Counting singular curves with prescribed tangency

Abstract

We obtain an explicit formula for the characteristic number of degree d curves in P2 with prescribed singularities (of type Ak) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. We are not aware of any explicit formula to enumerate plane curves of degree d with any number of Ak singularities (beyond codimension 8); however, combined with the results of S. Basu and R. Mukherjee ([1], [2], and [3]), this gives us a complete formula for the characteristic number of curves with δ-nodes and one singularity of type Ak, tangent to a given line, provided δ+k≤8. We use a topological method to compute the degenerate contribution to the Euler class. We have made several low degree checks to verify special cases of our result. When the singularities are only nodes, we have verified that our numbers are logically consistent with those computed by L. Caporaso and J. Harris ([7]). We also verify that our answers for the characteristic number of cubics with a cusp tangent to a given line and the characteristic number of quartics with two nodes and a cusp, tangent to a given line is logically consistent with the characteristic number of rational cubics and quartics tangent to a given line that was computed by L. Ernström and G. Kennedy ([9]).