JournalsrmiVol. 36, No. 7pp. 2147–2182

Newton–Okounkov bodies of exceptional curve valuations

  • Carlos Galindo

    Universitat Jaume I, Castellón de la Plana, Spain
  • Julio José Moyano-Fernández

    Universitat Jaume I, Castellón de la Plana, Spain
  • Francisco Monserrat

    Universitat Politècnica de València, Spain
  • Matthias Nickel

    Universität Frankfurt, Germany
Newton–Okounkov bodies of exceptional curve valuations cover
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Abstract

We prove that the Newton–Okounkov body associated to the flag E:={X=XrEr{q}}E_{\bullet}:= \{ X=X_r \supset E_r \supset \{q\} \}, defined by the surface XX and the exceptional divisor ErE_r given by any divisorial valuation of the complex projective plane P2\mathbb{P}^2, with respect to the pull-back of the line-bundle OP2(1)\mathcal{O}_{\mathbb{P}^2} (1) is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton–Okounkov bodies which turn out to be triangular.

Cite this article

Carlos Galindo, Julio José Moyano-Fernández, Francisco Monserrat, Matthias Nickel, Newton–Okounkov bodies of exceptional curve valuations. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2147–2182

DOI 10.4171/RMI/1195