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

Let Z be a finite set of double points in P1 × P1 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 IZ, the defining ideal of Z. We then relate the total Betti numbers of IZ to the shifts in the graded resolution, thus answering a special case of a question of Römer. © 2007 Elsevier Ltd. All rights reserved.