SOME RESULTS OF REGULARITY FOR SEVERI VARIETIES OF PROJECTIVE SURFACES

Abstract

For a linear system |C| on a smooth projective surface S, whose general member is a smooth, irreducible curve, the Severi variety V _|C|,δ is the locally closed subscheme of |C| which parametrizes curves with only δ nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper bounds on δ, ensuring that the corresponding Severi variety is smooth of codimension δ, Our result generalizes what is proven in [7] and [10]. We also consider examples of smooth Severi varieties on surfaces of general type in P ³ which contain a line.

The author is a member of GNSAGA-CNR.

Keywords:

• 1991 Mathematics Subject Classification: 14J29, 14H10

ACKNOWLEDGMENTS

This work is part of my Ph.D. thesis. I wish to thank many people who helped me during its preparation, especially L. Chiantini, D. Franco, M. Franciosi, A. F. Lopez, G. Pareschi, E. Sernesi and A. Verra. My special thanks to my Ph.D. advisor E. Sernesi for having introduced me in such an interesting research area.

Notes

The author is a member of GNSAGA-CNR.

^*We remark that, in the case of nodes, this condition is the same of [10]; moreover, their hypotheses (1.2) and (1.3) coincide in the case of nodes and become F(δ₀) < 0.