# Augmented base loci and restricted volumes on normal varieties, II

### What is it about?

Let X be a normal projective variety defined over an algebraically closed field and let Z be a subvariety. Let D be an $R-Cartier R-divisor on X. Given an expression (*) D \sim_R t_1 H_1 + ... + t_s H_s with t_i in R$and H_i$very ample, we define the (*) restricted volume of D to Z and we show that it coincides with the usual restricted volume when Z is not contained in B_+(D). Then, using some recent results of Birkar, we generalize to R-divisors the two main results of [BCL]: The first, proved for smooth complex projective varieties by Ein, Lazarsfeld, Mustata, Nakamaye and Popa, is the characterization of B_+(D) as the union of subvarieties on which the (*)-restricted volume vanishes; the second is that X - B_+(D) is the largest open subset on which the Kodaira map defined by large and divisible (*)-multiples of D is an isomorphism.

http://dx.doi.org/10.1017/s030500411500050x

The following have contributed to this page: Professor Angelo Felice Lopez