The finiteness of smooth curves of degree $\le 11$ and genus $\le 3$ on a general complete intersection of a quadric and a quartic in $\mathbb{P}^5$

Abstract

Let $W\subset \mathbb{P}^5$ be a general complete intersection of a quadric hypersurface and a quartic hypersurface. In this paper we prove that $W$ contains only finitely many smooth curves $C\subset \mathbb{P}^5$ such that $d:= \deg ({C}) \le 11$, $g:= p_a({C}) \le 3$ and $h^1(\mathcal{O} _C(1)) =0$.