The Minkowski equality of big divisors (Q6590152)

The paper under review is a continuation of a program by the author to extend results about volumes and linear series from algebraically closed fields to arbitrary fields.\N\NIn this context, the author proves the Minkowski inequality and characterizes the case of equality for big \(\mathbb R\)-divisors over arbitrary fields.\N\NThe author proves two versions of the Diskant inequality over arbitrary fields of characteristic \(0\). They are used to characterize the case of equality in the Khovanskii-Teissier inequalities for big \(\mathbb R\)-divisors over arbitrary fields of characteristic 0.\N\NAlong the way other classical results are extended over arbitrary fields: The injectivity of the cycle class map \(N^1(X)\to N_{d-1}(X)\) for projective varieties \(X\) of dimension \(d\), the theory of the positive intersection products, projection formula for volumes of big divisors under generically finite maps, the main results of [\textit{M. Fulger} et al., Mich. Math. J. 65, No. 2, 371--387 (2016; Zbl 1353.14011)] on equality of volumes vs. equality of Hilbert functions.