Newton-Okounkov bodies and Picard numbers on surfaces

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Julio José Moyano Fernández
Universitat Jaume I. Departamento de Matem´aticas
https://orcid.org/0000-0002-9128-4488
Matthias Nickel
Johann Wolfgang Goethe-Universit¨at. Institut f¨ur Mathematik
Joaquim Roé Vellvé
Universitat Aut`onoma de Barcelona. Departament de Matem`atiques
https://orcid.org/0000-0003-0033-8442

We study the shapes of all Newton–Okounkov bodies ∆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 ∆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 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.

Paraules clau
valuation, blowup, Newton–Okounkov body, algebraic surface, Picard number

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Moyano Fernández, Julio José et al. «Newton-Okounkov bodies and Picard numbers on surfaces». Publicacions Matemàtiques, 2025, vol.VOL 69, núm. 1, p. 3-25, doi:10.5565/PUBLMAT6912501.
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(c) 2024

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