On the Severi varieties of surfaces in P^3

Details On the Severi varieties of surfaces in P^3 On the Severi varieties of surfaces in P^3

1998-02-02

arxiv.org/abs/math/9802009v1

Mathematics

Algebraic Geometry

AMSTeX, AMSppt style, 14 pages

Scientific paper

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.

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