The analogue of Izumi's Theorem for Abhyankar valuations

16 pagesA well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we generalize this theorem to the case of Abhyankar valuations with archimedian value semigroup. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. Then we show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the Krull topology and the topology defined by the symbolic powers of an arbitrary ideal

Minimal plane valuations

[EN] We consider the value (mu) over cap(nu) = lim(m -> infinity) 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 O-P2,(p), L (respectively, p) being a line (respectively, a point) of the projective plane 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 (mu) over cap(nu) >= root 1/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.The authors were partially supported by the Spanish Government Ministerio de Economia, 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.Galindo Pastor, C.; Monserrat Delpalillo, FJ.; Moyano-Fernández, J. (2018). Minimal plane valuations. Journal of Algebraic Geometry. 27(4):751-783. https://doi.org/10.1090/jag/722S75178327