Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X . Given globally generated line bundles B1, . . . , Bl on X and m1, . . . , ml ∈ N, consider the line bundle L := B m1 1 ⊗ · · · ⊗ Bl l . We give conditions on the mi which guarantee that the ideal of X in P(H (X, L)∗) is generated by quadrics and the first p syzygies are line...

We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo–Mumford regularity. The f...

For a smooth projective variety X ⊂ P embedded by the complete linear system, Property Np has been studied for a long time([5], [11], [12], [10] etc). On the other hand, Castelnuovo-Mumford regularity conjecture and related problems have been focused for a projective variety which is not necessarily linearly normal([2], [13], [15], [17], [20] etc). This paper aims to explain the influence of Pr...

This paper contributes to the solution of the Poincaré problem, which is to bound the degree of a (generalized algebraic) leaf of a (singular algebraic) foliation of the complex projective plane. The first theorem gives a new sort of bound, which involves the Castelnuovo–Mumford regularity of the singular locus of the leaf. The second theorem gives a bound in terms of two singularity numbers of...

The problem of computing a minimal set of generators for the ideal of a rational parametric projective curve has been tackled by elimination theory and can be solved by the computation of Gröbner bases (see, for example Kalkbrener, 1991; Licciardi and Mora, 1994; Gao and Chou, 1992; Fix et al., 1996, and references therein). These methods have been implemented in CoCoA 3.6 (Capani et al., 1995)...

Introduction 2 1. A Guide to Duality 3 1.1. Local Duality 3 1.2. Dualizing Complexes and Some Vanishing Theorems 10 1.3. Cohomological Annihilators 17 2. A Few Applications of Local Cohomology 21 2.1. On Ideal Topologies 21 2.2. On Ideal Transforms 25 2.3. Asymptotic Prime Divisors 28 2.4. The Lichtenbaum-Hartshorne Vanishing Theorem 35 2.5. Connectedness results 37 3. Local Cohomology and Syzy...

Let I denote an R+-primary homogeneous ideal in a normal standard-graded Cohen-Macaulay domain over a field of positive characteristic p. We give a linear degree bound for the Frobenius powers I [q] of I, q = p, in terms of the minimal slope of the top-dimensional syzygy bundle on the projective variety ProjR. This provides an inclusion bound for tight closure. In the same manner we give a line...

For a rooted tree Γ, we consider path ideals of Γ, which are ideals that are generated by all directed paths of a fixed length in Γ. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of Γ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as p...