代数几何
Given an invertible sheaf, does it come from a Cartier divisor? This might fail in presence of embedded components. I give some examples and characterize those invertible sheaves that allow a Cartier divisor.
We describe birational models and decide the rationality/unirationality of moduli spaces AA_{d} (and AA^{lev}_{d}) of (1,d)-polarized abelian surfaces (with canonical level structure, respectively) for small values of d. The projective…
We classify irreducible d-semistable degenerations of primary Kodaira surfaces. As an application we construct a canonical completion for the moduli space of primary Kodaira surfaces.
We prove an algebraic formula for the Euler characteristic of the Milnor fibres of functions with critical locus a smooth curve on a space which is a weighted homogeneous complete intersection with isolated singularity.
The main object of this note is to study the conormal module $M$ and the computation of the second symbolic power $\bar I^{(2)}$ of an ideal $\bar I$ in the residue ring $R/H$ of a polynomial ring $R$ over a field of characteristic zero.…
This paper is withdrawn by the authors.
The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k<x_1, ...,…
It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like `triangulated category', not to mention derived functors. I will give some motivation for such things from algebraic geometry,…
Talk given at Harvard, January 1999, published in the Proceedings of the Harvard Winter School on mirror symmetry, vector bundles and lagrangian cycles, 1999, International Press. Surveys the joint work [ST, KS] with Paul Seidel and Mikhail…
This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group…
We develop an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure…
We construct a functor from the derived category of homotopy Gerstenhaber algebras with finite-dimensional cohomology to the purely geometric category of so-called $F_{\infty}$-manifolds. The latter contains Frobenius manifolds as a…
We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for Kac-Moody groups. For a pair consisting of a surface and a curve on it, we consider a…
This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition…
It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero. In this paper we…
In his famous book, Felix Klein describes a complex variable for the quotients of the ordinary sphere by the finite groups of rotations and in particular for the most complex situation of the quotient by the symmetry group of the…
We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the…
We construct an action of the big cactus group (the fundamental group of the Deligne-Mumford compactification of the moduli space of real curves of genus zero with n undistinguished marked points) on Fock-Goncharov's SL_m analog of the…
We give two characterizations of hyperquadrics: one as non-degenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as $LQEL$-manifolds with large secant defects.
We define and study a class of codes obtained from scrolls over curves of any genus over finite fields. These codes generalize Goppa codes in a natural way, and the orthogonal complements of these codes belong to the same class. We show how…