Abstract
A few years ago Okounkov associated a convex set (Newton–Okounkov body) to a divisor, encoding the asymptotic vanishing behaviour of all sections of all powers of the divisor along a fixed flag. This brought to light the following guiding principle “use convex geometry, through the theory of these bodies, to study the geometrical/algebraic/arithmetic properties of divisors on smooth projective varieties”. The main goal of this survey article is to explain some of the philosophical underpinnings of this principle with a view towards studying local positivity and syzygetic properties of algebraic varieties.
Most of the material presented here is joint work with Alex Küronya. The author would like to thank him for the support, advices and many interesting conversations on the subject. The author would also like to thank the organizers of the “Workshop on Positivity and Valuations” in Barcelona for organizing such a beautiful event and for giving the opportunity to talk about the ideas above.
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References
H. Chen, Inegalité d’indice de Hodge en géométrie et arithmétique: une approche probabiliste (2014)
J. Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs. J. Am. Math. Soc. 25(3), 907–927 (2012)
K. Kaveh, A.G. Khovanskii, Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. (2) 176(2), 925–978 (2012)
J. Kollár, Lectures on Resolution of Singularities, vol. 166. Annals of Mathematics Studies (Princeton University Press, Princeton, 2007), p. vi+208
A. Küronya, V. Lozovanu, Local positivity of linear series on surfaces (2014), arXiv:1411.6205
A. Küronya, V. Lozovanu, Positivity of line bundles and Newton–Okounkov bodies (2015), arXiv:1506.06525
A. Küronya, V. Lozovanu, Infinitesimal Newton–Okounkov bodies and jet separation (2015), arXiv:1507.04339
A. Küronya, V. Lozovanu, A Reider-type theorem for higher syzygies on Abelian surfaces (2015), arXiv:1509.08621
R. Lazarsfeld, Positivity in Algebraic Geometry I–II. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 48–49 (Springer, Berlin, 2004)
R. Lazarsfeld, M. Mustaţă, Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 783–835 (2009)
R. Lazarsfeld, G. Pareschi, M. Popa, Local positivity, multiplier ideals, and syzygies of Abelian varieties. Algebra Number Theory 5(2), 185–196 (2011)
A. Okounkov, Brunn–Minkowski inequality for multiplicities. Invent. Math. 125(3), 405–411 (1996)
A. Okounkov, Why would multiplicities be log-concave?, Progress in Mathematics, vol. 213 (Birkhäuser, Boston, 2003), pp. 329–347
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Lozovanu, V. (2018). From Convex Geometry of Certain Valuations to Positivity Aspects in Algebraic Geometry. In: Alberich-Carramiñana, M., Galindo, C., Küronya, A., Roé, J. (eds) Extended Abstracts February 2016. Trends in Mathematics(), vol 9. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-00027-1_3
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DOI: https://doi.org/10.1007/978-3-030-00027-1_3
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