A remark on the intersection of plane curves

C. Ciliberto, F. Flamini, M. Zaidenberg

Published 2017-04-02Version 1

Let $D$ be a very general curve of degree $d=2\ell-\epsilon$ in $\mathbb{P}^2$, with $\epsilon\in \{0,1\}$. Let $\Gamma \subset \mathbb{P}^2$ be an integral curve of geometric genus $g$ and degree $m$, $\Gamma \neq D$, and let $\nu: C\to \Gamma$ be the normalization. Let $\delta$ be the degree of the reduction modulo 2 of the divisor $\nu^*(D)$ of $C$. In this paper we prove the inequality $4g+\delta\geqslant m(d-8+2\epsilon)+5$. We compare this with similar inequalities due to Geng Xu and Xi Chen (cf. bibliography).