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Register a new userNon-positive and negative at infinity divisorial valuations of Hirzebruch surfaces
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Carlos-Jes\\'us Moreno-\\'Avila
Publication date
2018
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and research's language is
English
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We consider rational surfaces $Z$ defined by divisorial valuations $ u$ of Hirzebruch surfaces. We introduce the concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when $ u$ is non-positive at infinity, the extremal rays of the cone of curves of $Z$ can be explicitly given.
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We consider flags $E_bullet={Xsupset Esupset {q}}$, where $E$ is an exceptional divisor defining a non-positive at infinity divisorial valuation $ u_E$ of a Hirzebruch surface $mathbb{F}_delta$ and $X$ the surface given by $ u_E,$ and determine an analogue of the Seshadri constant for pairs $( u_E,D)$, $D$ being a big divisor on $mathbb{F}_delta$. The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs $(E_bullet,D)$ as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.
Non-positive at infinity valuations are a class of real plane valuations which have a nice geometrical behavior. They are divided in three types. We study the dual graphs of non-positive at infinity valuations and give an algorithm for obtaining them. Moreover we compare these graphs attending the type of their corresponding valuation.
We study foliations $mathcal{F}$ on Hirzebruch surfaces $S_delta$ and prove that, similarly to those on the projective plane, any $mathcal{F}$ can be represented by a bi-homogeneous polynomial affine $1$-form. In case $mathcal{F}$ has isolated singularities, we show that, for $ delta=1 $, the singular scheme of $mathcal{F}$ does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For $delta eq 1$, we prove that the singular scheme of $mathcal{F}$ does not determine the foliation. However we prove that, in most cases, two foliations $mathcal{F}$ and $mathcal{F}$ given by sections $s$ and $s$ have the same singular scheme if and only if $s=Phi(s)$, for some global endomorphism $Phi $ of the tangent bundle of $S_delta$.
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