Volume functions of linear series

Küronya, Alex and Victor, Lozovanu and Catriona, Maclean (2013) Volume functions of linear series. MATHEMATISCHE ANNALEN, 356 (2). pp. 635-652. ISSN 0025-5831

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Abstract

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 in- variant 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 Mustat¸˘a [16] on Ok- ounkov bodies, we show that any continuous, homogeneous, and log-concave function ap- pears 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.

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