym模空间中的类似除子也是有理的。

A note on syzygies and normal generation for trigonal curves

Let $C$ be a trigonal curve of genus $g\ge 5$ and let $T$ be the unique trigonal line bundle inducing a map $π: C \stackrel{3:1}{\longrightarrow} \mathbb{P}^1$. This note provides a short and easy proof of the normal generation for the residual line bundle $K_C\otimes T^{-1}$ for curves of genus $g\ge 7$. Moreover, we compute the minimal free resolution of the embedded curve $C\subset \mathbb{P}(H^0(C,K_C\otimes T^{-n})^*)$ for the residual line bundle $K_C\otimes T^{-n}$ for $n\ge 1$ and $g \ge 3n+4$.

Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

We develop the theory of Abelian functions associated with cyclic trigonal curves by considering two new cases. We investigate curves of genus six and seven and consider whether it is the trigonal nature or the genus which dictates certain areas of the theory. We present solutions to the Jacobi inversion problem, sets of relations between the Abelian function, links to the Boussinesq equation and a new addition formula.

一个参数化三角亏格4曲线的球商

A ball quotient parametrizing trigonal genus 4 curves

本文证明带 $g^1_3$ 的亏格4曲线模空间是9维Deligne-Mostow球商的 $\ rac{1}{2}(3^{10}-1)$ 次覆盖，并给出不依赖K3曲面和Torelli定理的初等构造。

热带三元曲线

Tropical trigonal curves

研究热带三元曲线与调和映射的关系，证明其模空间的维度与算术三元曲线相同。

Mapping Class Groups of Trigonal Loci

In this paper we study the topology of the stack $\mathcal{T}_g$ of smooth trigonal curves of genus g, over the complex field. We make use of a construction by the first named author and Vistoli, that describes $\mathcal{T}_g$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $\mathcal{T}_g$, of its substrata with prescribed Maroni invariant and describe their relation with the mapping class group $\mathcal{M}ap_g$ of Riemann surfaces of genus g.

Rationality of the Moduli Variety of Curves of Genus 3

We prove rationality of the moduli variety of curves of genus 3.

The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then…

三重曲线与低亏格嵌入的热带曲线

Trigonal and embedded tropical curves of low genus

本文研究了低亏格热带曲线的三重性表征及其与希尔伯特表面嵌入的关系，探讨了三重映射与嵌入障碍的联系，并通过实例展示非光滑嵌入的展开过程。

Geometry of genus 8 Nikulin surfaces and rationality of their moduli

Let S be a general complex Nikulin surface of genus 8, a geometric construction of S is given as follows. Consider a smooth 3-fold linear section T of the Grassmannian G(1,4) and the Hilbert scheme of rational normal sextic curves of T. In it consider the special family of sextics A which are also contained in the congruence of bisecant lines to a rational normal quartic curve of P^4. We show that S is biregular to a quadratic section of T containing a sextic A. In particular A admits a 1-dimensional family of bisecant lines contained in G(1,4) and 8 of them are in S. This explicit constructio…

Rationality of Mukai varieties over non-closed fields

We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hook…

Genus-five hyperelliptic or trigonal curves with many rational points in characteristic three

The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and…

具有两个 $\frac12(1,1,1)$-点的亏格 $8$ Fano $3$-折的模空间：有理性和 K-稳定性

The moduli space of genus $8$ Fano $3$-folds with two $\frac12(1,1,1)$-points: rationality and K-stability

研究具有两个 $\frac12(1,1,1)$ 型终端商奇点的亏格 $8$ Fano $3$-折的模空间，证明其有理性和一般成员的 K-稳定性。

Some remarks on bielliptic and trigonal curves

We prove some results on algebraic curves $X$ of genus $g\geq 2$ in characteristic $0$. For example: Assume that $X$ has an automorphism $σ$ of prime order $p\geq 5$. If $σ$ has no fixed points, then $X$ cannot be trigonal. On the other hand, if $σ$ has fixed points, then $X$ is bielliptic only if it belongs to one of three extremal types of curves of small genus.

The al function of a cyclic trigonal curve of genus three

A cyclic trigonal curve of genus three is a $\mathbb{Z}_3$ Galois cover of $\mathbb{P}^1$, therefore can be written as a smooth plane curve with equation $y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)$. Following Weierstrass for the hyperelliptic case, we define an ``$\mathrm{al}$'' function for this curve and $\mathrm{al}^{(c)}_r$, $c=0,1,2$, for each one of three particular covers of the Jacobian of the curve, and $r=1,2,3,4$ for a finite branchpoint $(b_r,0)$. This generalization of the Jacobi $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ functions satisfies the relation: $$ \sum_{r=1}^…

