Related papers: Veech surfaces and complete periodicity in genus 2
We introduce a "tremor" deformation on strata of translation surfaces. Using it, we give new examples of behaviors of horocycle flow orbits in strata of translation surfaces. In the genus two stratum with two singular points, we find orbits…
In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p…
Infinitesimal bendings for classes of two-dimensional surfaces in $\mathbb{R}^3$ are investigated. The techniques used to construct the bending fields include reduction to solvability of Bers-Vekua type equations and systems of differential…
We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.
Meromorphic differentials on Riemann surfaces are said to be real-normalized if all their periods are real. Moduli spaces of real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles and residues…
We investigate complex surfaces that fiber over Teichm\"uller curves where the generic fiber is a Veech surface. When the fiber has genus one, these surfaces are elliptic fibrations; for higher genus fibers, they are typically minimal…
We show that every surface in H^hyp(4) is either a Veech surface or a generic surface, i.e. its GL^+(2,R)-orbit is either a closed or a dense subset of H^hyp(4) . The proof develops new techniques applicable in general to the problem of…
We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1,…
This is an expanded version of our work [AN88], 1988, in Russian. We classify del Pezzo surfaces over C with log terminal singularities of index \le 2. By classification, we understand a description of the intersection graph of all…
This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous…
A translation surface in the Heisenberg group is constructed as the product of two planar curves. We classify a type of such surfaces with vanishing intrinsic curvature by analyzing the determinant of their Gauss map
We study Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. There are exactly 9 types of such…
We study algebraic subvarieties of strata of differentials in genus zero satisfying algebraic relations among periods. The main results are Ax-Schanuel and Andr\'e-Oort-type theorems in genus zero. As a consequence, one obtains several…
We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic…
We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov…
We construct new constant mean curvature surfaces in H2xR. They arise as sister surfaces of Plateau solutions. It is a family of MC 1/2 surfaces with k ends, genus 1 and k-fold dihedral symmetry, k greater 2. The surfaces are Alexandrov-…
We give examples of isospectral non-isometric surfaces of genus 2 and 3 with variable curvatures and apply the result to construct isospectral potentials on Riemann surfaces of genus 2.
We describe an algorithm to decide whether two genus-two surfaces embedded in the 3-sphere are isotopic or not. The algorithm employs well-known techniques in 3-manifolds topology, as well as a new algorithmic solution to a problem on free…
We classify all rank two affine manifolds in strata in genus three with two zeros. This confirms a conjecture of Maryam Mirzakhani in these cases. Several technical results are proven for all strata in genus three, with the hope that they…
Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group…