Extends abelian differentials to log twisted differentials with spin and hyperelliptic structures.
problem Compactify the moduli space of abelian differentials with spin and hyperelliptic structures.
method Introduce log twisted differentials and hyperelliptic differentials using stable log maps and admissible covers.
result Proves the existence of up to three connected components in the open strata of log twisted differentials.
In hyperelliptic components, GL(2,R) orbits are either closed, dense, or cover loci.
problem Understanding GL(2,R) orbits in hyperelliptic components of abelian differentials.
method Analyzing orbits in hyperelliptic components of abelian differentials.
result Orbits are either closed, dense, or cover loci.
Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.
problem Classifying Teichmüller curves in hyperelliptic components of meromorphic strata.
method Non-existence criterion based on intersections with moduli space boundary.
result Contradiction to algebraicity of candidate Teichmüller curves outside Hurwitz covers.
Study the boundary of Riemann surfaces with abelian automorphisms.
problem Characterize the boundary of Riemann surfaces with abelian automorphisms.
method Analyze the moduli space and its Deligne-Mumford compactification, focusing on equisymmetric loci.
result Describe the topological strata at the boundary for hyperelliptic and cyclic p-gonal actions. Consider the moduli space Mg of Riemann surfaces of genus g≥2 and its Deligne-Munford compactification Mgˉ. We are interested in the branch locus Bg for g>2, i.e., the subset of Mg consisting of surfaces with automorphisms. It is well-known that…
A closed formula is obtained for the integral ∫Hˉg1κ1ψ2g−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…
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
Study of trigonal curves in abelian differentials with specific divisor properties.
problem Characterizing locally closed subspaces of abelian differentials.
method Using linear systems on Segre-Hirzebruch surfaces to describe orbifold structure and orbifold fundamental groups.
result Identified the orbifold fundamental group of a specific subspace as a quotient of the Artin group of type E8. 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 prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping cla…
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy L(logL)d−2 growth rate. Classifies periodic diffeomorphisms and hyperelliptic involutions on surfaces.
problem Classifying periodic diffeomorphisms and involutions on surfaces.
method Refinement of Ishizaka's result using Dehn twist presentations.
result Dehn twist presentations of hyperelliptic periodic mapping classes.
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…
The Whitham flow for hyperelliptic curves has singularities that can be smoothly extended.
problem Singularities in the Whitham flow for hyperelliptic spectral curves.
method Analysis of deformations preserving periods of a meromorphic differential.
result Stable and unstable manifolds are non-empty, and the flow can be extended through the singularity.
The study finds a unique systole maximum in non-hyperelliptic surfaces.
problem Understanding systole functions on translation surfaces.
method Analyzing local and global maxima of systole functions.
result Local maxima are not global in non-hyperelliptic components.
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…
Tool for contracting subcurves of hyperelliptic curves, proving differential implications.
problem Understanding differentials on hyperelliptic curves and their limits.
method Flexible tool for contracting subcurves, proving Gorenstein contractions and dualising bundles.
result Hyperelliptic multiscale differentials determine Gorenstein contractions of nodal curves.
Study of hyperelliptic mapping class groups with applications and profinite completions.
problem Understanding hyperelliptic mapping class groups and their properties.
method Defined and studied hyperelliptic mapping class groups, applied theory to counterexamples, and examined profinite completions.
result Found a counterexample to a conjecture about mapping class groups and extended congruence subgroup property.
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. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As…
New Einstein-Weyl spaces derived from hyperelliptic curves.
problem Constructing new Lorentzian Einstein-Weyl spaces.
method Using hyperelliptic curves and minitwistors.
result Spaces are diffeomorphic to 3D deSitter space with closed geodesics.
Algorithms compute the topology of hyperelliptic curves in 2D and 3D.
problem Computing the topology of hyperelliptic curves in higher dimensions.
method Birational mapping of the plane or space to compute the topology of the curve.
result Algorithms implemented in { t Maple} for computing the topology of hyperelliptic curves.
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…
Discrete PU(1,1) representations of hyperelliptic groups are proven.
problem Characterizing PU(1,1) representations of hyperelliptic groups.
method Proving representations are basic if and only if they are discrete and faithful.
result A conjecture by S. Anan'in and E. Bento Gonçalves is partially proven.
Study on hyperelliptic Lefschetz fibrations, finding singular fiber counts.
problem Determining the minimal number of singular fibers in hyperelliptic Lefschetz fibrations.
method Analyzing complex surfaces and their Lefschetz fibrations over the 2-sphere.
result Minimal number of singular fibers is 2g+4 for even g≥4 and 2g+6 for odd g≥7.
We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve y2=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…
Finite groups with a hyperelliptic involution have a 2-rank of at most 4.
problem Finite groups acting on hyperelliptic 3-manifolds and their sectional 2-rank.
method Analyzing sectional 2-rank of finite groups containing hyperelliptic involutions.
result The sectional 2-rank of such groups is at most 4, with 4 being the best possible upper bound.
