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Volume functions of linear series

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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 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.

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Notes

  1. In their interesting paper [3], Boucksom et al. found a nice formula for the derivative of \({\text{ vol}}_X\) in any direction.

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Acknowledgments

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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Authors and Affiliations

  1. Department of Algebra, Budapest University of Technology and Economics, P.O. Box 91, Budapest, 1521, Hungary

    Alex Küronya

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1109, USA

    Victor Lozovanu

  3. UFR de Mathématiques, Université Joseph Fourier, 100 rue des Maths, BP 74, 38402 , St Martin d’Héres, France

    Catriona Maclean

Authors
  1. Alex Küronya
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  2. Victor Lozovanu
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  3. Catriona Maclean
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Correspondence to Alex Küronya.

Additional information

During this project A. Küronya was partially supported by the CNRS, the DFG-Leibniz program, the SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”, the OTKA Grants 61116, 77476, 77604, and a Bolyai Fellowship by the Hungarian Academy of Sciences.

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Küronya, A., Lozovanu, V. & Maclean, C. Volume functions of linear series. Math. Ann. 356, 635–652 (2013). https://doi.org/10.1007/s00208-012-0859-0

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