9,951 research outputs found
Tropical approach to Nagata's conjecture in positive characteristic
Suppose that there exists a hypersurface with the Newton polytope ,
which passes through a given set of subvarieties. Using tropical geometry, we
associate a subset of to each of these subvarieties. We prove that a
weighted sum of the volumes of these subsets estimates the volume of
from below.
As a particular application of our method we consider a planar algebraic
curve which passes through generic points with prescribed
multiplicities . Suppose that the minimal lattice width
of the Newton polygon of the curve is at least
. Using tropical floor diagrams (a certain degeneration of
on a horizontal line) we prove that
In the case this estimate becomes
. That
rewrites as for the curves
of degree .
We consider an arbitrary toric surface (i.e. arbitrary ) 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 of the maximal stable state
is obtained by adding extra grains at several points. It appears, that the
result of the relaxation of coincides with almost
everywhere; the set where 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
. Le r\'esultat de la relaxation est presque partout
\'egal \`a . On appelle lieu de d\'eviation l'ensemble des points o\`u
. 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
- …