Stacks of trigonal curves

Dihedral coverings of trigonal curves

We classify and study trigonal curves in Hirzebruch surfaces admitting dihedral Galois coverings. As a consequence, we obtain certain restrictions on the fundamental group of a plane curve~$D$ with a singular point of multiplicity $(°D-3)$.

Counting the number of trigonal curves of genus 5 over finite fields

The trigonal curves of genus 5 can be represented by projective plane quintics that have 1 singularity of delta invariant 1. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this then gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

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.

The Alexander module of a trigonal curve. II

We complete the enumeration of the possible roots of the Alexander polynomial (both conventional and over finite fields) of a trigonal curve. The curves are not assumed proper or irreducible.

The Alexander module of a trigonal curve

We describe the Alexander modules and Alexander polynomials (both over $\Q$ and over finite fields $\FF{p}$) of generalized trigonal curves. The rational case is closed completely; in the case of characteristic $p>0$, a few points remain open.

具有最大秩三角变形的三角曲线

Trigonal Curve with Trigonal Deformation of Maximal Rank

研究由希尔伯特曲面丰富截面给出的非退化曲线在线性系统内秩为\(g\)的一阶变形，通过扩展方法给出计算准则，计算实例证明对于\(g = 2k + 1\)且\(k\geq2\)存在具该变形的三角曲线。

三重曲线上的高斯映射

Gaussian maps on trigonal curves

本文研究了循环三重曲线的高斯映射，确定了秩的下界，并给出了第二高斯映射核的显式描述，证明了某些空间中不存在额外的渐近方向。

Thomae's Derivative Formulae for Trigonal Curves

In this paper we prove a Thomae derivative formula for trigonal curves admitting a non-singular affine model. This formula relates the derivatives of theta functions with rational characteristics on the curve to explicit expressions in the branching values.

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。

The fundamental group of a generalized trigonal curve

We develop a modification of the Zariski--van Kampen approach for the computation of the fundamental group of a trigonal curve with improper fibers. As an application, we list the deformation families and compute the fundamental groups of all irreducible maximizing simple sextics with a type $\bold D$ singular point.

Abelian Functions for Cyclic Trigonal Curves of Genus Four

We discuss the theory of generalized Weierstrass $σ$ and $\wp$ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the "purely trigonal" (or "cyclic trigonal") curve $y^3=x^5+λ_4 x^4 +λ_3 x^3+λ_2 x^2 +λ_1 x+λ_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the $\wp$ functions, and the derivation of an addition formulae.

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.

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.

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.

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 $g \ge 3n+4$.

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.

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。

Sharp slope bounds for sweeping families of trigonal curves

We establish sharp bounds for the slopes of curves in $\bar{M}_g$ that sweep the locus of trigonal curves, proving Stankova-Frenkel's conjectured bound of $7+6/g$ for even $g$ and obtaining the bound $7+20/(3g+1)$ for odd $g$. For even $g$, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd $g$, we describe the analogous extremal effective divisor and give a similar explicit expression.

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.

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。

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。

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.

Abelian functions for trigonal curves of degree four and determinantal formulae in purely trigonal case

This paper gives a natural extension of Frobenius-Stickelberger formula and Kiepert formula to Abelian functions for "Purely Trigonal Curves", especially, of degree four. A description on the theory of Abelian functions for general trigonal curves of degree four is also included.

Modular compactifications of the space of marked trigonal curves

We construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent. Beginning with the standard admissible cover compactification, the sequence first proceeds through contractions of the boundary divisors and then through flips of the so-called Maroni strata, culminating in a Fano model for even genera and a Fano fibration for odd genera. While the sequence of divisorial contractions arises from a more general construction, the sequence of flips uses the particular geometry of triple covers. We explicitly desc…

Abelian Functions for Trigonal Curves of Genus Three

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations satisfied by the \wp-functions, a proof that the coefficients of the power series expansion of the sigma-function are polynomials of moduli parameters, and the derivation of two addition formulae.

Trigonal curves and Galois Spin(8)-bundles

Let SU_C(2) denote the moduli variety of rank 2 semistable vector bundles with trivial determinant on an algebraic curve C. We prove that if C is trigonal then there exists a projective moduli variety N_C containing SU_C(2) as a subvariety and smooth of dimension 7g-14 away from SU_C(2). N_C parametrises Galois Spin(8)-bundles on the Galois closure of C over P^1. Moreover, if x in J_C[2] is a 2-torsion point let R(x) be the Recillas tetragonal curve whose Jacobian is isomorphic to Prym(C,x). Then there is an injection of SU_R(x)(2) into N_C giving a `nonabelian Schottky configuration' in N_C s…

A note on local trigonal fibrations

We show that some of centeral fibers of degenerations of hyperelliptic curves are realized as those trigonal curves. In particular, any hyperelliptic curve can be the central fiber of a degeneration of trigonal curves.

