Positive cones of $b$-divisor classes

TL;DR

The paper develops a birationally robust framework for positivity of $b$-divisor classes by defining ample Cartier $b$-divisor classes through Seshadri constants and showing they form a convex cone inside the nef cone. It introduces a volume-based criterion to characterize big Cartier $b$-divisor classes, proving that big $b$-divisor classes also form a convex cone whose interior matches the interior of the pseudo-effective cone. The work builds a comprehensive numerical theory on the Riemann-Zariski space, including Weil and Cartier $b$-cycle classes, a well-behaved intersection product with a perfect pairing in top degree, and a clear description of how nef and ample $b$-divisor classes pull back along morphisms. Together, these results provide structural insights into positivity in birational geometry and establish tools for comparing positivity across models via the $b$-divisor framework.

Abstract

In this article, we define the notion of ample Cartier $b$-divisor classes by using the notion of Seshadri constants for Cartier $b$-divisor classes. In particular, we have shown that the set of all ample Cartier $b$-divisor classes forms a convex cone inside the nef cone of Cartier $b$-divisor classes. Furthermore, we have studied various properties of these Cartier ample $b$-divisor classes. We have also given an equivalent characterization of big Cartier $b$-divisor classes in terms of volume function of the pseudo-effective Cartier $b$-divisor classes. More specifically, we prove that the set of all big Cartier $b$-divisor classes form a convex cone. Finally we have investigated how the nef Cartier $b$-divisor classes behave under the pullback.

Theorems & Definitions (36)

• Remark 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Definition 3.4
• Definition 3.5
• Definition 4.1
• Lemma 4.2
• proof
• Definition 4.3