Minimal plane valuations

We consider the value μ ^ ( ν ) = lim m → ∞ m − 1 a ( m L ) \hat {\mu } (\nu ) = \lim _{m \rightarrow \infty } m^{-1} a(mL) , where a ( m L ) a(mL) is the last value of the vanishing sequence of H 0 ( m L ) H^0(mL) along a divisorial or irrational valuation ν \nu centered at O P 2 , p \mathcal {O}_{\mathbb {P}^2,p} , L L (respectively, p p ) being a line (respectively, a point) of the projective plane P 2 \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 μ ^ ( ν ) ≥ 1 / v o l ( ν ) \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.