热带三次曲线：一般情形

Tropical trigonal curves: the general case

本文将度为3的因子存在性与调和态射的等价性从3-边连通热带曲线推广到不含项链的低边连通热带曲线。

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…

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 $p \leq 13$.

A Viro Theorem without convexity hypothesis for trigonal curves

A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coherent way we prove that no convexity hypothesis is required to patchwork such curves.

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.

Absolutely simple Prymians of trigonal curves

Using Galois Theory, we construct explicitly absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. Our approach is based on the previous papers math/0610138 [math.AG] and math/0605028 [math.AG] .

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.

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…

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.

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.

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 …

三角曲线的图与双曲面上双次数$(4,3)$的奇异实代数曲线的刚性同痕

Graphs of trigonal curves and rigid isotopies of singular real algebraic curves of bidegree $(4,3)$ on a hyperboloid

本文利用 Hirzebruch 曲面上实三角曲线的图，补全双曲面上双次数$(4,3)$奇异实代数曲线刚性同痕分类中16类曲线连通分量唯一性的缺失证明。

One Step Degeneration of Trigonal Curves and Mixing of Solitons and Quasi-Periodic Solutions of the KP Equation

We consider certain degenerations of trigonal curves and hyperelliptic curves, which we call one step degeneration. We compute the limits of corresponding quasi-periodic solutions using the Sato Grassmannian. The mixing of solitons and quasi-periodic solutions is clearly visible in the obtained solutions.

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。

Families of n-gonal curves with maximal variation of moduli

We study families of n-gonal curves with maximal variation of moduli, which have a rational section. Certain numerical results on the degree of the modular map are obtained for such families of hyperelliptic and trigonal curves. In the last case we use the description of the relative Picard group of the universal family of trigonal curves.

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。

Trigonal Morsifications on Hirzebruch Surfaces with an appendix by E. Shustin

In this paper we obtain a classification of rigid isotopy classes of totally reducible trigonal curves lying on a Hirzebruch surface $Σ_n$, and having a maximal number of non-degenerated double points. Such curves correspond to morsifications of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ (the union of three smooth real branches intersecting each other with multiplicity $n$). We obtain this classification by studying combinatorial properties of dessins. In the appendix, we prove that any morsification of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ …

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…

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}^…

Trigonal curves and algebro-geometric solutions to soliton hierarchies

Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hier…

Addition via reduction algorithm on trigonal curves

In this paper we propose a direct and explicit realization of addition of divisors by means of an iterative reduction algorithm. Each iteration of the algorithm is the reduction of a degree $g+1$ divisor to a divisor of degree~$g$. Such an approach allows to carry out all computations explicitly in a symbolic form, which is done for curves $C_{3,4}$, $C_{3,5}$ in this paper, and also for curves of higher genera up to $C_{3,13}$, $C_{3,14}$.

三角纤维曲面及高维纤维化的斜率

Slopes of trigonal fibred surfaces and of higher dimensional fibrations

本文推广Cornalba-Harris方法，在GIT半稳定条件下给出高维纤维化斜率下界，并证明偶数亏格三角纤维化斜率的锐利下界，从而证实Harris-Stankova猜想。

Degeneration of Trigonal Curves and Solutions of the KP-Hierarchy

It is known that soliton solutions of the KP-hierarchy corresponds to singular rational curves with only ordinary double points. In this paper we study the degeneration of theta function solutions corresponding to certain trigonal curves. We show that, when the curves degenerate to singular rational curves with only ordinary triple points, the solutions tend to some intermediate solutions between solitons and rational solutions. They are considered as cerain limits of solitons. The Sato Grassmannian is extensively used here to study the degeneration of solutions, since it directly connects sol…

The sigma function for trigonal cyclic curves

A recent generalization of the "Kleinian sigma function" involves the choice of a point $P$ of a Riemann surface $X$, namely a "pointed curve" $(X, P)$. This paper concludes our explicit calculation of the sigma function for curves cyclic trigonal at $P$. We exhibit the Riemann constant for a Weierstrass semigroup at $P$ with minimal set of generators $\{3, 2r+s,2s+r\}$, $r<s$, equivalently, non-symmetric, we construct a basis of $H^1(X, \mathbb{C})$ and a fundamental 2-differential on $X\times X$, we give the order of vanishing for sigma on Wirtinger strata of the Jacobian of $X$, and a solut…

Spin curves and Scorza quartics

In the previous paper, we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. In this paper, we show that these quartics coincide with the Scorza quartics of general pairs of trigonal curves and ineffective theta characteristics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartics for any general pairs of curves and ineffective theta characteristics. We also give descriptions of the moduli spaces of trigonal even spin curves. For curves of genus 4, we deepen this descri…

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 …

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.