American Institute of Mathematical Sciences

Advanced
  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, J. Algebra, 324 (2010), 92-101. 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, J. Reine Angew. Math., 576 (2004), 209-233. 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, Ann. of Math. (2), 176 (2012), 925-978. 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, Adv. Math., 229 (2012), 2622-2639. doi: 10.1016/j.aim.2012.01.013.  Google Scholar

[5]

Yu. Manin, Cubic Forms. Algebra, Geometry, Arithmetic, North-Holland Mathematical Library, Vol. 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974.  Google Scholar

[6]

M. Mustaţă and R. Lazarsfeld, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 783-835.  Google Scholar

[7]

P. Łuszcz-Świdecka, On Minkowski Decompositions of Okounkov bodies on a Del Pezzo surface, Ann. Univ. Paedagog. Crac. Stud. Math., 10 (2011), 105-115.  Google Scholar

[8]

P. Łuszcz-Świdecka and D. Schmitz, Minkowski decomposition of Okounkov bodies on surfaces, J. Algebra, 414 (2014), 159-174. doi: 10.1016/j.jalgebra.2014.05.024.  Google Scholar

[9]

M. Nakamaye, Stable base loci of linear series, Math. Ann., 318 (2000), 837-847. 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, J. Algebra, 324 (2010), 92-101. 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, J. Reine Angew. Math., 576 (2004), 209-233. 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, Ann. of Math. (2), 176 (2012), 925-978. 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, Adv. Math., 229 (2012), 2622-2639. doi: 10.1016/j.aim.2012.01.013.  Google Scholar

[5]

Yu. Manin, Cubic Forms. Algebra, Geometry, Arithmetic, North-Holland Mathematical Library, Vol. 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974.  Google Scholar

[6]

M. Mustaţă and R. Lazarsfeld, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 783-835.  Google Scholar

[7]

P. Łuszcz-Świdecka, On Minkowski Decompositions of Okounkov bodies on a Del Pezzo surface, Ann. Univ. Paedagog. Crac. Stud. Math., 10 (2011), 105-115.  Google Scholar

[8]

P. Łuszcz-Świdecka and D. Schmitz, Minkowski decomposition of Okounkov bodies on surfaces, J. Algebra, 414 (2014), 159-174. doi: 10.1016/j.jalgebra.2014.05.024.  Google Scholar

[9]

M. Nakamaye, Stable base loci of linear series, Math. Ann., 318 (2000), 837-847. 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

[1]

Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155
[2]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
[3]

Giovanni Panti. Billiards on pythagorean triples and their Minkowski functions. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4341-4378. doi: 10.3934/dcds.2020183
[4]

Patricia Gaitan, Hiroshi Isozaki, Olivier Poisson, Samuli Siltanen, Janne Tamminen. Probing for inclusions in heat conductive bodies. Inverse Problems & Imaging, 2012, 6 (3) : 423-446. doi: 10.3934/ipi.2012.6.423
[5]

Meiyue Jiang, Chu Wang. The Orlicz-Minkowski problem for polytopes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1917-1930. doi: 10.3934/dcdss.2021043
[6]

Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439
[7]

Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543
[8]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675
[9]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[10]

Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems & Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009
[11]

Alexander Plakhov, Vera Roshchina. Fractal bodies invisible in 2 and 3 directions. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1615-1631. doi: 10.3934/dcds.2013.33.1615
[12]

Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223
[13]

Liran Rotem. Banach limit in convexity and geometric means for convex bodies. Electronic Research Announcements, 2016, 23: 41-51. doi: 10.3934/era.2016.23.005
[14]

Angelo Alberti, Claudio Vidal. Singularities in the gravitational attraction problem due to massive bodies. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 805-822. doi: 10.3934/dcds.2010.26.805
[15]

Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40
[16]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
[17]

Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115
[18]

Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453
[19]

Scott W. Hansen, Rajeev Rajaram. Riesz basis property and related results for a Rao-Nakra sandwich beam. Conference Publications, 2005, 2005 (Special) : 365-375. doi: 10.3934/proc.2005.2005.365
[20]

Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403

2020 Impact Factor: 0.929

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences