The minimal resolutions of double points in [formula omitted] 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.