Twenty Points in P^3

TL;DR

This paper uses computational methods to classify Gorenstein linkages of point sets in P^3 over algebraically closed fields, resolving a long-standing problem for 20 points and establishing new bounds for glicci sets.

Contribution

It provides a complete classification of Gorenstein linkages for general points in P^3 and demonstrates the effectiveness of computer algebra in solving longstanding geometric problems.

Findings

All possibilities for Gorenstein linkages between general points in P^3 are determined.

A general set of up to 33 points, or 37-38 points, is glicci.

The case of 20 points, an open problem for over a decade, is resolved.

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 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.