A remark on the intersection of plane curves

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 \emph{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. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.
Comment: 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)