On a conjecture of Harris

Abstract

For $d\ge 4$, the Noether-Lefschetz locus ${\mathrm{NL}}_d$ parametrizes smooth, degree $d$ surfaces in ${\mathbb{P}}^3$ with Picard number at least $2$. A conjecture of Harris states that there are only finitely many irreducible components of the Noether-Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large $d$, but is true for $d\le 5$. She also showed that for $d=6,7$, there are finitely many \emph{reduced}, irreducible components of ${\mathrm{NL}}_d$ of non-maximal codimension. In this article, we prove that for any $d\ge 6$, there are infinitely many \emph{non-reduced} irreducible components of ${\mathrm{NL}}_d$ of non-maximal codimension.

Published in Communications in Contemporary Mathematics