Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps
Marco Andreatta, Ciro Ciliberto, Roberto Pignatelli
TL;DR
The paper studies the birational geometry of the Hilbert scheme of length-2 subschemes of Hirzebruch surfaces through secant maps to the Grassmannian: for S_{a,b} of degree r, the secant map yields X_{a,b} ⊂ G(1,r+1). It proves a uniform degree formula $deg(X_{a,b})=3r^2-8r+6$ and shows X_{a,b}$ is smooth except at (a,b)=(1,r−1), where the unique singularity has tangent cone given by a cone over a Segre-section, with multiplicity $ inom{r}{2}$ . In the crucial r=4 case, X_{2,2}$ is a smooth Fano 4-fold of degree 22, while X_{1,3}$ has a singular point; nevertheless their smooth hyperplane sections are Fano 3-folds (the FlF) and every FlF arises as a hyperplane section of X_{1,3}$. The work also connects these geometric constructions to a degeneration framework from X_{2,2}$ to X_{1,3}$, and provides a GIT analysis for the stability of smooth hyperplane sections under the action of the automorphism group G of S_{1,3}, indicating potential moduli interpretations for Fano–like families.
Abstract
In this paper, extending some ideas of Fano, we study the birational geometry of the Hilbert scheme of 0-dimensional subschemes of length 2 of a rational normal scroll. 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., $S_{a,b}$ is the relative Proj of the vector bundle $O_{{\mathbb P}^1}(a)\oplus O_{{\mathbb P}^1}(b)$ embedded in ${\mathbb P}^{r+1}$ with its O(1) line bundle (from an abstract viewpoint $S_{a,b}\cong {\mathbb F}_{b-a}$), we consider the subvariety $X_{a,b}$ of the Grassmannian $G(1,r+1)$ 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, 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 $S_{1,3}.$