Non-reduced components of the Noether-Lefschetz locus

Abstract: For a fixed integer $d \ge 5$, the Noether-Lefschetz locus parametrizes smooth degree $d$ complex hypersurfaces in $\mathbb{P}<sup>3$ with Picard number greater than $1$. There are infinitely many irreducible components of this locus. The aim of this article is to understand the scheme structure of these components. In particular, we study when is such a component non-reduced. We give a criterion for such a component to be non-reduced. Finally, we give examples of such components.