Let
be a discrete valuation on the function field of a normal projective variety
.
Ein, Lazarsfeld, Mustaţă, Nakamaye, and Popa showed that
induces a nonnegative real-valued continuous function on the big cone
of ,
which they called the asymptotic order of vanishing
along . The
case where
is given by the order of vanishing along a prime divisor was studied
earlier by Nakayama, who extended the domain of the function to the
pseudoeffective cone and investigated the continuity of the extended
function.
Here we generalize Nakayama’s results to any discrete
valuation ,
using an approach inspired by Lazarsfeld and Mustaţă’s construction of the
global Okounkov body, which has a quite different flavor from the arguments
employed by Nakayama.
A corollary is that the asymptotic order-of-vanishing function
can be extended continuously to the pseudoeffective cone
of
if
is polyhedral (note
that we do
not require
to be
rational polyhedral).
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