A remark on the intersection of plane curves

TL;DR

This paper establishes a new inequality relating the genus, degree, and intersection properties of curves on a very general plane curve, comparing it with existing inequalities and discussing subvariety genera in hypersurfaces.

Contribution

It introduces a novel inequality involving the genus, degree, and reduction modulo 2 of curves intersecting a very general plane curve, expanding understanding of intersection theory.

Findings

Proves the inequality 4g + δ ≥ m(d - 8 + 2ε) for certain curves.

Provides comparisons with inequalities by Geng Xu and Xi Chen.

Includes a brief discussion on genera of subvarieties in projective hypersurfaces.

Abstract

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.