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 Néron–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, inspired by the work of Lazarsfeld and Mustaţă (Ann Inst Fourier (Grenoble) 56(6):1701–1734, 2006) 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.
KeywordsVolume Function Projective Variety Closed Convex Cone Linear Series Cartier Divisor
Part of this work was done while the first and the second authors were enjoying the hospitality of the Université Joseph Fourier in Grenoble. We would like to use this opportunity to thank Michel Brion and the Department of Mathematics for the invitation. We are grateful to Sebastien Boucksom, Rob Lazarsfeld, and Mircea Mustaţă for many helpful discussions. Special thanks are due to an anonymous referee for suggestions leading to significant expository improvements and notable strengthening of the results of Sect. 2.
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