On the Severi varieties of surfaces in P^3

TL;DR

This paper investigates the structure of Severi varieties of surfaces in P^3, establishing the existence of expected-dimensional components for general surfaces and providing examples of reducible Severi varieties.

Contribution

It proves the existence of reduced, expected-dimension components of Severi varieties for general surfaces and constructs examples of reducible Severi varieties on surfaces of degree greater than 7.

Findings

Existence of a unique reduced component of expected dimension for Severi varieties.

Construction of reducible Severi varieties on surfaces of degree >7.

Validation of enumerative geometry approaches for singular curves.

Abstract

The Severi variety V_{n,d} of a smooth projective surface S is defined as the subvariety of the linear system |O_S(n)|, which parametrizes curves with d nodes. We show that, for a general surface S of degree k in P^3 and for all n>k-1, d=0,...,dim(|O_S(n)|), there exists one component of V_{n,d} which is reduced, of the expected dimension dim(|O_S(n)|)-d. Components of the expected dimension are the easiest to handle, trying to settle an enumerative geometry for singular curves on surfaces. On the other hand, we also construct examples of reducible Severi varieties, on general surfaces of degree k>7 in P^3.