Some results on fake quadrics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913746086002688 |
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| author | Yang, Jianqiang |
| author_facet | Yang, Jianqiang |
| contents | In this paper, we give a criterion to assess the effectiveness and ampleness of divisors on a fake quadric surface $S$, and then we establish a relationship between the cones: \[\mathring{\Eff}(S)=\Amp(S)\subset \SAmp(S)=\Mov(S) \subset \Nef(S)=\Eff(S)=\overline{\Amp(S)}. \]
In particular, we prove that any fake quadric of odd type does not contain a negative curve. This result is central to our manuscript.
As applications, first we give that any fake quadric is a fibration over $\mathbb P^1;$ Subsequently, we show that no fake quadric can be embedded in $\mathbb P^4$; Finally, we prove that the fake quadric $S$ possesses the bounded cohomology property. This property is characterized by the existence of a positive constant $c_{S}$ such that $$h^1(\CO_S(C))\leq c_S h^0(\CO_S(C))$$ for any curve $C \subset S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11175 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some results on fake quadrics Yang, Jianqiang Algebraic Geometry 14J29, 32J15 In this paper, we give a criterion to assess the effectiveness and ampleness of divisors on a fake quadric surface $S$, and then we establish a relationship between the cones: \[\mathring{\Eff}(S)=\Amp(S)\subset \SAmp(S)=\Mov(S) \subset \Nef(S)=\Eff(S)=\overline{\Amp(S)}. \] In particular, we prove that any fake quadric of odd type does not contain a negative curve. This result is central to our manuscript. As applications, first we give that any fake quadric is a fibration over $\mathbb P^1;$ Subsequently, we show that no fake quadric can be embedded in $\mathbb P^4$; Finally, we prove that the fake quadric $S$ possesses the bounded cohomology property. This property is characterized by the existence of a positive constant $c_{S}$ such that $$h^1(\CO_S(C))\leq c_S h^0(\CO_S(C))$$ for any curve $C \subset S$. |
| title | Some results on fake quadrics |
| topic | Algebraic Geometry 14J29, 32J15 |
| url | https://arxiv.org/abs/2307.11175 |