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Register a new userDiscrete equivalence of non-positive at infinity plane valuations
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Carlos-Jes\\'us Moreno-\\'Avila
Publication date
2019
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and research's language is
English
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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.
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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.
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.
We study the normal map for plane projective curves, i.e., the map associating to every regular point of the curve the normal line at the point in the dual space. We first observe that the normal map is always birational and then we use this fact to show that for smooth curves of degree higher than four the normal map uniquely determines the curve. Our proof works in characteristic zero and in positive characteristic higher than the degree of the curve. We notice also that in high characteristic strange curves provide examples of different plane curves with same curve of normal lines. We will reinterpret our results also in the modern terminology of bottlenecks of algebraic curves.
We characterize plane curve germes non-degenerate in Kouchnirenkos sense in terms of characteristics and intersection multiplicities of branches.
Let X be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over X. We prove that every terminal valuation over X is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of X. In dimension two, this result gives a new proof of the theorem of Fernandez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.
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