Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Minimal plane valuations


Authors: Carlos Galindo, Francisco Monserrat and Julio-José Moyano-Fernández
Journal: J. Algebraic Geom. 27 (2018), 751-783
DOI: https://doi.org/10.1090/jag/722
Published electronically: July 16, 2018
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Abstract: We consider the value $ \hat {\mu } (\nu ) = \lim _{m \rightarrow \infty } m^{-1} a(mL)$, where $ a(mL)$ is the last value of the vanishing sequence of $ H^0(mL)$ along a divisorial or irrational valuation $ \nu $ centered at $ \mathcal {O}_{\mathbb{P}^2,p}$, $ L$ (respectively, $ p$) being a line (respectively, a point) of the projective plane $ \mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $ \hat {\mu } (\nu ) \geq \sqrt {1 / \mathrm {vol}(\nu )}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in [Comm. Anal. Geom. 25 (2017), pp. 125-161] to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original Nagata Conjecture. We also provide infinitely many families of minimal very general valuations with an arbitrary number of Puiseux exponents and an asymptotic result that can be considered as evidence in the direction of the above-mentioned conjecture.


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Carlos Galindo
Affiliation: Instituto Universitario de Matemáticas y Aplicaciones de Castellón and Departamento de Matemáticas, Universitat Jaume I, Campus de Riu Sec. s/n, 12071 Castelló de la Plana, Spain
Email: galindo@mat.uji.es

Francisco Monserrat
Affiliation: Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain
Email: framonde@mat.upv.es

Julio-José Moyano-Fernández
Affiliation: Instituto Universitario de Matemáticas y Aplicaciones de Castellón and Departamento de Matemáticas, Universitat Jaume I, Campus de Riu Sec. s/n, 12071 Castelló de la Plana, Spain
Email: moyano@uji.es

DOI: https://doi.org/10.1090/jag/722
Received by editor(s): May 3, 2017
Published electronically: July 16, 2018
Additional Notes: The authors were partially supported by the Spanish Government Ministerio de Economía, Industria y Competitividad/FEDER, grants MTM2012-36917-C03-03, MTM2015-65764-C3-2-P, and MTM2016-81735-REDT, as well as by Universitat Jaume I, grant P1-1B2015-02.
Article copyright: © Copyright 2018 University Press, Inc.

American Mathematical Society