Enumeration of $n$-fold tangent hyperplanes to a surface

TL;DR

This paper derives formulas for counting n-nodal curves in linear systems on surfaces, providing explicit counts for rational curves on K3 surfaces and singular plane quintic curves in threefolds.

Contribution

It introduces explicit formulas for enumerating n-nodal curves in n-dimensional linear systems on surfaces, including special cases like K3 surfaces and quintic threefolds.

Findings

Formulas for n-nodal curves for 1 ≤ n ≤ 6

Number of rational curves on K3 surfaces

Count of singular plane quintic curves in a threefold

Abstract

For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete system of curves of genus $n$ as well as the number {\bf17,601,000} of rational ({\em singular}) plane quintic curves in a generic quintic threefold.