Blowups, Gale duality, and moduli spaces
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911720748875776 |
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| author | Araujo, Carolina Castravet, Ana-Maria Kaur, Inder Martinelli, Diletta |
| author_facet | Araujo, Carolina Castravet, Ana-Maria Kaur, Inder Martinelli, Diletta |
| contents | The goal of this paper is to describe the birational geometry of the blowup of $\mathbb{P}^n$ at $n+4$ points in very general position. To achieve this, we follow an idea of Mukai and explore a special instance of Gale duality, namely, a correspondence between configurations of $n+4$ points in the projective spaces $\mathbb{P}^n$ and $\mathbb{P}^2$. We first prove that the blowup $X$ of $\mathbb{P}^n$ at $n+4$ general points is isomorphic to a certain Gieseker moduli space of rank $2$ vector bundles on the surface $S$ obtained by blowing up $\mathbb{P}^2$ at the $n+4$ Gale dual points. We then study the variation of these moduli spaces as we vary the polarization $L$ on $S$, and translate this variation into a partial Mori chamber decomposition of $\overline{Eff}(X)$, describing to some extent the birational geometry of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27152 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blowups, Gale duality, and moduli spaces Araujo, Carolina Castravet, Ana-Maria Kaur, Inder Martinelli, Diletta Algebraic Geometry The goal of this paper is to describe the birational geometry of the blowup of $\mathbb{P}^n$ at $n+4$ points in very general position. To achieve this, we follow an idea of Mukai and explore a special instance of Gale duality, namely, a correspondence between configurations of $n+4$ points in the projective spaces $\mathbb{P}^n$ and $\mathbb{P}^2$. We first prove that the blowup $X$ of $\mathbb{P}^n$ at $n+4$ general points is isomorphic to a certain Gieseker moduli space of rank $2$ vector bundles on the surface $S$ obtained by blowing up $\mathbb{P}^2$ at the $n+4$ Gale dual points. We then study the variation of these moduli spaces as we vary the polarization $L$ on $S$, and translate this variation into a partial Mori chamber decomposition of $\overline{Eff}(X)$, describing to some extent the birational geometry of $X$. |
| title | Blowups, Gale duality, and moduli spaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2605.27152 |