The minimal resolutions of double points in P1×P1P1×P1 with ACM support

Let ZZ be a finite set of double points in P1×P1P1×P1 and suppose further that XX, the support of ZZ, is arithmetically Cohen–Macaulay (ACM). We present an algorithm, which depends only upon a combinatorial description of XX, for the bigraded Betti numbers of IZIZ, the defining ideal of ZZ. We then relate the total Betti numbers of IZIZ to the shifts in the graded resolution, thus answering a special case of a question of Römer.