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

We compute the dimension of certain components of the family of smooth determinantal degree $d$ surfaces in ${\mathbb{P}}^3$, and show that each of them is the closure of a component of the Noether-Lefschetz locus $NL(d)$. Our computations exhibit that smooth determinantal surfaces in ${\mathbb{P}}^3$ of degree 4 form a divisor in $\mathrm{\mid }{\mathcal{O}}_{{\mathbb{P}}^3}(4)\mathrm{\mid }$ with 5 irreducible components. We will compute the degrees of each of these components: $320,2508,136512,38475$ and $320112$.