相关论文: Riemann surfaces, ribbon graphs and combinatorial …

We show how the Abel-Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel-Jacobi map is just the St\"{a}ckel matrix,…

This expository article is an introduction to logarithmic Gromov--Witten (GW) theory. We discuss how to study the GW theory of a smooth projective variety via simple normal crossings degenerations. We survey several approaches to…

The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold…

We exploit the properties of the three-dimensional hyperbolic space to discuss a simplicial setting for open/closed string duality based on (random) Regge triangulations decorated with null twistorial fields. We explicitly show that the…

In this paper, we study the birational geometry of the Hilbert scheme of n points on P^2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the…

Stable Hamiltonian structures generalize contact forms and define a volume-preserving vector field known as the Reeb vector field. We study two aspects of Reeb vector fields defined by stable Hamiltonian structures on 3-manifolds: on one…

Topology of the Generic Hamiltonian Dynamical Systems on the Riemann Surfaces given by the real part of the generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB). This…

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…

The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points.…

These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy,…

We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by…

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane…

We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…

We describe typical degenerations of quadratic differentials thus describing ``generic cusps'' of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not…

The modular curves serve as excellent objects for testing conjectures in arithmetic geometry. They possess a natural geometric definition in contrast with rather nontrivial structure. On the other hand, they are well-studied from the…

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology…

Using the orbicell decomposition of the Deligne-Mumford compactification of the moduli space of Riemann surfaces studied previously, a chain complex based on semistable ribbon graphs is constructed which is an extension of Konsevich's graph…

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…

We begin with a comprehensive discussion of the punctual Hilbert scheme of the regular two-dimensional local ring in terms of the Gr\"obner cells. These schemes are the most degenerate fibers of the Grothendieck-Deligne norm map (the…

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm\"uller space…