Newton-Okounkov bodies and Picard numbers on surfaces
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913652319191040 |
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| author | Moyano-Fernández, Julio José Nickel, Matthias Roé, Joaquim |
| author_facet | Moyano-Fernández, Julio José Nickel, Matthias Roé, Joaquim |
| contents | We study the shapes of all Newton-Okounkov bodies $Δ_{v}(D)$ of a given big divisor $D$ on a surface $S$ with respect to all rank 2 valuations $v$ of $K(S)$. We obtain upper bounds for, and in many cases we determine exactly, the possible numbers of vertices of the bodies $Δ_{v}(D)$. The upper bounds are expressed in terms of Picard numbers and they are birationally invariant, as they do not depend on the model $\tilde{S}$ where the valuation $v$ becomes a flag valuation. We also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor $D$ determines the Picard number of $S$, and prove that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_05338 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Newton-Okounkov bodies and Picard numbers on surfaces Moyano-Fernández, Julio José Nickel, Matthias Roé, Joaquim Algebraic Geometry Commutative Algebra 14C20, 14E15, 14C22, 13A18 We study the shapes of all Newton-Okounkov bodies $Δ_{v}(D)$ of a given big divisor $D$ on a surface $S$ with respect to all rank 2 valuations $v$ of $K(S)$. We obtain upper bounds for, and in many cases we determine exactly, the possible numbers of vertices of the bodies $Δ_{v}(D)$. The upper bounds are expressed in terms of Picard numbers and they are birationally invariant, as they do not depend on the model $\tilde{S}$ where the valuation $v$ becomes a flag valuation. We also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor $D$ determines the Picard number of $S$, and prove that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies. |
| title | Newton-Okounkov bodies and Picard numbers on surfaces |
| topic | Algebraic Geometry Commutative Algebra 14C20, 14E15, 14C22, 13A18 |
| url | https://arxiv.org/abs/2101.05338 |