We show that most of the genus-zero subgroups of the braid group $\mathbb{B}_3$ (which are roughly the braid monodromy groups of the trigonal curves on the Hirzebruch surfaces) are irrelevant as far as the Alexander invariant is concerned: there is a very restricted class of \enquote{primitive} genus-zero subgroups such that these subgroups and their genus-zero intersections determine all the Alexander invariants. Then, we classify the primitive subgroups in a special subclass. This result implies the known classification of the dihedral covers of irreducible trigonal curves.
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.
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.
In this paper, we focus on properties of dessins d'enfants associated to trigonal curves. Degtyarev studied dessins d'enfants to compute braid monodromies and fundamental groups of trigonal curves using their combinatorial data. We first classify all possible combinatorial data that can occur for trigonal curves of low degree, as well as bounds on the number of possibilities for all degree. We also study deformations of trigonal curves and corresponding deformations of their dessins. Of special interest to Degtyarev was the case when the dessins are maximal. We give a sufficient condition for …
Trigonal curves provide an example of Brill-Noether special curves. Theorem 1.3 of [9] characterizes the Brill-Noether theory of general trigonal curves and the refined stratification by Brill-Noether splitting loci, which parametrize line bundles whose push forward to $\mathbb{P}^1$ has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill-Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected…
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 $(°D-3)$.
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.
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.
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.
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.
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.
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.
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.
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 $g \ge 3n+4$.
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.
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.
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.
In this paper, we compute Alexander polynomials of a torus curve C of type (2, 5), C : f(x, y) = f_2(x, y)^5 + f_5(x, y)^2 = 0, under the assumption that the origin O is the unique inner singularity and f2 = 0 is an irreducible conic. We show that the Alexander polynomial remains the same with that of a generic torus curve as long as C is irreducible.
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.
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…
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.
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…
We prove two formulae which express the Alexander polynomial $Δ^C$ of several variables of a plane curve singularity $C$ in terms of the ring ${\cal O}_{C}$ of germs of analytic functions on the curve. One of them expresses $Δ^C$ in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring ${\cal O}_{C}$. The other one gives the coefficients of the Alexander polynomial $Δ^C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes). The final version of this article will be published in the Duke Math…
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.
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…
We show that the degree of the Alexander polynomial of an irreducible plane algebraic curve with nodes and cusps as the only singularities does not exceed ${5 \over 3}d-2$ where $d$ is the degree of the curve. We also show that the Alexander polynomial $Δ_C(t)$ of an irreducible curve $C=\{F=0\}\subset \mathbb P^2$ whose singularities are nodes and cusps is non-trivial if and only if there exist homogeneous polynomials $f$, $g$, and $h$ such that $f^3+g^2+Fh^6=0$. This is obtained as a consequence of the correspondence, described here, between Alexander polynomials and ranks of Mordell-Weil gr…
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.
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.
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 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.
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.
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] .
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.
We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with $θ$^Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration $C_t \to D_0+jL$. We show that there exists a certain surjectivity of the fundamental groups and divisibility among their Alexander polynomials.
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.
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 $(3,3n)$ …
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…
The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface $S=\{(z_1,z_2,z_3)\in {\mathbb C}^3: z_1^5+z_2^3+z_3^2=0\}$ with the 5-dimensional sphere ${\mathbb S}_{\varepsilon}^5=\{(z_1,z_2,z_3)\in {\mathbb C}^3: \vert z_1\vert^2+\vert z_2\vert^2+\vert z_3\vert^2=\varepsilon^2\}$. An algebraic link in the Poincaré sphere is the intersection of a germ $(C,0)\subset (S,0)$ of a complex analytic curve in $(S,0)$ with the sphere ${\mathbb S}_{\varepsilon}^3$ of radius $\varepsilon$ …
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 …
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.
We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the classical case, this gives a geometrical interpretation of Libgober's divisibility Theorem. We calculate twisted polynomials for some algebraic curves and show how they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degeneration…
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.
We give two formulae which express the Alexander polynomial $Δ^C$ of several variables of a plane curve singularity $C$ in terms of the ring ${\cal O}_{C}$ of germs of analytic functions on the curve. One of them expresses $Δ^C$ in terms of dimensions of some factorspaces corresponding to a (multi-indexed) filtration on the ring ${\cal O}_{C}$. The other one gives the coefficients of the Alexander polynomial $Δ^C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).
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…
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…
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.
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}^…
