Room 440, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 440室)

Inequalities of Mobility Functions between the Cycle Classes and their Intersections with Divisors

Wei-Chung Chen (University of Tokyo)

Abstract

The volume of a Cartier divisor on a projective integral variety measures the asymptotic rate of growth of the dimension of the global sections of the multiples of the divisor. It provides a good way to understand the big divisor classes. As a generalization, the mobility defined on the cone of pseudo-effective cycle classes measures the asymptotic rate of growth of the dimensions of the global sections of the multiples of the cycle classes. It provides a way to understand the big cycle classes. The mobility function for divisors coincides with the volume function, and the mobility function for 0-cycles is just n! times the degree function deg. However, the mobility is difficult to compute in general. None of a non-trivial example of mobility is known precisely. The main purpose of this article is to provide an inequality between the mobility of a cycle class and its intersection with a nef divisor. Using this inequality, we can find an upper bound of the mobility of a cycle in terms of the degree of its intersection with divisors.