On the Severi problem in arbitrary characteristic

Abstract

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi's original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.

36 pages, 9 figures. v2: Minor changes throughout the paper. v3: Added two references. Final version, to appear in Publ. Math. IHÉS

References in corpus (4)

• On the structure of nonarchimedean analytic curves
• Brief introduction to tropical geometry
• Monodromy of rational curves on toric surfaces
• Degeneration of curves on some polarized toric surfaces

Cited by in corpus (3)

• Degeneration of curves on some polarized toric surfaces
• A note on the Severi problem for toric surfaces
• Severi dimensions for unicuspidal curves