Primitive collections and toric varieties
David A. Cox and Christine von Renesse
Source: Tohoku Math. J. (2) Volume 61, Number 3 (2009), 309-332.
Abstract
This paper studies Batyrev's notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with an open problem.
Primary Subjects: 14M25
Keywords: Toric varieties; primitive collections
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.tmj/1255700197
Digital Object Identifier: doi:10.2748/tmj/1255700197
References
V. Batyrev, On the classification of smooth projective toric varieties, Tohoku Math. J. 43 (1991), 569--585.
Mathematical Reviews (MathSciNet):
MR1133869
Digital Object Identifier: doi:10.2748/tmj/1178227429
Project Euclid: euclid.tmj/1178227429
C. Casagrande, Contractible classes in toric varieties, Math. Z. 243 (2003), 99--126.
Mathematical Reviews (MathSciNet):
MR1953051
Zentralblatt MATH:
1077.14070
Digital Object Identifier: doi:10.1007/s00209-002-0453-3
D. Cox, Recent developments in toric geometry, Algebraic geometry---Santa Cruz 1995, (J. Kollár, R. Lazarsfeld and D. Morrison, eds.), 389--436, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 1997.
Mathematical Reviews (MathSciNet):
MR1492541
Zentralblatt MATH:
0899.14025
W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, NJ, 1993.
Mathematical Reviews (MathSciNet):
MR1234037
Zentralblatt MATH:
0813.14039
O. Fujino, Notes on toric varieties from Mori theoretic viewpoint, Tohoku Math. J. 55 (2003), 551--564.
Mathematical Reviews (MathSciNet):
MR2017225
Digital Object Identifier: doi:10.2748/tmj/1113247130
Project Euclid: euclid.tmj/1113247130
O. Fujino and H. Sato, Introduction to toric Mori theory, Michigan Math. J. 52 (2004), 649--665.
Mathematical Reviews (MathSciNet):
MR2097403
Digital Object Identifier: doi:10.1307/mmj/1100623418
Project Euclid: euclid.mmj/1100623418
I. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Math. Theory Appl., Birkhäuser Boston Inc., Boston, MA, 1994.
Mathematical Reviews (MathSciNet):
MR1264417
G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math. 339, Springer-Verlag, Berlin-New York, 1973.
Mathematical Reviews (MathSciNet):
MR335518
Zentralblatt MATH:
0271.14017
A. Kresch, Gromov-Witten invariants of a class of toric varieties, Michigan Math. J. 48 (2000), 369--391.
Mathematical Reviews (MathSciNet):
MR1786497
Digital Object Identifier: doi:10.1307/mmj/1030132725
Project Euclid: euclid.mmj/1030132725
K. Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002.
Mathematical Reviews (MathSciNet):
MR1875410
Zentralblatt MATH:
0988.14007
K. Miyake and T. Oda, Almost homogeneous algebraic varieties under algebraic torus action, Manifolds---Tokyo 1973, (A. Hattori, ed), 373--381, Univ. of Tokyo Press, Tokyo, 1975.
Mathematical Reviews (MathSciNet):
MR379501
Zentralblatt MATH:
0306.14008
T. Oda and H. S. Park, Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions, Tohoku Math. J. 43 (1991), 375--399.
Mathematical Reviews (MathSciNet):
MR1117211
Digital Object Identifier: doi:10.2748/tmj/1178227461
Project Euclid: euclid.tmj/1178227461
T. Oda, Convex bodies and algebraic geometry, Ergeb. Math. Grenzgeb. (3) 15, Springer-Verlag, Berlin, 1988.
Mathematical Reviews (MathSciNet):
MR922894
Zentralblatt MATH:
0628.52002
M. Reid, Decomposition of toric morphisms, Arithmetic and Geometry, (M. Artin and J. Tate, eds.), 395--418, Prog. in Math. 36, Birkhäuser Boston Inc., Boston, MA, 1983.
Mathematical Reviews (MathSciNet):
MR717617
Zentralblatt MATH:
0571.14020
H. Sato, Combinatorial descriptions of toric extremal contractions, Nagoya Math. J. 180 (2005), 111--120.
Mathematical Reviews (MathSciNet):
MR2186671
Project Euclid: euclid.nmj/1134569898
A. Scaramuzza, Smooth complete toric varieties: an algorithmic approach, Ph. D. Thesis, Università degli Studi di “Roma Tre”, 2007.
H. M. Thompson, Comments on toric varieties, math.AG/0310336.
C. von Renesse, Combinatorial aspects of toric varieties, Ph. D. Thesis, University of Massachusetts, 2007.
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