The distribution of bidegrees of smooth surfaces in Gr(1, P3)

Abstract

The study of algebraic surfaces in Gr (1, p3), the Grassmann variety parametrising lines in projective three-space, was a popular one algebraic geometers of the late nineteenth and early twentieth centuries. Calling them line congruences, researchers such as Kummer, Fano, Roth and many others published many papers on the topic, classifying congruences and studying their invariants. Since that time, the field has lain dormant until very recently. The classical geometers identified two numbers associated with a given congruence: the order and the class. Thinking of a congruence as a two dimensional family of lines, the order is the number of lines in the family passing through a general point in p3, and the class the number of lines in the family contained in a general plane. Together, these two numbers make up the bidegree of the congruence. In modern terms, the bidegree gives the class of the congruence in the Chow ring of Gr (1, p3). In this paper, we consider the question: for what values of a and b does there exist (or not exist) a smooth congruence of bidegree (a, b)? In particular, we try to find restrictions on the bidegree, using an approach suggested by Dolgachev and Reider in [8]. This approach is to study the restriction of the universal bundle g of Gr (1, p3), which appears in the exact sequence