Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps
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arXiv
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| Natura: | Preprint |
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2025
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| author | Andreatta, Marco Ciliberto, Ciro Pignatelli, Roberto |
| author_facet | Andreatta, Marco Ciliberto, Ciro Pignatelli, Roberto |
| contents | 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 $S_{1,3}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02920 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps Andreatta, Marco Ciliberto, Ciro Pignatelli, Roberto Algebraic Geometry Primary 14N05, 14N10, Secondary 14B05 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 $S_{1,3}.$ |
| title | Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps |
| topic | Algebraic Geometry Primary 14N05, 14N10, Secondary 14B05 |
| url | https://arxiv.org/abs/2501.02920 |