A rank-$2$ vector bundle on ${\mathbb P}^2\times {\mathbb P}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915253805121536 |
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| author | Ran, Ziv Rathmann, Jürgen |
| author_facet | Ran, Ziv Rathmann, Jürgen |
| contents | We construct a rank-$2$ indecomposable vector bundle on $\mathbb P^2\times\mathbb P^2$ in characteristic $2$ that
does not come from a bundle on $\mathbb P^2$ by factor projection nor from a bundle on $\mathbb P^{m} $
by central projection. We show that the zero-sets of a suitable twist of $E$ form
a family of nonclassical smooth Enriques surfaces of bidegree (4, 4) whose general member is
'singular' in the sense that Frobenius acts isomorphically on $H^1$, and there is a smooth divisor consisting of smooth supersingular surfaces (Frobenius acts as zero). Every nonclassical Enriques surface of bidegree (4, 4) in $\mathbb P^2\times\mathbb P^2$ that is bilinearly normal arises as a zero-set in this way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A rank-$2$ vector bundle on ${\mathbb P}^2\times {\mathbb P}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2 Ran, Ziv Rathmann, Jürgen Algebraic Geometry 14n05, 14j60 We construct a rank-$2$ indecomposable vector bundle on $\mathbb P^2\times\mathbb P^2$ in characteristic $2$ that does not come from a bundle on $\mathbb P^2$ by factor projection nor from a bundle on $\mathbb P^{m} $ by central projection. We show that the zero-sets of a suitable twist of $E$ form a family of nonclassical smooth Enriques surfaces of bidegree (4, 4) whose general member is 'singular' in the sense that Frobenius acts isomorphically on $H^1$, and there is a smooth divisor consisting of smooth supersingular surfaces (Frobenius acts as zero). Every nonclassical Enriques surface of bidegree (4, 4) in $\mathbb P^2\times\mathbb P^2$ that is bilinearly normal arises as a zero-set in this way. |
| title | A rank-$2$ vector bundle on ${\mathbb P}^2\times {\mathbb P}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2 |
| topic | Algebraic Geometry 14n05, 14j60 |
| url | https://arxiv.org/abs/2504.16174 |