Trigonal Curves Research Articles

Explicit moduli of superelliptic curves with level structure

In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine.

Quasi-periodic solutions of a discrete integrable equation with a finite-dimensional integrable symplectic structure

Through the paper, we research the quasi-periodic solutions to a semi-discrete hierarchy which has Hamiltonian structure and integrable symplectic map. Firstly, a semi-discrete hierarchy is derived by use of the discrete zero-curvature representation and then its integrability is proved under the Liouville condition. Using the binary nonlinearization approach, the integrable symplectic map and finite-dimensional Hamiltonian system of the hierarchy are obtained. Moreover, the trigonal curve Kq−1 is denoted through the characteristic polynomials for the Lax pair, as well as the related Baker–Akhiezer function and meromorphic function are introduced, from which the asymptotic properties and divisors of the two functions mentioned above are analyzed. Finally, we introduce the three kinds of Abel differentials and straighten out of corresponding continuous and discrete flows, the quasi-periodic solutions of the equations are received via the Riemann theta function.

Application of the trigonal curve to a hierarchy of generalized Toda lattices

Application of the trigonal curve to a hierarchy of generalized Toda lattices

Application of the trigonal curve to a hierarchy of generalized Toda lattices

Из уравнения нулевой кривизны и рекуррентных соотношений Ленарда получена иерархия обобщенной цепочки Тоды. Через характеристический полином пары Лакса для дискретной иерархии вводится тригональная кривая; в результате уравнения расщепляются, и получается система уравнений типа Дубровина. Проведен анализ асимптотик функции Бейкера-Ахиезера и мероморфной функции, а также обсуждаются дивизоры этих функций. Кроме того, определено отображение Абеля, соответствующие потоки на многообразии Якоби выпрямляются, таким образом, окончательные алгебро-геометрические решения иерархии находятся с помощью тета-функций Римана.

Stable Cohomology of the Moduli Space of Trigonal Curves

Abstract 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.

Quasi‐periodic solutions to a hierarchy of integrable nonlinear differential–difference equations

Using the discrete zero‐curvature equation, we derive a hierarchy of new nonlinear differential–difference equations associated with a discrete matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix, we introduce a trigonal curve and the associated three‐sheeted Riemann surface, from which we derive the Baker–Akhiezer function and meromorphic function. By comparing the asymptotic expansions of the meromorphic function and its Riemann theta function representations, we obtain quasi‐periodic solutions for a hierarchy of nonlinear differential–difference equations.

Quasi-periodic solutions of three-component Burgers hierarchy

<abstract><p>Starting from a $ 3\times3 $ matrix spectral problem and the characteristic polynomial of the Lax matrix, we propose a trigonal curve, the associated meromorphic functions and three kinds of Abelian differentials. By discussing the asymptotic properties for the Baker-Akhiezer functions and their Riemann theta function expressions, we get quasi-periodic solutions of the three-component Burgers hierarchy. Finally, we straighten out the three-component Burgers flows.</p></abstract>

The trigonal construction in the ramified case

AbstractTo every double cover ramified in two points of a general trigonal curve of genus , one can associate an étale double cover of a tetragonal curve of genus . We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties. This extends Recillas' trigonal construction to covers ramified in two points.

Finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy

Resorting to the zero-curvature equation and the Lenard recursion equations, the generalized Merola–Ragnisco–Tu lattice hierarchy associated with a 3 × 3 discrete matrix spectral problem is derived. With the aid of the characteristic polynomial of the Lax matrix for the generalized Merola–Ragnisco–Tu lattice hierarchy, a trigonal curve is defined, on which we construct the Baker–Akhiezer function, two meromorphic functions, three kinds of Abelian differentials, and Riemann theta function. By analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions, especially their asymptotic expansions near three infinite points and three zero points, we obtain their essential singularities and divisors. Finally, we obtain the finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy in terms of the Riemann theta function.

