The characteristic numbers of cuspidal plane cubics in $\mathbb P^3$

Abstract

We obtain the characteristic numbers of the variety of non degenerate cuspidal plane cubics in $\mathbb P^3$, namely, the non-zero intersection numbers which arise from considering 10 (possibly repeated) conditions from among the following: $P$, that the cuspidal cubic go through a point; $\nu$, that the cuspidal cubic intersect a line; and $\rho$, that the cuspidal cubic be tangent to a plane. In order to reach this goal, we consider a suitable compactification of the variety of non degenerate cuspidal plane cubics in $\mathbb P^3$ and we calculate, using several degeneration formulae, some of its non-zero intersection numbers, including all the characteristic ones.