School of Mathematics Seminars and Lectures
Newton-Okounkov bodies and Picard numbers on surfaces
by Dr. Aditya Subramaniam (TIFR, Mumbai)
Wednesday, January 11, 2023 from to (Asia/Kolkata)
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.