Newton-Okounkov bodies and Picard numbers on surfaces
| Autor: | Moyano-Fernández, Julio José, Nickel, Matthias, Roé, Joaquim |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We study the shapes of all Newton-Okounkov bodies $\Delta_{v}(D)$ 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 $\Delta_{v}(D)$. 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. Comment: 25 pages. Revised version: the proof of Theorem 4.6 (Theorem C) has been rewritten to overcome a gap in (former) Lemma 4.4. Exposition has been improved throughout |
| Databáze: | arXiv |
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