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  2014, 21: 126-131. doi: 10.3934/era.2014.21.126

Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone

Piotr Pokora 1, and  Tomasz Szemberg 1,

1. 

Instytut Matematyki, Pedagogical University of Cracow, Podchorążych 2, PL-30-084 Kraków, Poland, Poland

Received  May 2014 Revised  July 2014 Published  August 2014

The purpose of this note is to study the number of elements in Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone.
Keywords: Minkowski basis, del Pezzo surfaces., Okounkov bodies.
Mathematics Subject Classification: 14C20, 14J2.
Citation: Piotr Pokora, Tomasz Szemberg. Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone. Electronic Research Announcements, 2014, 21: 126-131. doi: 10.3934/era.2014.21.126
References:
[1]

Th. Bauer, M. Funke and S. Neumann, Counting Zariski chambers on Del Pezzo surfaces,, \emph{J. Algebra}, 324 (2010), 92.  doi: 10.1016/j.jalgebra.2010.02.037.  Google Scholar

[2]

Th. Bauer, A. Küronya and T. Szemberg, Zariski chambers, volumes, and stable base loci,, \emph{J. Reine Angew. Math.}, 576 (2004), 209.  doi: 10.1515/crll.2004.090.  Google Scholar

[3]

K. Kaveh and A. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory,, \emph{Ann. of Math. (2)}, 176 (2012), 925.  doi: 10.4007/annals.2012.176.2.5.  Google Scholar

[4]

A. Küronya, V. Lozovanu and C. Maclean, Convex bodies appearing as Okounkov bodies of divisors,, \emph{Adv. Math.}, 229 (2012), 2622.  doi: 10.1016/j.aim.2012.01.013.  Google Scholar

[5]

Yu. Manin, Cubic Forms. Algebra, Geometry, Arithmetic,, North-Holland Mathematical Library, (1974).   Google Scholar

[6]

M. Mustaţă and R. Lazarsfeld, Convex bodies associated to linear series,, \emph{Ann. Sci. Éc. Norm. Supér. (4)}, 42 (2009), 783.   Google Scholar

[7]

P. Łuszcz-Świdecka, On Minkowski Decompositions of Okounkov bodies on a Del Pezzo surface,, \emph{Ann. Univ. Paedagog. Crac. Stud. Math.}, 10 (2011), 105.   Google Scholar

[8]

P. Łuszcz-Świdecka and D. Schmitz, Minkowski decomposition of Okounkov bodies on surfaces,, \emph{J. Algebra}, 414 (2014), 159.  doi: 10.1016/j.jalgebra.2014.05.024.  Google Scholar

[9]

M. Nakamaye, Stable base loci of linear series,, \emph{Math. Ann.}, 318 (2000), 837.  doi: 10.1007/s002080000149.  Google Scholar

[10]

P. Pokora, D. Schmitz and S. Urbinati, Minkowski decomposition and generators of the moving cone for toric varieties,, \arXiv{1310.8505}., ().   Google Scholar

show all references

References:
[1]

Th. Bauer, M. Funke and S. Neumann, Counting Zariski chambers on Del Pezzo surfaces,, \emph{J. Algebra}, 324 (2010), 92.  doi: 10.1016/j.jalgebra.2010.02.037.  Google Scholar

[2]

Th. Bauer, A. Küronya and T. Szemberg, Zariski chambers, volumes, and stable base loci,, \emph{J. Reine Angew. Math.}, 576 (2004), 209.  doi: 10.1515/crll.2004.090.  Google Scholar

[3]

K. Kaveh and A. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory,, \emph{Ann. of Math. (2)}, 176 (2012), 925.  doi: 10.4007/annals.2012.176.2.5.  Google Scholar

[4]

A. Küronya, V. Lozovanu and C. Maclean, Convex bodies appearing as Okounkov bodies of divisors,, \emph{Adv. Math.}, 229 (2012), 2622.  doi: 10.1016/j.aim.2012.01.013.  Google Scholar

[5]

Yu. Manin, Cubic Forms. Algebra, Geometry, Arithmetic,, North-Holland Mathematical Library, (1974).   Google Scholar

[6]

M. Mustaţă and R. Lazarsfeld, Convex bodies associated to linear series,, \emph{Ann. Sci. Éc. Norm. Supér. (4)}, 42 (2009), 783.   Google Scholar

[7]

P. Łuszcz-Świdecka, On Minkowski Decompositions of Okounkov bodies on a Del Pezzo surface,, \emph{Ann. Univ. Paedagog. Crac. Stud. Math.}, 10 (2011), 105.   Google Scholar

[8]

P. Łuszcz-Świdecka and D. Schmitz, Minkowski decomposition of Okounkov bodies on surfaces,, \emph{J. Algebra}, 414 (2014), 159.  doi: 10.1016/j.jalgebra.2014.05.024.  Google Scholar

[9]

M. Nakamaye, Stable base loci of linear series,, \emph{Math. Ann.}, 318 (2000), 837.  doi: 10.1007/s002080000149.  Google Scholar

[10]

P. Pokora, D. Schmitz and S. Urbinati, Minkowski decomposition and generators of the moving cone for toric varieties,, \arXiv{1310.8505}., ().   Google Scholar

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