Arithmetic properties of volumes of divisors

Abstract: The volume of a Cartier divisor on an irreducible projective variety describes the asymptotic rate of growth of the number of its global sections. As such, it is  a non-negative real number, which happens to be rational whenever the section ring of the divisor in question is finitely generated.

In a joint work with Catriona Maclean and Victor Lozovanu we study the multiplicative semigroup of volumes of divisors. We prove that this set  is countable on the one hand, on the other hand it contains transcendental elements.