Algebraic Geometry

In this revised form, the proof of the principal lemma has been simplified and the main theorem has been extended to all characteristics for those varieties which are smooth in codimension one. This principal theorem essentially says the following : given an ample line bundle O(1) on a projective variety X and a fixed upper bound M on the multiplicities, there exists a lower bound D such that any generic union of multiple points of multiplicity at most M imposes independent conditions on the sections of O(d) for d>D. Here a multiple point is the closed subscheme defined by a power m of the ideal of a smooth point in X and m is its multiplicity.

We study the nonnegative part B_{\ge 0} of the flag variety of a reductive algebraic group G, as defined by Lusztig. Using positivity properties of the canonical basis it is shown that B_{\ge 0} has an algebraic cell decomposition indexed by pairs w\le w' of the Weyl group. This result was conjectired by Lusztig in [Lu; Progress in Math 123].

In this article we give an algorithm for computing the integral closure of a reduced Noetherian ring R, in case this integral closure is finitely generated over R.

Let M(v) be the moduli of stable sheaves on K3 surfaces X of Mukai vector v. If v is primitive, than it is expected that M(v) is deformation equivalent to some Hilbert scheme and weight two hogde structure can be described by H^*(X,Z). These are known by Mukai, O'Grady and Huybrechts if rank is 1 or 2, or the first Chern class is primitive. Under some conditions on the dimension of M(v), we shall show that these assertion are true. For the proof, we shall use Huybrechts's results on symplectic manifolds.

In this note we show the existence of a family of elliptic conic bundles in P^4 of degree 8. This family has been overlooked and in fact falsely ruled out in a series of classification papers. Our surfaces provide a counterexample to a conjecture of Ellingsrud and Peskine. According to this conjecture there should be no irregular m-ruled surface in P^4 for m at least 2.

Let S be a smooth projective surface, and consider the following two subvarieties of the Hilbert scheme parameterizing closed subschemes of S of length n: A = {subschemes with support in a fixed point of S} B = {subschemes with support in one (variable) point of S} A and B have complementary dimensions in the Hilbert scheme. We prove that the intersection number [A].[B] = n(-1)^(n-1), answering a question by H. Nakajima.

In this paper we construct a projective action of certain arithmetic group on the derived category of coherent sheaves on an abelian scheme A , which is analogous to Weil representation of the symplectic group. More precisely, the arithmetic group in question is a congruence subgroup in the group of "symplectic" automorphisms of A× A ^ where A ^ is the dual abelian scheme. The "projectivity" of this action refers to shifts in the derived category and tensorings with line bundles pulled from the base. In particular, if A is an abelian scheme over S equipped with an ample line bundle L of degree 1 then we construct an action of a central extension of S p 2n (Z) by Z×Pic(S) on the derived category of coherent sheaves on A n (the n -th fibered power of A over S ). We describe the corresponding central extension explicitly using the the canonical torsion line bundle on S associated with L . As a main technical result we prove the existence of a representation of rank d for a symmetric finite Heisenberg group scheme of odd order d 2 .

Every smooth minimal complex algebraic surface of general type, X , may be mapped into a moduli space, $\MM_{c_1^2(X), c_2(X)}$, of minimal surfaces of general type, all of which have the same Chern numbers. Using the braid group and braid monodromy,we construct infinitely many new examples of pairs of minimal surfaces of general type which have the same Chern numbers and non-isomorphic fundamental groups. Unlike previous examples, our results include X for which | π 1 (X)| is arbitrarily large. Moreover, the surfaces are of positive signature. This supports our goal of using the braid group and fundamental groupsto decompose $\MM_{c_1^2(X),c_2(X)}$ into connected components.

We show that through a point of an affine variety there always exists a smooth plane curve inside the ambient affine space, which has the multiplicity of intersection with the variety at least 3. This result has an application to the study of affine schemes.

Algebraic curve branches can be classified according to their multiplicity sequences. Arf's solution to this problem using Arf closures and possible implementations of Henselization are discussed.