Nets of quadrics, and theta-characteristics of singular curves

Abstract

The major part of this paper is devoted to enumerating all the many types of nets of quadrics in C 4 . An introductory section puts the invariant theory in context, and gives a framework for the classification. A quadric x T (λA 0 +μA 1 + νA 2 )x of the net has dual equation X T adj (λA 0 +μA 1 + νA 2 ) X = 0. The adjugate system is the system of curves in the (λ,μ,ν) plane given by these equations. The set B of base points of this system on the curve 0 = Δ ≡ det (λA 0 +μA 1 + νA 2 ), together with the curve Δ , give system to the enumeration. In a final section of the paper the calculations are used to provide evidence for conjectures of the following type (generalizing results known when Δ = 0 is non-singular): each net determines and is determined by a square root of the canonical bundle on the curve r obtained from Δ by blowing B up; the set of square roots is an affine space over F 2 , and those arising are the zeros of a certain quadratic map.