Abstract
We consider the question what can be said about the rank of the Picard group Pic Xσ of a compact toric variety Xσ if we know only the combinatorial type of the associated fan σ. We establish upper and lower bounds for the rank of Pic Xσ and give conditions for Pic Xσ to be determined by the combinatorial type of σ. Furthermore, we show that for simple fans Pic Xσ is necessary isomorphic to {0} or Z and give an example for a compact toric variety having a trivial Picard group. Moreover in the projective case we study the relation between addition of T-invariant Cartier divisors on Xσ, taking tensor product of elements of Pic Xσ and piecewise linear functions on σ with Minkowski-addition of polytopes, where the latter operation is extended to a group operation. Finally, we explain the relation to strong cohomology in the projective case.
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References
BjÖrner, Anders: Face numbers of complexes and polytopes. Proc. Int. Congress of Math., Vol. 2, (Berkeley, Calif., 1986), 1408–1416
Danilov, V. I.: The geometry of toric varieties. Russ. math. Surveys 33:2 (1978), 97–154
Eikelberg, Markus: Zur Homologie torischer Varietäten. Ph. D. thesis Ruhr-Univ. Bochum 1989
Elkelberg, Markus: The Picard group of a compact toric variety. to appear in Results in Mathematics
Ewald, Günter: Algebraic Geometry and Convexity. to be published in Handbook [P. Gruber, J. Wills ed.]
Ewald, Günter: Combinatorial Convexity and Algebraic Geometry. To appear
Griffiths, Phillip Harris, Joseph: Pinciples of Algebraic Geometry. New York, Chichester, Brisbane, Toronto: Wiley 1978
GrÜnbaum, Branko: Convex polytopes. London, New York, Sydney: Wiley 1967
Hartshorne, Robin: Algebraic Geometry. New York, Heidelberg, Berlin: Springer 31983
Hironaka, Heisuke: Triangulations of Algebraic Sets. Proceedings of Symposia in Pure Mathematics 29 (1975), 165–185
Jurkiewicz, Jerzy: Chow ring of projective non-singular torus embedding. Colloq. Math. 43 (1980), 261–270
Leichtwei\, Kurt: Konvexe Mengen. Berlin, Heidelberg, New York: Springer 1980
Meyer, Walter: Indecomposable polytopes. Trans. Amer. math. Soc. 190 (1974), 77–86
McConnell, Marc: The Rational Homology of Toric Varieties is not a Combinatorial Invariant. Proc. Amer. math. Soc. 105 (1989) 4, 986–991
McMullen, Peter: Representations of polytopes and polyhedral sets. Geometriae dedicata 2 (1973), 83–99
Oda, Tadao: Convex bodies and algebraic geometry. Berlin, Heidelberg, New York: Springer 1988
Shephard, G. C: Decomposable convex polyhedra. Mathematika 10 (1963), 89–95
Shephard, G. C: Reducible convex sets. Mathematika 13 (1966), 49–50
Smilansky, Zeev: Decomposability of Polytopes and Polyhedral Sets. Geometriae dedicata 24 (1987), 29–49
Smilansky, Zeev: An Indecomposable Polytope all of whose Facets are Decomposable. Mathematika 33 (1986), 192–196
StÖcker, Ralph Zieschang, Heiner: Algebraische Topologie. Stuttgart: Teubner 1988
Teissier, Bernard: Variétés toriques et polytopes. Séminaire Bourbaki 33e année (1980/81) No. 565, 565-01-565-14
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Eikelberg, M. Picard Groups of compact toric Varieties and combinatorial Classes of Fans. Results. Math. 23, 251–293 (1993). https://doi.org/10.1007/BF03322301
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DOI: https://doi.org/10.1007/BF03322301