三角典范曲线的Harder-Narasimhan滤过

The Harder-Narasimhan Filtration of a Trigonal Canonical Curve

本研究利用三角典范曲线位于有理正规面scroll上的性质，计算了一般三角典范曲线及亏格6典范曲线法丛的Harder-Narasimhan滤过，为代数几何中相关丛的结构研究提供了具体结果。

亏格4中Prym映射的消解

Resolution of the Prym map in genus 4

研究亏格4中Prym映射的消解问题，核心方法是热带三角构造，描述了相关最小环面消解。

Scorza quartics of trigonal spin curves and their varieties of power sums

Our fundamental result is the construction of new subvarieties in the varieties of power sums for the Scorza quartic of any general pairs of trigonal curves and non-effective theta characteristics. This is a generalization of Mukai's description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartic for any general pairs of curves and non-effective theta characteristics.

Stable cohomology of the moduli space of trigonal curves

We prove that the rational cohomology $H^i(\mathcal{T}_g;\mathbf{Q})$ of the moduli space of trigonal curves of genus $g$ is independent of $g$ in degree $i<\lfloor g/4\rfloor.$ This makes possible to define the stable cohomology ring as $H^\bullet(\mathcal{T}_g;\mathbf{Q})$ for a sufficiently large $g.$ We also compute the stable cohomology ring, which turns out to be isomorphic to the tautological ring. This is done by studying the embedding of trigonal curves in Hirzebruch surfaces and using Gorinov-Vassiliev's method.

Maximally inflected real trigonal curves on Hirzebruch surfaces

In 2014 A. Degtyarev, I. Itenberg and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type~I (over a base $ B $ of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for $ B=\mathbb{P}^1 $ obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type~II over $ B=\mathbb{P}^1 $. As usual, the results for such curves are extended to real Jacobian elliptic surfaces with all singular fibers real.

Real trigonal curves and real elliptic surfaces of type I

We study real trigonal curves and elliptic surfaces of type $\I$ (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's \emph{dessins d'enfants}. We give a description of maximally inflected trigonal curves of type $\I$ in terms of the combinatorics of sufficiently simple graphs and, in the case of the rational base, obtain a complete classification of such curves. As a consequence, these results lead to conclusions concerning real Jacobian elliptic surfaces of type $\I$ with all singular fibers real.

The universal K3 surface of genus 14 via cubic fourfolds

Using the isomorphism between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. Precisely, we show that the universal K3 surface of genus 14 is a projective bundle over a certain moduli space of nodal scrolls in P^5, whose rationality we prove using a degenerate version of Mukai's structure theorem for curves of genus 8.

Rationality of some tetragonal loci

We prove that the moduli space of tetragonal curves of genus g>6 is rational when g is congruent to 1, 2, 5, 6, 9, 10 modulo 12 and not equal to 9, 45.

The connectedness of the moduli space of maps to homogeneous spaces

We prove the connectedness of the moduli space of maps (of fixed genus and homology class) to the homogeneous space G/P by degeneration via the maximal torus action. In the genus 0 case, the irreducibility of the moduli of maps is a direct consequence of connectedness. An analysis of a related Bialynicki-Birula stratification of the map space yields a rationality result: the (coarse) moduli space of genus 0 maps to G/P is a rational variety. The rationality argument depends essentially upon rationality results for quotients of SL2 representations proven by Katsylo and Bogomolov.

实三重曲线上的特殊除子

Special Divisors on Real Trigonal Curves

本文研究了实三重曲线相关Brill-Noether簇的拓扑结构，在Maroni不变量等特定关系约束下，计算了参数化特定次数和维数线性系统簇的实轨迹连通分量数量。

亏格5非特殊曲线作为平面六次曲线的表示及其在寻找具有许多有理点的曲线中的应用

Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points

本文为亏格5非超椭圆非三叶草曲线提供有效参数化，并构造算法完整枚举有理点超过给定界限的此类曲线，在特征3素域上用MAGMA实现。

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 $g \geq 2$.

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 $g\geq 2$.

A Note on a Brill-Noether Locus over a Non-hyperelliptic curve of genus 4

We prove that a certain Brill-Noether locus over a non-hyperelliptic curve $C$ of genus 4, is isomorphic to the \textit{Donagi-Izadi cubic threefold} in the case when the pencils of the two trigonal line bundles of $C$ coincide.

