×

zbMATH — the first resource for mathematics

The functor of a smooth toric variety. (English) Zbl 0828.14035
A map \(Y \to \mathbb{P}^n\) is determined by a line bundle \({\mathcal L}\) on \(Y\) and \(n + 1\) global sections of \({\mathcal L}\) which do not vanish simultaneously on \(Y\). In this paper, we generalize this description to the case of maps from \(Y\) to an arbitrary smooth toric variety. The data needed to determine such a map consist of a collection of line bundles on \(Y\) together with a section of each line bundle. Further, the line bundles must satisfy certain compatibility conditions, and the section must be nondegenerate in an appropriate sense. These data characterize maps to the toric variety in the strong sense that the toric variety represents the functor assigning to \(Y\) all collections as above. In the case of maps from \(\mathbb{P}^m\) to a smooth toric variety, we get an especially simple description that generalizes the usual way of specifying maps between projective spaces in terms of homogeneous polynomials of the same degree which do not vanish simultaneously.
Reviewer: D.A.Cox (Amherst)

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14E99 Birational geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. ASH, D. MUMFORD, M. RAPOPORT AND Y. -S. TAI, Smooth Compactiicationsof Locally Symmetric Varieties, Math Sci Press, Brookline, MA, 1975. · Zbl 0334.14007
[2] D. Cox, Thehomogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17-50 · Zbl 0846.14032
[3] V. DANILOV, Thegeometry of toric varieties, Russian Math. Surveys 33 (1978), 97-154 · Zbl 0425.14013 · doi:10.1070/RM1978v033n02ABEH002305
[4] J. DIEUDONNE AND A. GROTHENDIECK, Elements de Geometrie Algebrique I, Springer-Verlag, Berli Heidelberg, NewYork, 1971.
[5] W. FULTON, Introduction to Toric Varieties, Princeton University Press, Princeton, 1993 · Zbl 0813.14039
[6] M. GUEST, Thetopology of the space of rational curves on a toric variety, Acta Math. 174 (1995), 119-145. · Zbl 0826.14035 · doi:10.1007/BF02392803
[7] K. JACZEWSKI, Generalized Euler sequence andtoric varieties, in Classification of Algebraic Varieties, edited by C. Cilberto, E. Livorni and A. Sommese, AMS, Providence, 1994, 227-247. · Zbl 0837.14042
[8] D. MORRISON AND R. PLESSER, Summing the instantons: quantum cohomology and miroor symmetry toric varieties, · Zbl 0908.14014
[9] T. ODA, Convex Bodies and Algebraic geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1988 · Zbl 0628.52002 · eudml:203658
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.