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Let $n=\frac{r(r+1)}{2}$ or $n=r(r+1)$. We prove that the property of being extremal is preserved under residuality on the Hilbert scheme of $n$ points in the plane.

We classify ACM curves contained in a surface of degree d in $\mathbb{P}^{3}$ in terms of weak admissible pairs. In the case of a very general smooth determinantal quartic surface, we provide a geometric description of these curves and compute their Picard classes on the surface. Finally, we present a generalization to ACM closed subvarieties of codimension $1$ on a hypersurface in $\mathbb{P}^{n}$.

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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 $|\mathcal{O}_{\mathbb{P}^3}(4)|$ with 5 irreducible components. We will compute the degrees of each of these components: $320,2508,136512,38475$ and $320112$.

A general linear determinantal quartic in $\mathbb{P}^4$ is nodal, non-$\mathbb{Q}$-factorial and rational. We show that the family $\mathcal{F}$ of such quartics also contains rational $\mathbb{Q}$-factorial quartics, and that a generic member of $\mathcal{F}$ can specialize to a rational non-$\mathbb{Q}$-factorial double quadric. We describe the birational geometry of these three types of 3-folds, showing that it is governed by the extrinsic geometry of a curve $C\subset \mathbb{P}^3$.