arXiv subjects
Publications and source records attributed to Adam Van Tuyl.
Let I = p_1^{m_1} \cap ... \cap p_s^{m_s} be the defining ideal of a scheme of fat points in P^{n_1} x ... x P^{n_k} with support in generic position. When all the m_i's are 1, we explicitly calculate the Castelnuovo-Mumford regularity of I. In general, if at least one m_i >= 2, we give an upper bound for the regularity of I, which extends the result of Catalisano, Trung and Valla to the multi-projective case.
The Hilbert functions of sets of distinct points in P^n have been characterized. We show that if we restrict to sets of distinct of points in P^{n_1} x ... x P^{n_k} that are also arithmetically Cohen-Macaulay (ACM for short), then there is a natural generalization of this result. We begin by determining the possible values for the invariants K-dim R/Ix and depth R/Ix, where R/Ix is the coordinate ring associated to a set of distinct points X in P^{n_1} x ... x P^{n_k}. At the end of this paper we give a new characterization of ACM sets of points in P^1 x P^1.
We study the Hilbert functions of fat points in P^1 x P^1. If Z is an arbitrary fat point subscheme of P^1 x P^1, then it can be shown that for every i and j the values of the Hilbert function H_Z(l,j) and H_Z(i,l) eventually become constant for l >> 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in P^1 x P^1. This enables us to compute all but a finite number values of H_Z without using the coordinates of points. We also characterize the ACM fat points schemes using our description of the eventual behaviour. In fact, in the case that Z is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.
We describe the eventual behaviour of the Hilbert function of a set of distinct points in P^{n_1} x ... x P^{n_k}. As a consequence of this result, we show that the Hilbert function of a set of points in P^{n_1} x ... x P^{n_k} can be determined by computing the Hilbert function at only a finite number of values. Our result extends the result that the Hilbert function of a set of points in P^n stabilizes at the cardinality of the set of points. Motivated by our result, we introduce the notion of the_border_ of the Hilbert function of a set of points. By using the Gale-Ryser Theorem, a classical result about (0,1)-matrices, we characterize all the possible borders for the Hilbert function of a set of distinct points in P^1 x P^1.