Algebraic Geometry

Let C be a smooth projective irreducible curve of genus g . And let G α (n,d,l) be the moduli space of α stable pairs of a vector bundle of $\rank n, °d$ and a subspace of H 0 (C,E) of dim=l . We find an explicit birational map from G α (n,d,n+1) to G α (1,d,n+1) for C general, 1/α≫0 and g≥ n 2 −1 . Because of this and other examples, we conjecture G α (a,d,a+z) maps birationally to G α (z,d,a+z) for 1/α≫0 and C general with g>2 .

This note outlines the Borel-Weil-Bott theory for (arbitrary-genus) flag varieties of loop groups, following ideas of G. Segal. Recent revivalist interest in the "WZW fusion rules" suggests that my original (CMP 1995) treatment of the matter is not all that well understood, so a brief account thereof has been recycled along with the more recent results. Proofs are only outlined, with full details given in the references.

We give an effective form of the theorem of Mazur-Kamienny-Merel on the torsion of elliptic curves over number fields.

In this paper we describe the complement of real line arrangements in the complex plane in terms of the boundary three-manifold of the line arrangement. We show that the boundary manifold of any line arrangement is a graph manifold with Seifert fibered vertex manifolds, and depends only on the incidence graph of the arrangement. When the line arrangement is defined over the real numbers, we show that the homotopy type of the complement is determined by the incidence graph together with orderings on the edges emanating from each vertex.

Arrondo, Sols and De Cataldo proved that there are only finitely many families of codimension two subvarieties not of general type in the smooth quadric of dimension n+2 for n≥2 , n≠4 . In this paper we drop the assumption n≠4 from the previous result (obviously the assumption n≥2 cannot be removed).

The Castelnuovo-Mumford regularity reg(X) of a projective scheme X was introduced by Mumford by generalizing ideas of Castelnuovo. The interest in this concept stems partly from the fact that X is m-regular if and only if for every p \geq 0 the minimal generators of the p-th syzygy module of the defining ideal I of X occur in degree \leq m + p. There are some bounds in the case that X is a locally Cohen-Macaulay scheme. The aim of this paper is to extend and improve these results for so-called (k,r)-Buchsbaum schemes. In order to prove our theorems, we need to apply a spectral sequence. We conclude by describing two sharp examples and open problems.

This paper will appear in the Santa Cruz proceedings. An overview of the braid group techniques in the theory of algebraic surfaces, from Zariski to the latest results, is presented. An outline of the Van Kampen algorithm for computing fundamental groups of complements of curves and the modification of Moishezon-Teicher regarding branch curves of generic projections are given. The paper also contains a description of a quotient of the braid group, namely B ~ n which plays an important role in the description of fundamental groups of complements of branch this http URL turns out that all such groups are ``almost solvable'' $\tildeB_n$-groups. Finally, possible applications to the study of moduli spaces of surfaces of general type are described and new examples of positive signature spin surfaces whose fundamental groups can be computed using the above algorithm (Galois cover of Hirzebruch surfaces) are presented.

This paper begins by introducing and characterizing Buchsbaum-Rim sheaves on $Z = \Proj R$ where R is a graded Gorenstein K-algebra. They are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free R-modules. Then we study multiple sections of a Buchsbaum-Rim sheaf $\cBf$, i.e, we consider morphisms $\psi: \cP \to \cBf$ of sheaves on Z dropping rank in the expected codimension, where $H^0_*(Z,\cP)$ is a free R-module. The main purpose of this paper is to study properties of schemes associated to the degeneracy locus S of ψ . It turns out that S is often not equidimensional. Let X denote the top-dimensional part of S . In this paper we measure the ``difference'' between X and S , compute their cohomology modules and describe ring-theoretic properties of their coordinate rings. Moreover, we produce graded free resolutions of X (and S ) which are in general minimal. Among the applications we show how one can embed a subscheme into an arithmetically Gorenstein subscheme of the same dimension and prove that zero-loci of sections of the dual of a null correlation bundle are arithmetically Buchsbaum.

In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of \KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality \KachruVafa\KachruSilverstein.

We consider the operation of intersecting with a locally principal Cartier divisor (i.e., a Cartier divisor which is principal on some neighborhood of its support). We describe this operation explicitly on the level of cycles and rational equivalences and as a corollary obtain a formula for rational equivalence between intersections of two locally principal Cartier divisors. Such canonical rational equivalence applies quite naturally to the setting of algebraic stacks. We present two applications: (i) a simplification of the development of Fulton-MacPherson-style intersection theory on Deligne-Mumford stacks, and (ii) invariance of a key rational equivalence under a certain group action (which is used in developing the theory of virtual fundamental classes via intrinsic normal cones).