Twenty Points in P^3
  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. 
David Eisenbud, Robin Hartshorne, Frank-Olaf Schreyer
math.AG math.AC