Limits of pluri-tangent planes to quartic surfaces

Abstract

We describe, for various degenerations $S\to Δ$ 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 $t\in Δ^*$ 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.

v3: a few minor changes, final version. Warning: numbering is different in the published version. Algebraic and Complex Geometry, A. Frühbis-Krüger, R. Kloosterman, M. Schütt editors, Springer Proceedings in Mathematics & Statistics 71, Springer International Publishing Switzerland 2014