The Noether–Lefschetz locus of surfaces in P 3 ${\mathbb {P}}^3$ formed by determinantal surfaces

Abstract

We compute the dimension of certain components of the family of smooth determinantal degree $d$ surfaces in $P^3$, and show that each of them is the closure of a component of the Noether–Lefschetz locus $NL\left(d\right)$. Our computations exhibit that smooth determinantal surfaces in $P^3$ of degree 4 form a divisor in $|O_{P^3}\left(4\right)|$ with five irreducible components. We will compute the degrees of each of these components: $320,2508,136512,38475$, and $320112$.