Jürgen Stückrad

Towards a structure theory for projective varieties of degree = codimension + 2

Abstract The problem under consideration in this paper is that of finding a structure theory for varieties X of P n k ( k is an algebraically closed field of arbitrary characteristic) with degree ( X ) = codimension( X ) + 2. Takao Fujita has a satisfactory classification theory for projective varieties of Δ-genus zero and one. In either case the singularities of X turn out to be of very special type. Our approach also sheds some light on the structure of these singularities.

On endomorphism rings of local cohomology modules

Let R be a local complete ring. For an R-module M the canonical ring map R → End R (M) is in general neither injective nor surjective; we show that it is bijective for every local cohomology module M:= H h I (R) if H l I (R) = 0 for every I ≠ h (= height(I)) (I an ideal of R); furthermore the same holds for the Matlis dual of such a module. As an application we prove new criteria for an ideal to be a set-theoretic complete intersection.

Matlis duals of top Local cohomology modules

In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of H d-1 J (R), where R is a d-dimensional local ring and J ⊆ R an ideal such that dim(R/J) = 1 and H d J (R) = 0.

Monomial conjecture and complete intersections

The Monomial Conjecture is an assertion about all systems of parameters in an arbitrary noetherian local ringA. We reduce this to a statement about a closely related complete intersection ringR of the same dimension. This statement again involves parameters, but in an ostensibly different way, raising new questions.

Castelnuovo–Mumford regularity of simplicial toric rings

Abstract Bounds for the Castelnuovo–Mumford regularity of simplicial toric rings are given which are close to the bound stated in Eisenbud–Gotos Conjecture.

Quasi-complete intersection ideals of height 2

Abstract The paper examines some relationships between minimal generating sets of prime ideals of height 2, which are quasi-complete intersections. By characterizing the ideals p(n1, n2, n3) of monominal curves in P k3 with μ(p(n1, n2, n3)) = 4, which are quasi-complete intersections, it is shown that our results are best possible.

Reduction of everywhere convergent power series with respect to Gröbner bases

Abstract We introduce a notion of Grobner reduction of everywhere convergent power series over the real or complex numbers with respect to ideals generated by polynomials and an admissible term ordering. The presented theory is situated somewhere between the known theories for polynomials and formal power series. Our main theorem states the existence of a formula for the division of everywhere convergent power series over the real or complex numbers by a finite set of polynomials. If the set of polynomials is a Grobner basis then the remainder of that division depends only on the equivalence class of the power series modulo the ideal generated by the polynomials. When the power series which shall be divided is a polynomial the division formula leads to a usual Grobner representation well known from polynomial rings. Finally, the results are applied to prove the closedness of ideals generated by polynomials in the ring of everywhere convergent power series and to give a very simple proof of the affine version of Serres graph theorem.

Quasi-complete intersections of monomial curves in projective three-space

We classify completely all quasi-complete intersections of monomial curves in K an infinite field, see Theorem 4.1 and Theorem 4.2. This completes the investigations started in [4].

Intersections of sequences of ideals generated by polynomials

Abstract We present a method for determining the reduced Grobner basis with respect to a given admissible term order of order type ω of the intersection ideal of an infinite sequence of polynomial ideals. As an application we discuss the Lagrange type interpolation on algebraic sets and the “approximation” of the ideal I of an algebraic set by zero dimensional ideals, whose affine Hilbert functions converge towards the affine Hilbert function of I .