arXiv research

Tool for contracting subcurves of hyperelliptic curves, proving differential implications.

New Einstein-Weyl spaces derived from hyperelliptic curves.

New examples of translation surfaces on hyperelliptic curves with many automorphisms.

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.

Let $X$be a complex hyperelliptic curve of genus two equipped with the canonical metric $ds^2$. We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of $(X,ds^2)$ determines an explicit solution to a mean field equation.

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundament…

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 determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the spac…

We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus $g$, the number $N$ of non-separating vanishing cycles and the number $D$ of singular fibers satisfy the inequality $N \…

We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as solutions of sine-Gordon equation using Weierstrass's local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gord…

In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop solitons in a plane, \it i.e., loops whose curvatures obey the modified Korteweg-de Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous article …

Study of closed real plane curves with hyperelliptic genus three solutions.

We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.

A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…

We consider the Whitham equations for deformations of hyperelliptic spectral curves, which preserve all periods of a meromorphic differential. If the meromorphic differential has a root at a fixed point of the hyperelliptic involution, then the Whitham flow has a singularity. We prove that the stable and unstable manif…

New invariant links graph structure to tropical curve properties.

Study on quadratic L-functions using hyperelliptic curves and homology.

In this paper, we study the monodromy of the Hitchin fibration for rank 2 vector bundles over hyperelliptic curves. We reduce the problem to studying a surface braid group generalization of the classical Burau representation, and give a combinatorial method for computing this representation.

Researchers compute spin structures on hyperelliptic curves using braid groups.

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by t…

Minimal constructions of meanders and hyperelliptic pillowcase covers help in understanding ratio-optimizing pseudo-Anosovs.

New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.

New insights into algebraic geometry of a conjecture, leading to origami curves.

The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus $4$ with unique trigonal structure.

The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant subman…

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…

Analytic curves linked to algebraic ones via Schottky groups.

Study on second homology group of genus 3 hyperelliptic Torelli group.

A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of ${\mathbb C} P^1$ branched over 3 points. The algorithm yields a previously unknown symplectic basis of the hyperelliptic curve defined by the affine eq…

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first part of the paper, we show that, under mild restrictions, the moduli spaces of smoo…

Machine learning accurately distinguishes Sato-Tate groups for hyperelliptic curves.

Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over the 2-sphere. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of a 2-sphere bundle over the 2-sphere, branched over an embedded surface. If the …

Genus $g$ Torelli space is the moduli space of genus $g$ curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when $g\geq 3$ these com…

In this paper, we give an algebraic construction of the solution to the following mean field equation $Δψ+e^ψ=4π\sum_{i=1}^{2g+2}δ_{P_{i}},$ on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$ Furt…

Unique Teichmüller curve found in complex geometry.

Researchers describe a Ceresa class for tropical and topological curves, linking algebraic and cohomological perspectives.

Study Bergman kernel metrics on degenerating hyperelliptic surfaces.

We show that for each genus there are only finitely many algebraically primitive Teichmueller curves C, such that i) C lies in the hyperelliptic locus and ii) C is generated by an abelian differential with two zeros of order g-1. We prove moreover that for these Teichmueller curves the trace field of the affine group i…

Complex manifold describes solvable Pell-Abel equations with fixed degrees.

Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $ψ$. We show that there exists a sequence $(R_n, ψ_n)$ of fibers and monodromies contained in the fibered cone of $(S,ψ)$ such that the asymptotic translation length of $ψ_n$ on the curve complex $\mathca…

We characterize how to vary the Abel-Jacobi map in terms of Schiffer variation. From this characterization, we will interpret the relation of hyperellipticity of curves with Schiffer variation and describe the deformation of elliptic solitons under Schiffer variation.

We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determin…

A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}κ_{1}ψ^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstraß points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained. The calculation utilizes the homeomorphism between the moduli…

We resume the study initiated in \cite{CL}. For a generic curve $C$ in an ample linear system $\vert \mathcal{L} \vert$ on a toric surface $X$, a vanishing cycle of $C$ is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of $C$ to a nodal curve in $\vert \mathcal{L} \vert$.…

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingr…

The paper calculates period matrices for specific algebraic curves.

The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in $S ^ 3$ result when these spectral curves satisfy periodicity conditions. We prove that the spectra…