TY - JOUR

T1 - Tropical Approach to Nagata’s Conjecture in Positive Characteristic

AU - Kalinin, Nikita

PY - 2017/7/1

Y1 - 2017/7/1

N2 - Suppose that there exists a hypersurface with the Newton polytope Δ , which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points p1, … , pn with prescribed multiplicities m1, … , mn. Suppose that the minimal lattice width ω(Δ) of the Newton polygon Δ of the curve C is at least max (mi). Using tropical floor diagrams (a certain degeneration of p1, … , pn on a horizontal line) we prove that area(Δ)≥12∑i=1nmi2-S,whereS=12max(∑i=1nsi2|si≤mi,∑i=1nsi≤ω(Δ)).In the case m1= m2= ⋯ = m≤ ω(Δ) this estimate becomes area(Δ)≥12(n-ω(Δ)m)m2. That rewrites as d≥(n-12-12n)m for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary Δ) 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.

AB - Suppose that there exists a hypersurface with the Newton polytope Δ , which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points p1, … , pn with prescribed multiplicities m1, … , mn. Suppose that the minimal lattice width ω(Δ) of the Newton polygon Δ of the curve C is at least max (mi). Using tropical floor diagrams (a certain degeneration of p1, … , pn on a horizontal line) we prove that area(Δ)≥12∑i=1nmi2-S,whereS=12max(∑i=1nsi2|si≤mi,∑i=1nsi≤ω(Δ)).In the case m1= m2= ⋯ = m≤ ω(Δ) this estimate becomes area(Δ)≥12(n-ω(Δ)m)m2. That rewrites as d≥(n-12-12n)m for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary Δ) 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.

KW - Floor diagrams

KW - m-Fold point

KW - Nagata’s conjecture

KW - Tropical geometry

UR - http://www.scopus.com/inward/record.url?scp=85018677602&partnerID=8YFLogxK

U2 - 10.1007/s00454-017-9894-7

DO - 10.1007/s00454-017-9894-7

M3 - Article

AN - SCOPUS:85018677602

VL - 58

SP - 158

EP - 179

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -