Limits of pluritangent planes to quartic surfaces,

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 \begin{inparaenum}[(i)] \item the counting of nodal plane curves through base points in special position, \item the irreducibility of Severi varieties of a general quartic surface, and \item the monodromy of the universal family of rational curves on quartic $K3$ surfaces.