We prove a BGG-type correspondence describing coherent sheaves on complete intersections in toric varieties, and a similar assertion for the stable categories of related complete intersection singularities. 2000 Math. Subj. Class. Primary: 18E30; Secondary: 14F05, 14M25, 14M10.

In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) onto it, where M is the direct sum of powers of m. We compute a Gröbne...

We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Rep n A, of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is anothe...

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting ...

In this paper we examine the role of four Hilbert functions in the determination of the defining relations of the Rees algebra of almost complete intersections of finite colength. Because three of the corresponding modules are Artinian, some of these relationships are very effective, opening up tracks to the determination of the equations and also to processes of going from homologically define...

Let R be a commutative ring, (f) an ideal of R, and E = K(f ;R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T , the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.

In this paper, we develop the theory of Jacobian rings of open complete intersections, which mean a pair (X,Z) where X is a smooth complete intersection in the projective space and and Z is a simple normal crossing divisor in X whose irreducible components are smooth hypersurface sections on X . Our Jacobian rings give an algebraic description of the cohomology of the open complement X − Z and ...

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions. 0. Introduction. Let X denote a non-singular compact Kähler toric variety with the Picard number k. The variety X can be obtained by the sympl...

Following Newton, Ivory and Arnold, we study the Newtonian potentials of algebraic hypersurfaces in R. The ramification of (analytic continuations of) these potential depends on a monodromy group, which can be considered as a proper subgroup of the local monodromy group of a complete intersection (acting on a twisted vanishing homology group if n is odd). Studying this monodromy group we prove,...