The rationality of quaternionic Darmon points over genus fields of real quadratic fields

Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in previous joint works with Rotger as generalizations of Darmon's Stark-Heegner points. In this article we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner poi…

稳定映射的Kontsevich模空间的合理性及其伽罗瓦下降

Rationality of Kontsevich moduli of stable maps, and their Galois descents

本文研究域k上几何有理簇X的低次数零亏格稳定映射的Kontsevich模空间的k-合理性，定义其扭曲伽罗瓦下降，针对不同类型的X展开分析，复数域齐次簇情形对应Kim-Pandharipande的已知结论。

Deciding trigonality of algebraic curves

Let C be a non-hyperelliptic algebraic curve of genus at least 3. Enriques and Babbage proved that its canonical image is the intersection of the quadrics that contain it, except when C is trigonal (that is, it has a linear system of degree 3 and dimension 1) or C is isomorphic to a plane quintic (genus 6). We present a method to decide whether a given algebraic curve is trigonal, and in the affirmative case to compute a map from C to the projective line whose fibers cut out the linear system. It is based on the Lie algebra method presented in Schicho (2006). Our algorithm is part of a larger …

Moduli of Trigonal Curves

We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any fibration $X\to B$ of stable curves with smooth general member is the ratio $δ_B/λ_B$ of the restrictions of the boundary class $δ$ and the Hodge class $λ$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base $B$. We associate to a trigonal family $X$ a canonical rank two vector bundle $V$, and show that for Bogomolov-semistable $V$ the slope satisfies the stronger inequality ${δ_B}/{λ_B}\leq 7+{6}/{g}$. We further describe the rational…

Computing binary curves of genus five

Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in $\mathbb{P}^4$. We present and explain algorithms we used to determine, up to isomorphism over $\mathbb{F}_2$, all genus 5 curves defined over $\mathbb{F}_2$, and we do that separately for each of the three mentioned types. We consider these curves in terms of isogeny classes over $\mathbb{F}_2$ of their Jacobians or their Newton polygons, and for each of the three types, we compute the number of curves over $\mathbb{F}_2$ weighted by the size of their $…

Some results on the second Gaussian map for curves

We study the second Gaussian map for a curve X of genus g, in relation with the second fundamental form of the period map. We exhibit a class of infinitely many curves with surjective second Gaussian map. We compute its rank on the hyperelliptic and trigonal locus and show global generation of its image for X not hyperelliptic nor trigonal.

An arithmetic invariant theory of curves from $E_8$

Let $k$ be a field of characteristic 0, let $C/k$ be a uniquely trigonal genus 4 curve, and let $P \in C(k)$ be a simply ramified point of the uniquely trigonal morphism. We construct an assignment of an orbit of an algebraic group of type $E_8$ acting on a specific variety to each element of $J_C(k)/2$. The algebraic group and variety are independent of the choice of $(C,P)$. We also construct a similar identification for uniquely trigonal genus 4 curves $C$ with $P \in C(k)$ a totally ramified point of the trigonal morphism. Our assignments are analogous to the assignment of a genus 3 curve …

Finite Hilbert stability of canonical curves, II. The even-genus case

This paper is a sequel to arXiv:1109.4986, where we proved that a general smooth curve of odd genus, canonically or bicanonically embedded, has semistable finite Hilbert points. Here, we prove that a generic canonically embedded curve of even genus has semistable finite Hilbert points. More precisely, we prove that a generic canonically embedded trigonal curve of even genus has semistable finite Hilbert points. Furthermore, we show that the analogous result fails for bielliptic curves. Namely, the Hilbert points of bielliptic curves are asymptotically semistable but become non-semistable below…

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。

Effective radical parametrization of trigonal curves

Let $C$ be a non-hyperelliptic algebraic curve. It is known that its canonical image is the intersection of the quadrics that contain it, except when $C$ is trigonal (that is, it has a linear system of degree 3 and dimension 1) or isomorphic to a plane quintic (genus 6). In this context, we present a method to decide whether a given algebraic curve is trigonal, and in the affirmative case to compute a map from $C$ to the projective line whose fibers cut out the linear system.

特殊化下代数曲线的有理性与参数化

Rationality and Parametrizations of Algebraic Curves under Specializations

本研究针对含参数的有理代数曲线特殊化问题，通过划分参数仿射空间明确亏格变化与曲线可约性的参数区域，在亏格零情形下优化参数化描述并给出应用实例，完善了代数曲线特殊化的理论刻画。

On the existence of curves with prescribed $a$-number

We study the existence of Artin-Schreier curves with large $ a$-number. We also give bounds on the $a$-number of trigonal curves of genus $5$ in small characteristic.

Rationality and Chow-Kuenneth decompositions for some moduli stacks of curves

In this paper, we show the existence of a Chow--Kuenneth decomposition for the moduli stack of stable curves of genus g with r marked points, for low values of g,r. We also look at the moduli space R of double covers of genus 3 curves, branched along 4 distinct points. We obtain a birational model of the moduli space R as a group quotient of a product of two Grassmanian varieties. This provides a Chow-Kuenneth decomposition over an open subset of R. The question of rationality of R is also discussed.

A census of all genus 4 curves over the field with 2 elements

We explain how we computed equations for all genus 4 curves defined of the field with 2 elements, up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.