Title of Seminar: Algebraic Geometry Preprint Seminar 2025
Title of Talk: Newton-Okounkov bodies and Picard numbers on surfaces
Speaker: Aditya Subramaniam, TIFR, Mumbai
Date: January 11, 2023
Time: 16:00:00 Hours
Venue: AG-77

Abstract: In this talk, we will discuss a preprint of Fernández-Nickel-Roé 'Newton-Okounkov bodies and Picard numbers on surfaces' https://arxiv.org/pdf/2101.05338v1.pdf Newton-Okounkov bodies were introduced by A. Okounkov as a tool in representation theory; later Kaveh-Khovanskii and Lazarsfeld-Mustata developed a general theory with applications to both convex and algebraic geometry. In this preprint, the authors study the shapes of all Newton-Okounkov bodies of a given big divisor on a surface S with respect to all rank 2 valuations of K(S). They obtain upper bounds for, and in many cases determine exactly, the possible numbers of vertices of these bodies. The upper bounds are expressed in terms of Picard numbers. They also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor determines the Picard number of S, and proves that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies.