Arithmetic in the Boij S\"oderberg Cone

TL;DR

This paper advances understanding of Betti number bounds for graded modules by proving new cases in higher codimensions through arithmetic analysis in the Boij-S"oderberg cone, linking algebraic and number-theoretic methods.

Contribution

It introduces a novel approach to conjectures on Betti numbers by translating them into arithmetic problems within the Boij-S"oderberg cone, solving cases in codimensions five and six.

Findings

Classified Diophantine obstructions in codimension three.

Proved new bounds for Betti numbers in codimensions five and six.

Connected Betti tables with classical Diophantine equations.

Abstract

We study two long-standing conjectures concerning lower bounds for the Betti numbers of a graded module over a polynomial ring. We prove new cases of these conjectures in codimensions five and six by reframing the conjectures as arithmetic problems in the Boij-S\"oderberg cone. In this setting, potential counterexamples correspond to explicit Diophantine obstructions arising from the numerics of pure resolutions. Using number-theoretic methods, we completely classify these obstructions in the codimension three case revealing some delicate connections between Betti tables, commutative algebra and classical Diophantine equations. The new results in codimensions five and six concern Gorenstein algebras where a study of the variety determined by these Diophantine equations is sufficient to resolve the conjecture in this case.