We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the original graph is c...

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...

In recent years, two different multigraded variants of Castelnuovo-Mumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local cohomology modules, and the resolution regularity vector, defined by the multi-degrees in a minimal free resolution. In this paper, we study the relationship between multigraded regularity and the resolu...

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...

This article comes from our quest for bounds on the Castelnuovo-Mumford regularity of schemes in terms of their “defining equations”, in the spirit of [BM], [BEL], [CP] or [CU]. The references [BS], [BM], [V] or [C] explains how this notion of regularity is a mesure of the algebraic complexity of the scheme, and provides several computational motivations. It was already remarked by several auth...