Limits of pluri-tangent planes to quartic surfaces

TL;DR

This paper investigates the limits of Severi varieties of nodal plane sections on degenerating quartic K3 surfaces, with applications to counting nodal curves, irreducibility of Severi varieties, and monodromy of rational curves.

Contribution

It describes the limits of Severi varieties during degenerations of quartic K3 surfaces and applies these results to several enumerative and geometric problems.

Findings

Limits of Severi varieties are characterized for various degenerations.

Results imply irreducibility of Severi varieties of a general quartic surface.

Applications include counting nodal curves and understanding monodromy of rational curves.

Abstract

We describe, for various degenerations $S\to \Delta$ 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 $t\in \Delta^*$ tends to 0 of the Severi varieties $V_\delta(S_t)$, parametrizing irreducible $\delta$-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.