A remark on the intersection of plane curves

Abstract

Let $D$ be a very general curve of degree $d=2\ell-ε$ in $\mathbb{P}^2$, with $ε\in \{0,1\}$. Let $Γ\subset \mathbb{P}^2$ be an integral curve of geometric genus $g$ and degree $m$, $Γ\neq D$, and let $ν: C\to Γ$ be the normalization. Let $δ$ be the degree of the \emph{reduction modulo 2} of the divisor $ν^*(D)$ of $C$. In this paper we prove the inequality $4g+δ\geqslant m(d-8+2ε)+5$. We compare this with similar inequalities due to Geng Xu and Xi Chen. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.

to appear in Cont. Math. ("Selim Krein Centennial"), pp. 1-19. Collaboration has benefitted by "MIUR Excellence Department Project" Dep. of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, grant 346300 for "IMPAN" from the "Simons Foundation"(code: BCSim-2018-s09), funds "Mission Sustainability 2017 - Fam Curves" CUP E81-18000100005 (Tor Vergata)