Abstract

Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in ℙ3 over an algebraically closed field of characteristic 0. As a consequence, we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d ≤ 33 or d = 37, 38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years [8].

For Rob Lazarsfeld on the occasion of his 60th birthday

1 Introduction

The theory of liaison (linkage) is a powerful tool in the theory of curves in ℙ3 with applications, for example, to the question of the unirationality of the moduli spaces of curves (e.g., [3, 26, 29]). One says that two curves C, D ⊂ ℙ3 (say, reduced and without common components) are directly linked if their union is a complete intersection, and evenly linked if there is a chain of curves C = C0, C1, …, C2m = D such that Ci is directly linked to Ci+1 for all i. The first step in the theory is the result of Gaeta that any two arithmetically Cohen-Macaulay curves are evenly linked, and in particular are in the linkage class of a complete intersection, usually written licci. Much later Rao [23] showed that even linkage classes are in bijection with graded modules of finite length up to shift, leading to an avalanche of results (reported, e.g., in [19, 20]). However, in codimension > 2 linkage yields an equivalence relation that seems to be very fine, and thus not so useful; for example, the scheme consisting of the four coordinate points in ℙ3 is not licci.

A fundamental paper of Peskine and Szpiro [22] laid the modern foundation for the theory of linkage.