Tropical Approach to Nagata’s Conjecture in Positive Characteristic

Abstract

Suppose that there exists a hypersurface with the Newton polytope \(\Delta \), which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of \(\Delta \) to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of \(\Delta \) from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points \(p_1,\ldots ,p_n\) with prescribed multiplicities \(m_1,\ldots ,m_n\). Suppose that the minimal lattice width \(\omega (\Delta )\) of the Newton polygon \(\Delta \) of the curve C is at least \(\max (m_i)\). Using tropical floor diagrams (a certain degeneration of \(p_1,\ldots , p_n\) on a horizontal line) we prove that $$\begin{aligned} {{\mathrm {area}}}(\Delta )\ge & {} \frac{1}{2}\sum _{i=1}^n m_i^2-S,\ \ \text {where } \\ S= & {} \frac{1}{2}\max \left( \sum _{i=1}^n s_i^2\, \Big |\, s_i\le m_i, \sum _{i=1}^n s_i\le \omega (\Delta )\right) . \end{aligned}$$In the case \(m_1=m_2=\cdots =m\le \omega (\Delta )\) this estimate becomes \({\mathrm {area}}(\Delta )\ge \frac{1}{2}\bigl (n-\frac{\omega (\Delta )}{m}\bigr )m^2\). That rewrites as \(d\ge \bigl (\sqrt{n}-\frac{1}{2}-\frac{1}{2\sqrt{n}}\bigr )m\) for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary \(\Delta \)) 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.