Toric vector bundles, branched covers of fans, and the resolution property
Author:
Sam Payne
Journal:
J. Algebraic Geom. 18 (2009), 1-36
DOI:
https://doi.org/10.1090/S1056-3911-08-00485-2
Published electronically:
April 22, 2008
MathSciNet review:
2448277
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Abstract |
References |
Additional Information
Abstract: We associate to each toric vector bundle on a toric variety $X(\Delta )$ a “branched cover” of the fan $\Delta$ together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently.
References
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References
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- W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. MR 1234037 (94g:14028)
- ---, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
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- L. Illusie, Existence de résolutions globales, Théorie des intersections et théorème de Riemann-Roch, SGA 6 1966/67, Springer Lect. Notes Math., vol. 225, 1971, pp. 160–221.
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- K. Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725 (95g:14056)
- G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lect. Notes in Math., vol. 339, Springer-Verlag, Berlin, 1973. MR 0335518 (49:299)
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- B. Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1–22. MR 2108211 (2005j:14002)
Additional Information
Sam Payne
Affiliation:
Stanford University, Department of Mathematics, Building 380, Stanford, California 94305
MR Author ID:
652681
Email:
spayne@stanford.edu
Received by editor(s):
September 26, 2006
Received by editor(s) in revised form:
March 14, 2007
Published electronically:
April 22, 2008
Additional Notes:
The author was supported by a Graduate Research Fellowship from the NSF
