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 $$\mathrm{area}(\Delta )\ge \frac{1}{2}\sum _{i=1}^n{m}_i^2-S,\text{where }S=\frac{1}{2}\max \left(\sum _{i=1}^n{s}_i^2|s_i\le m_i,\sum _{i=1}^n{s}_i\le \omega (\Delta )\right).$$ In the case $m_1=m_2=\dots =m\le \omega (\Delta )$ this estimate becomes $\mathrm{area}(\Delta )\ge \frac{1}{2}(n-\frac{\omega (\Delta )}{m})m^2$. That rewrites as $d\ge (\sqrt{n}-\frac{1}{2}-\frac{1}{2\sqrt{n}})m$ 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 à 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.

major revision, many typos and mistakes are corrected

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Cited by in corpus (3)

• Computing tropical varieties over fields with valuation
• The Newton polygon of a planar singular curve and its subdivision
• A guide to tropical modifications