Algebraic Geometry

The aim of this note is to give a simple definition of genus zero virtual orientation classes (or fundamental classes) for projective complete intersections or, more generally, for complete intersections in convex varieties, and to prove a push forward formula for them.

We prove that a recent construction of a numerical Godeaux surface due to P. Craighero and R. Gattazzo is simply connected, and show how to realize their construction as a double plane. By proving that the surface contains no (-2)-curves, we obtain that this is the first example of a simply connected surface with vanishing geometric genus and ample canonical class.

This is the continuation of papers by Braun and Floystad, Cook, Braun and Cook. We use Generic Initial Ideal Theory in conjunction with Liaison Theory to further restrict the possible generic initial ideals of hyperplane sections of smooth surfaces not of general type in P4.

We continue the work of Braun and Floystad, and Cook bounding the degree of smooth surfaces in P4 not of general type using generic initial ideal theory.

We present bounds for the sparseness and for the degrees of the polynomials in the Nullstellensatz. Our bounds depend mainly on the unmixed volume of the input polynomial system. The degree bounds can substantially improve the known ones when this polynomial system is sparse, and they are, in the worst case, simply exponential in terms of the number of variables and the maximum degree of the input polynomials.

Let E be a vector bundle of rank r≥2 on a smooth projective curve C of genus g≥2 over an algebraically closed field K of arbitrary characteristic. For any integer with 1≤k≤r−1 we define {\se}_k(E):=k°E-r\max°F. where the maximum is taken over all subbundles F of rank k of E . The s k gives a stratification of the moduli space M(r,d) of stable vector bundles of rank r and degree on d on C into locally closed subsets ${\calM}(r,d,k,s)$ according to the value of s and k . There is a component M 0 (r,d,k,s) of M(r,d,k,s) distinguish by the fact that a general E∈ M 0 (r,d,k,s) admits a stable subbundle F such that E/F is also stable. We prove: {\it For g≥ r+1 2 and 0<s≤k(r−k)(g−1)+(r+1) , s≡kdmodr, M 0 (r,d,k,s) is non-empty,and its component M 0 (r,d,k,s) is of dimension} dim M 0 (r,d,k,s)={ ( r 2 + k 2 −rk)(g−1)+s−1 s<k(r−k)(g−1) if r 2 (g−1)+1 s≥k(r−k)(g−1)

On the basis of analysis on the adele ring of any algebraic numbers field (Tate's formula) a regularization for divergent adelic products of gamma- and beta-functions for all completions of this field are proposed, and corresponding regularized adelic formulas are obtained.

Let A be an arrangement of complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism from a finitely generated free group to the pure braid group. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.

There is a natural stratification of the character variety of a finitely presented group coming from the jumping loci of the first cohomology of one-dimensional representations. Equations defining the jumping loci can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Simpson, Arapura and others show that if Γ is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori. If follows that for Kähler groups the jumping loci must be defined by binomial ideals. We discuss properties of the jumping loci of general finitely presented groups and apply the ``binomial criterion" to obtain new obstructions for one-relator groups to be Kähler.

We propose a generalization of Artin's definition of algebraic stack, which we call {\em geometric n -stack}. The main observation is that there is an inductive structure to the definition whereby the ingredients for the definition of geometric n -stack involve only n−1 -stacks and so are already previously defined. We use this inductive structure to obtain some basic properties. We look at maps from a projective variety into certain such n -stacks, and obtain an interpretation of the Brill-Noether locus as the set of points of a geometric n -stack. At the end we explain how this provides a context for looking at de Rham theory for higher nonabelian cohomology, how one can define the Hodge filtration and so on.