Irreducibility of Severi varieties on K3 surfaces
Abstract
Let (S,L) be a general primitively polarized K3 surface of genus g. For every 0≤ δ ≤ g we consider the Severi variety parametrizing integral curves in |L| with exactly δ nodes as singularities. We prove that its closure in |L| is connected as soon as δ≤ g-1. If δ≤ g-4, we obtain the stronger result that the Severi variety is irreducible, as predicted by a well-known conjecture. The results are obtained by degeneration to Halphen surfaces.
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