Study Fuchsian loci in mPSLn(R)-Hitchin components of a pair of pants.
problem Understanding Fuchsian loci in mPSLn(R)-Hitchin components. method Using Bonahon-Dreyer parametrization, explicit parametrization of Fuchsian loci of a pair of pants.
result Explicit parametrization of Fuchsian loci of a pair of pants.
Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.
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 \…
Let Xbe a complex hyperelliptic curve of genus two equipped with the canonical metric ds2. We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of (X,ds2) determines an explicit solution to a mean field equation.
The study finds bounds on homologically independent loops on hyperelliptic hyperbolic surfaces.
problem Finding bounds on the lengths of homologically independent loops on hyperelliptic hyperbolic surfaces.
method Analyzing the genus and using constant upper bounds on minimal length of non-zero period lattice vectors.
result For any λ∈(0,1), there exists a constant N(λ) such that every hyperelliptic hyperbolic surface has at least ⌈λ⋅32gceil homologically independent loops of length at most N(λ). Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.
problem Understanding the structure and properties of skein algebras of small surfaces.
method Constructed finite-dimensional representations at all roots of unity, using explicit formulas and analyzing reducibility.
result Azumaya loci of the surfaces contain the smooth loci of classical shadow varieties, with equality for the one-punctured torus and proper containment for the four-punctured sphere.
The study constructs and analyzes new symplectic 4-manifolds from hyperelliptic Lefschetz fibrations.
problem Understanding and constructing new symplectic 4-manifolds.
method Explicitly constructing Lefschetz pencils on hyperelliptic Lefschetz fibrations.
result Infinite families of symplectic 4-manifolds are diffeomorphic to fiber sums of standard hyperelliptic Lefschetz fibrations.
Teichmüller space and hyperelliptic surfaces parametrized by angles.
problem Parametrizing Teichmüller space and hyperelliptic surfaces using angles.
method Proved parametrization using 6g-5 and 4g-2 angle parameters for Teichmüller space and hyperelliptic surfaces respectively.
result Proved parametrization of Teichmüller space and hyperelliptic surfaces by angle parameters.
The paper studies maps on graphs and their fibers, focusing on hyperelliptic graphs.
problem Analyzing maps on graphs and their fibers.
method Analyzes maps Phi and Phi* on Culler--Vogtmann's outer space CV_n and its quotient T_n, focusing on hyperelliptic graphs.
result Fibers of Phi in T_n are aspherical and pi_1-injective subspaces.
Study identifies roots of hyperelliptic involutions and braid groups in mapping class groups.
problem Identifying roots of hyperelliptic involutions and braid groups in mapping class groups.
method Analyzes braid groups and mapping class groups on surfaces of genus nk. result Hyperelliptic involutions have infinitely many square and cubic roots.
We consider the hyperelliptic handlebody group on a closed surface of genus g. This is the subgroup of the mapping class group on a closed surface of genus g consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the …
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 introduces new indices for special fibered surfaces of genus 4.
problem Characterizing relatively minimal fibered surfaces of genus 4.
method Introduces Horikawa index and local signature for surfaces with unique trigonal structure.
result New indices provide insights into the structure of surfaces.
Paper computes rational cohomology of spin hyperelliptic mapping class groups.
problem Computing rational cohomology of spin hyperelliptic mapping class groups.
method Computes the G-invariant part of the rational cohomology of the pure braid group. result Includes rational cohomology of spin hyperelliptic mapping class groups of genus g. Study pseudo-Anosov homeomorphisms on translation surfaces to compute geodesic lengths.
problem Compute the geodesic lengths on hyperelliptic connected components of Teichmüller space.
method Developed a new framework to study pseudo-Anosov homeomorphisms on translation surfaces.
result Computed the systole of Teichmüller geodesic flow on hyperelliptic components.
For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Rie…
In this note, we describe a procedure to construct generalized complex structures with an arbitrarily large number of type change loci on products of the circle with a connected sum of closed 3-manifolds. The loci need not be isotopic.
We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.
Constructs correspondences on hyperelliptic surfaces combining orbifold groups and Blaschke products.
problem Combining geometric and dynamical properties on hyperelliptic surfaces.
method Analytic combinations on Riemann sphere, algebraic characterization, and Teichmüller spaces.
result Explicit description of correspondences and injection into Hurwitz spaces.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest arcs in sub-Riemannian metrics on Lie groups.
method Investigated left-invariant sub-Riemannian metrics on SU(1,1)imesR and SO0(2,1)imesR. result Found geodesics, shortest arcs, cut loci, and conjugate loci.
Let C be a hyperelliptic Riemann surface. We show that the hyperelliptic Weierstrass points of C are non-degenerated critical points of Morse index +2 of the curvature function K of the Theta metric on C (called also Bergman metric). When the genus of C is two, we compute all critical points of K and we show in this ca…
New examples of translation surfaces on hyperelliptic curves with many automorphisms.
problem Determining when translation surfaces are supported on the same algebraic curve.
method Analyzing eigenforms of automorphisms on hyperelliptic curves with many automorphisms.
result Presentation of infinitely many examples of translation surfaces on hyperelliptic curves.