  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.
