Tropical approach to Nagata's conjecture in positive characteristic

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.Comment: major revision, many typos and mistakes are correcte

Tropical curves in sandpiles

We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation $\psi$ of the maximal stable state $\mu\equiv 3$ is obtained by adding extra grains at several points. It appears, that the result $\psi^\circ$ of the relaxation of $\psi$ coincides with $\mu$ almost everywhere; the set where $\psi^\circ\ne \mu$ is called the deviation locus. The scaling limit of the deviation locus turns out to be a distinguished tropical curve passing through the perturbation points. Nous consid\'erons le mod\`ele du tas de sable sur l'ensemble des points entiers d'un polygone entier. En ajoutant des grains de sable en certains points, on obtient une perturbation mineure de la configuration stable maximale $\mu\equiv 3$. Le r\'esultat $\psi^\circ$ de la relaxation est presque partout \'egal \`a $\mu$. On appelle lieu de d\'eviation l'ensemble des points o\`u $\psi^\circ\ne \mu$. La limite au sens de la distance de Hausdorff du lieu de d\'eviation est une courbe tropicale sp\'eciale, qui passe par les points de perturbation.Comment: small correction