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 P 3 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]. * This paper reports on work done during the Commutative Algebra Program, 2012-13, at MSRI. We are grateful to MSRI for providing such an exciting environment, where a chance meeting led to the progress described here. The first author is also grateful to the National Science Foundation for partial support during this period.