Newton-Okounkov bodies and Picard numbers on surfaces
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| Titel: | Newton-Okounkov bodies and Picard numbers on surfaces |
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| Autoren: | Julio-José Moyano-Fernández, Mathias Nickel, Joaquim Roé |
| Quelle: | Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona |
| Publication Status: | Preprint |
| Verlagsinformationen: | Universitat Autonoma de Barcelona, 2025. |
| Publikationsjahr: | 2025 |
| Schlagwörter: | Mathematics - Algebraic Geometry, Algebraic surface, Newton-Okounkov body, Picard number, FOS: Mathematics, 14C20, 14E15, 14C22, 13A18, Blowup, 0101 mathematics, Mathematics - Commutative Algebra, Commutative Algebra (math.AC), 01 natural sciences, Algebraic Geometry (math.AG), Valuation |
| Beschreibung: | 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. 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 |
| Publikationsart: | Article |
| Dateibeschreibung: | application/pdf |
| Sprache: | English |
| ISSN: | 0214-1493 |
| DOI: | 10.5565/publmat6912501 |
| DOI: | 10.48550/arxiv.2101.05338 |
| Zugangs-URL: | http://arxiv.org/abs/2101.05338 https://ddd.uab.cat/record/304450 |
| Rights: | arXiv Non-Exclusive Distribution URL: http://rightsstatements.org/vocab/InC/1.0/ |
| Dokumentencode: | edsair.doi.dedup.....a72dcfe3413beda1e9ce3f1a9a624ec0 |
| Datenbank: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | 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.<br />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 |
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| ISSN: | 02141493 |
| DOI: | 10.5565/publmat6912501 |