Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps

In this paper, extending some ideas of Fano in [Fano, 1949] and of the first and last authors in [Andreatta–Pignatelli, 2023], we study the birational geometry of the Hilbert scheme of 0-dimensional subschemes of length 2 of a rational normal scroll \mathbb{F}_{n} . This fourfold has three elementary contractions associated to the three faces of its nef cone. We study natural projective realizations of these contractions. In particular, given a smooth rational normal scroll S_{a,b} of degree r in \mathbb{P}^{r+1} with 1\leq a\leq b and a+b=r , i.e., S_{a,b}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(a)\oplus \mathcal{O}_{\mathbb{P}^1}(b)) embedded in \mathbb{P}^{r+1} with its \mathcal{O}(1) line bundle (from an abstract viewpoint S_{a,b}\cong \mathbb{F}_{b-a} ), we consider the variety X_{a,b}\subset \mathbb{G}(1,r+1) described by all lines that are secant or tangent to S_{a,b} . The variety X_{a,b} is the image of some of the aforementioned contractions; it is smooth if a>1 , and it is singular at a unique point if a=1 . We compute the degree of X_{a,b} and the local structure of the singularity of X_{a,b} when a=1 . Finally, we discuss in some detail the case r=4 , originally considered by Fano in [Fano, 1949], because the smooth hyperplane sections of X_{2,2} and X_{1,3} are the Fano 3-folds that appear as number 16 in the Mori–Mukai list of Fano 3-folds with Picard number 2. We prove that any smooth hyperplane section of X_{2,2} is also a hyperplane section of X_{1,3} , and we discuss the GIT-stability of the smooth hyperplane sections of X_{1,3} , where G is the subgroup of the projective automorphisms of X_{1,3} coming from the ones of S_{1,3} .