Limits of Pluri–Tangent Planes to Quartic Surfaces

Abstract

We describe, for various degenerations S → Δ of quartic K3 surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as t ∈ Δ ∗ tends to 0 of the Severi varieties V δ (S t ), parametrizing irreducible δ-nodal plane sections of S t . We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic K3 surfaces.