Mathematics – Algebraic Geometry
Scientific paper
2010-08-24
Mathematics
Algebraic Geometry
15 pages, 1 figure
Scientific paper
The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole N\'eron--Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, relying on the work of Lazarsfeld and Musta\c t\u a \cite{LM08} on Okounkov bodies, we show that any continuous, homogeneous, and log-concave function appears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case.
Kuronya Alex
Lozovanu Victor
Maclean Catriona
No associations
LandOfFree
Volume functions of linear series does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Volume functions of linear series, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Volume functions of linear series will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-393372