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

Abstract

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 $1\le a\le b$ 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 $S_{1,3}$.