The minimal resolutions of double points in P1×P1 with ACM support

Abstract

Let Z be a finite set of double points in P 1 × P 1 and suppose further that X , the support of Z , is arithmetically Cohen-Macaulay (ACM). We present an algorithm, which depends only upon a combinatorial description of X , for the bigraded Betti numbers of I Z , the defining ideal of Z . We then relate the total Betti numbers of I Z to the shifts in the graded resolution, thus answering a special case of a question of Römer.Given a set of fat points Z in P n , it has been the goal of many authors to describe the homological invariants encoded in the graded minimal free resolution of I Z , the defining ideal of Z . A non-exhaustive list of references includes [1,[5][6][7][8]16,21]. Many interesting questions about these numerical characters remain open; Harbourne's survey [15] on these problems in P 2 provides a good entry point to this material.Recently, many authors have extended this circle of problems to include fat points in multiprojective spaces. The Hilbert function [11][12][13] and the Castelnuovo-Mumford regularity [14,20] are two such topics that have been investigated. Besides their intrinsic interest, motivation to study such points arises from a paper of Catalisano, Geramita, and Gimigliano [3] which exhibited a connection between specific values of the Hilbert function of a set of fat points in a multiprojective space and the dimensions of certain secant varieties of the Segre varieties. We contribute to this ongoing research program by providing an algorithm to compute the bigraded minimal free resolution of the ideal of double points in P 1 × P 1 whose support is arithmetically Cohen-Macaulay. The N 2 -graded polynomial ring S = k[x 0 , x 1 , y 0 , y 1 ] with deg x i = (1, 0) and deg y i = (0, 1) is the coordinate ring of P 1 × P 1 . If P = R × Q ∈ P 1 × P 1 is a point in P 1 × P 1 , then the defining ideal of P is I P = (L R , L Q )

Symbolic powers of codimension two Cohen-Macaulay ideals

“…In the final section, we use the minimal graded free resolution given in Theorem 2.9 to verify that for small powers of codimension two perfect ideals that are locally complete intersections, a question of R€ omer [36] has an affirmative answer. In the case of ACM sets of points in P 1 Â P 1 , we also have a new proof of a result of Guardo and Van Tuyl [21].…”

“…In addition, we show how Theorem 1.1 significantly simplifies earlier arguments of Guardo and Van Tuyl [21] and Guardo, Harbourne, and Van Tuyl [18]. In fact, the original motivation of this project was to prove [18,Conjecture 4.1].…”

“…In this section we apply the main results of Section 2, namely Theorem 2.3, to ACM sets of points in P 1 Â P 1 : In particular, we show how our new results give new short proofs to results in [18,21].…”

“…R€ omer showed that Question 5.1 is true for all codimension two Cohen-Macaulay ideals, while Guardo and Van Tuyl [21] verified the question for any set of double points in P 1 Â P 1 whose support is ACM. We now use Theorem 2.9 to extend the family of known positive answers to Question 5.1.…”