A compactification of the space of twisted cubics

We give an elementary, explicit smooth compactification of a parameter space for the family of twisted cubics. The construction also applies to the family of subschemes defined by determinantal nets of quadrics, e.g., cubic ruled surfaces in $\boldsymbol P^4$, Segre varieties in $\boldsymbol P^5$. It is suitable for applications of Bott's formula to a few enumerative problems.

Nets of quadrics and deformations of Σ3〈3〉 singularities

The 2-jet of a Σ3 map-germ f:(3, 0) → (3, 0) determines a net of quadratic maps from 3 to 3; for nets of general type this jet is sufficient for equivalence. The classification of such nets involves a single parameter c. It is shown in [7], also in [3], that the versai unfolding of f is topologically trivial over the parameter space. However, there are 4 connected components of this space of nets. The main object of this paper is to show that the corresponding unfolded maps are of different topological types.

Nets of quadrics, and theta-characteristics of singular curves

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.