IMPANGA Working Group

Okounkov Solids

Pedagogical University Cracow

15-17 December, 2011

Okounkov solids (bodies) form a new and rapidly developing research area. These are convex bodies associated to algebraic varieties in a very general setting, and may be viewed as a vast generalization of the theory of toric varieties. The idea is to associate to a variety a convex polytope, which is in particular a combinatorial object, in such a way that questions about the original variety can be answered from the geometry of the associated polytope.

The program was based on the following articles and preprints:

• [BC] Boucksom, S., Chen, H.: Okounkov bodies of filtered linear series, Compos. Math 147 (2011), 1205-1229
• [I] Ito, A.: Okounkov bodies and Seshadri constants, preprint 2011
• [J] Jow, S.-Y.: Okounkov bodies and restricted volumes along very general curves, Adv. Math. 223 (2010), 1356-1371
• [KLM] Kueronya, A., Lozovanu, V., Maclean, C.: Convex bodies appearing as Okounkov bodies of divisors, arxiv:1008.4431v1
• [KMS] Kueronya, A., Maclean, C., Szemberg, T.: Okounkov functions, preprint 2011
• [LM] Lazarsfeld, R., Mustata, M.: Convex bodies associated to linear series. Ann. del'ENS 42 (2009), 783-83
• [L] Luszcz-Swidecka, P.: Minkowski decomposition of Okounkov bodies on a Del Pezzo surface, preprint 2011
• [N] Nystroem, D. W.: Transforming metrics on a line bundle to the Okounkov body, preprint, arXiv:0903.5167v1.

Lectures held:

Thursday, 15 December, 2011:

13:00-14:00 Mateusz Michalek (Grenoble): Construction of Okounkov solids

14:30-15:30 Tomasz Swiderski (UP Cracow) Volumes of Okounkov solids

16:30-17:30 Ulrich Derenthal (LMU Muenchen) Slices of Okounkov solids and Okounkov solids on surfaces I

Friday, 16 December, 2011:

10:00-11:00 Tomasz Szemberg (UP Cracow) Okounkov solids on surfaces II

11:30-12:30 Slawomir Rams (UJ Cracow) Reversing the construction I

13:00-14:00 Slawomir Cynk (UJ Cracow) Reversing the construction II

15:30-16:30 Halszka Tutaj-Gasinska (UJ Cracow): Okounkov solids and Seshadri constants I

17:00-18:00 Marcin Dumnicki (UJ Cracow): Okounkov solids and Seshadri constants II

Saturday, 17 December, 2011:

10:00-11:00 Patrycja Luszcz-Swidecka (UJ/UP Cracow): Minkowski decomposition of Okounkov bodies on a Del Pezzo surface

11:30-12:30 Tomasz Szemberg (UP Cracow): Functions on Okounkov bodies

Last modified: 6 January, 17:19 by Tomasz Szemberg