High-dimensional convex sets arising in algebraic geometry

TL;DR

This paper introduces an asymptotic positivity concept in algebraic geometry linked to high-dimensional convex sets, providing a linear programming approach for complex surfaces to verify asymptotic log positivity.

Contribution

It proposes a new asymptotic positivity notion related to convex sets and connects it to linear programming for complex surfaces, advancing algebraic geometry methods.

Findings

Asymptotic positivity relates to high-dimensional convex sets.

Linear programming can verify asymptotic log positivity in complex surfaces.

The convex sets' dimension increases with birational operations.

Abstract

We introduce an asymptotic notion of positivity in algebraic geometry that turns out to be related to some high-dimensional convex sets. The dimension of the convex sets grows with the number of birational operations. In the case of complex surfaces we explain how to associate a linear program to certain sequences of blow-ups and how to reduce verifying the asymptotic log positivity to checking feasibility of the program.