Global properties of families of plane curves

Abstract

We describe degenerations of projective plane curves to curves containing a fixed line $l$ as a component, and show that $H^1({\overline V}_{n,d,m}, {\Cal O} (r))=0, r \in{\Bbb Z}$, where $V_{n,d,m}\subset {\Bbb P}^N (N = n(n+3)/2)$ is the subscheme consisting of irreducible plane curves having smooth contact of order at least m with $l$ at a fixed point p of $l$ and d nodes and no other singularities. Note: the notion of admissible schemes of plane curves, introduced for the proof of the vanishing theorem, allows us to give a recipe for calculating the Hilbert polynomial of ${\overline V}_{n,d}$ (see Sect. 4), in particular the quantum cohomology of the plane.