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Regular version of the site

Article

Tropical approach to Nagata's conjecture in positive characteristic

Discrete and Computational Geometry. 2017. Vol. 58. No. 1. P. 158-179.

Suppose that there exists a hypersurface with the Newton polytope Δ, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points p1,…,pn with prescribed multiplicities m1,…,mn. Suppose that the minimal lattice width ω(Δ) of the Newton polygon Δ of the curve C is at least max(mi). Using tropical floor diagrams (a certain degeneration of p1,…,pn on a horizontal line) we prove that

area(Δ)≥12n∑i=1m2i−S,  where S=12max(n∑i=1s2i|si≤mi,n∑i=1si≤ω(Δ)).

In the case m1=m2=⋯=m≤ω(Δ) this estimate becomes area(Δ)≥12(n−ω(Δ)m)m2. That rewrites as d≥(√n−12−12√n)m for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary Δ) and our ground field is an infinite field of any characteristic, or a finite field large enough. The latter constraint arises because it is not a priori clear what is a collection of generic points in the case of a small finite field. We construct such collections for fields big enough, and that may be also interesting for the coding theory.