Multigraded Regularity: Syzygies and Fat Points Jessica Sidman and Adam Van Tuyl

The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P n1 × · · · × P k . Introduction Let k be an algebraically closed field of characteristic zero. If M is a finitely generated graded module over a Z-graded polynomial ring over k, its CastelnuovoMumford regularity, denoted reg(M), is an invariant that measures the difficulty of computations involving M. Recently, several authors (cf. [1, 18, 19]) have proposed extensions of the notion of regularity to a multigraded context. Taking our cue from the study of the Hilbert functions of fat points in P (cf. [5, 11, 23]), we apply these new notions of multigraded regularity to study the coordinate ring of a scheme of fat points Z ⊆ P1 × · · · × Pk with the goal of understanding both the nature of regularity in a multigraded setting and what regularity may tell us about the coordinate ring of Z. This paper also complements the investigation in [16] of the Castelnuovo-Mumford regularity of fat points in multiprojective spaces. In the study of the coordinate ring of a fat point scheme in P many authors have found beautiful relationships between algebra, geometry, and combinatorics (cf. [17] for a survey when n = 2). Extensions and generalizations of such results to the multigraded setting are potentially of both theoretical and practical interest. Schemes of fat points in products of projective spaces arise in algebraic geometry in connection with secant varieties of Segre varieties (cf. [3, 4]). More generally, the base points of rational maps betweeen higher dimensional varieties may be non-reduced schemes of points, and in the case of maps between certain surfaces, the regularity of the ideals that arise may have implications for the implicitization problem in computer-aided design (cf. [8, 26]). It is well known that the Castelnuovo-Mumford regularity of a finitely generated Z-graded module M can be defined either in terms of degree bounds for the generators of the syzygy modules of M or in terms of the vanishing of graded pieces of local cohomology modules. (See [12].) Aramova, Crona, and De Negri define a 2000 Mathematics Subject Classification. 13D02, 13D40, 14F17.