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 $p_1,\dots,p_n$ with prescribed multiplicities $m_1,\dots,m_n$. 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 $p_1,\dots, p_n$ on a horizontal line) we prove that $$\mathrm{area}(\Delta)\geq \frac{1}{2}\sum_{i=1}^n m_i^2-S,\ \ \text{where } S=\frac{1}{2}\max \left(\sum_{i=1}^n s_i^2 \Big| s_i\leq m_i, \sum_{i=1}^n s_i\leq \omega(\Delta)\right).$$ In the case $m_1=m_2=\ldots =m\leq \omega(\Delta)$ this estimate becomes $\mathrm{area}(\Delta)\geq \frac{1}{2}(n-\frac{\omega(\Delta)}{m})m^2$. That rewrites as $d\geq (\sqrt{n}-\frac{1}{2}-\frac{1}{2\sqrt n})m$ 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 {\it \`a priori} clear what is {\it 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.