Counting Singularities of Quadratic Forms on Vector Bundles

Abstract

The study of surfaces in ℙ3, with many nodes (= ordinary double points) is a beautiful classical topic, which recently found much attention again [3, 4]. All systematic ways to produce such surfaces seem related to symmetric matrices of homogeneous polynomials or, more generally, to quadratic forms on vector bundles: If the form q on the bundle E is generic, then q is of maximal rank on an open set. The rank of q is one less on the discriminant hypersurface det q = o, which represents the class 2c1. (E*). This hypersurface is nonsingular in codimension one, but has ordinary double points in codimension two exactly where rank q drops one more step.