On a Harry–Dym-Type Hierarchy: Trigonal Curve and Quasi-Periodic Solutions

Resorting to the characteristic polynomial of Lax matrix for a Harry–Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker–Akhiezer function and meromorphic function are introduced. With the help of the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we obtain the quasi-periodic solutions for the entire Harry–Dym-type hierarchy.

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 Kodaira dimension of Hurwitz spaces

We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type.

A cellular structure of the space of branched coverings for the two-dimensional sphere

For a closed oriented surface Σ \Sigma , let X Σ , n X_{\Sigma , n} be the space of isomorphism classes of n n -fold orientation preserving branched coverings Σ → S 2 \Sigma \rightarrow S ^2 of the two-dimensional sphere. Earlier, the authors constructed a compactification X ¯ Σ , n \overline {X}_{\Sigma , n} of this space, which coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H ( Σ , n ) ⊂ X Σ , n H (\Sigma , n) \subset X_{\Sigma , n} that consists of isomorphism classes of branched coverings with all critical values being simple. With the help of Grothendieck’s dessins d’enfants, a cellular structure of this compactification is constructed. The results obtained are applied to the space of trigonal curves on an arbitrary Hirzebruch surface.

Algebro-geometric integration of a modified shallow wave hierarchy

Abstract By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

Refined Brill–Noether theory for all trigonal curves

Trigonal curves provide an example of Brill–Noether special curves. Theorem 1.3 of Larson (Invent Math 224(3):767–790, 2021) 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, and if furthermore the Maroni invariant is 0 or 1, they are irreducible.

Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

Generic pointed quartic curves in {mathbb {R}}{mathbb {P}}^{2} and uninodal dessins

In this article we obtain a rigid isotopy classification of generic pointed quartic curves (A, p) in {mathbb {R}}{mathbb {P}}^{2} by studying the combinatorial properties of dessins. The dessins are real versions, proposed by Orevkov (Ann Fac Sci Toulouse 12(4):517–531, 2003), of Grothendieck’s dessins d’enfants. This classification contains 20 classes determined by the number of ovals of A, the parity of the oval containing the marked point p, the number of ovals that the tangent line T_p A intersects, the nature of connected components of Asetminus T_p A adjacent to p, and in the maximal case, on the convexity of the position of the connected components of Asetminus T_p A. We study the combinatorial properties and decompositions of dessins corresponding to real uninodal trigonal curves in real ruled surfaces. Uninodal dessins in any surface with non-empty boundary can be decomposed in blocks corresponding to cubic dessins in the disk {mathbf {D}}^2, which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of generic pointed quartic curves in {mathbb {R}}{mathbb {P}}^{2}. This classification was first obtained in Rieken (Geometr Ded 185(1):171–203, 2016) based on the relation between quartic curves and del Pezzo surfaces.

Explicit Solutions of the Coupled Bogoyavlensky Lattice 1(2) Hierarchy

By means of the discrete zero curvature representation, the coupled Bogoyavlensky lattice 1(2) hierarchy related to a $$3\times 3$$ matrix problem is derived. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a trigonal curve $$\mathcal {K}_{m-1}$$ of arithmetic genus $$m-1$$ and construct the related Baker–Akhiezer function and meromorphic function. By analyzing the asymptotic expansion of the meromorphic function, the algebro-geometric solutions to the stationary coupled Bogoyavlensky lattice 1(2) are obtained. Moreover, the explicit theta function representations for the coupled Bogoyavlensky lattice hierarchy are given with the help of the Abelian differential.

Characterizing gonality for two-component stable curves

It is a well-known result that a stable curve of compact type over \({\mathbb {C}}\) having two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersection is a Weierstrass point for each of them. With the use of admissible covers, we generalize this characterization in two ways: for stable curves of higher gonality having two smooth components and one node; and for hyperelliptic and trigonal stable curves having two smooth non-rational components and any number of nodes.

On the Alexander invariants of trigonal curves

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.