Related papers: Curve obstruction for autonomous diffeomorphisms o…

Let $\Sigma_g$ be a closed hyperbolic surface of genus $g$ and let $Ham(\Sigma_g)$ be the group of Hamiltonian diffeomorphisms of $\Sigma_g$. The most natural word metric on this group is the autonomous metric. It has many interesting…

Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved…

We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean self-intersection. As a consequence, we show that the number of intersections of such holomorphic curves with the…

The fine curve graph of a surface is the graph whose vertices are simple closed essential curves in the surface and whose edges connect disjoint curves. In this paper, we prove that the automorphism group of the fine curve graph of a…

Let $\Sigma$ be a surface equipped with an area form. There is an long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the set of…

Let $N$ be a connected nonorientable surface of genus $g$ with $n$ punctures. Suppose that $g$ is odd and $g+n \geqslant 6$. We prove that the automorphism group of the complex of curves of $N$ is isomorphic to the mapping class group…

This paper studies the Hilbert scheme of a curve on a complete-intersection K-trivial threefold, in the case in which the curve is unobstructed in the ambient variety in which the threefold lives. The basic result is that the obstruction…

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in…

Let $\Sigma$ be a compact surface equipped with an area form. There is an long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the…

We consider a closed negatively curved surface $(M, g)$ with marked length spectrum sufficiently close (multiplicatively) to that of a hyperbolic metric $g_0$ on $M$. We show there is a smooth diffeomorphism $F:M \to M$ with derivative…

The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic…

We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of…

The group of automorphisms of the free group on two generators is known to act geometrically, in an essentially unique way, on a 2-dimensional CAT(0) space X. We prove that X contains precisely two Hamiltonian surfaces. By this we mean a…

Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite…

The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under…

We are interested in obstructions to the FIRST order deformation of a pair of a smooth hypersurface $f_0$ and a smooth curve $C_0$ contained in $f_0$. In the first half of the paper, we give necessary conditions for the pair to deform in…

We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with…

A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube graph $Q_n$ admits such an embedding on an orientable surface when $n$ is a power of 2.…

This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…

Let $f: X \to Y$ be a regular covering of a surface $Y$ of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$ to…