On projective toric varieties whose defining ideals have minimal generators of the highest degree
[Sur les variétés toriques projectives dont les idéaux annulateurs ont des générateurs minimaux de plus haut degré]
Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 2243-2255.

Il est connu que les générateurs de l’idéal annulateur d’une variété torique projective de dimension n, plongée par les sections globales d’un fibré en droites normalement engendré, sont de degré au plus n+1. Nous caractérisons les variétés projectives de dimension n dont un générateur au moins de l’idéal annulateur doit être de degré n+1.

It is known that generators of ideals defining projective toric varieties of dimension n embedded by global sections of normally generated line bundles have degree at most n+1. We characterize projective toric varieties of dimension n whose defining ideals must have elements of degree n+1 as generators.

DOI : 10.5802/aif.2005
Classification : 14M25, 14J40, 52B20
Keywords: toric varieties, convex polytopes, generators of ideals
Mot clés : variétés toriques, polytopes convexes, générateurs d'idéaux
Ogata, Shoetsu 1

1 Tohoku University, Mathematical Institute, Sendai 980 (Japon)
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Ogata, Shoetsu. On projective toric varieties whose defining ideals have minimal generators of the highest degree. Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 2243-2255. doi : 10.5802/aif.2005. http://www.numdam.org/articles/10.5802/aif.2005/

[A] T. Abe On the study of integral convex polytopes and toric varieties (2002) Master thesis, Tohoku University (Japanese)

[BGT] W. Bruns; J. Gubeladze; N. V. Trung Normal polytopes, triangulations, and Koszul algebras, J. reine angew. Math, Volume 485 (1997), pp. 123-160 | MR | Zbl

[ES] D. Eisenbud; B. Sturmfels Binomial ideals, Duke Math. J, Volume 84 (1996), pp. 1-45 | DOI | MR | Zbl

[EW] G. Ewald; U. Wessels On the ampleness of invertible sheaves in complete projective toric varieties, Results in Mathematics, Volume 19 (1991), pp. 275-278 | MR | Zbl

[Fj] T. Fujita; (edited by Bailly and Shioda) Defining Equations for Certain Types of Polarized Varieties, Complex Analysis and Algebraic Geometry (1977), pp. 165-173 | Zbl

[Fl] W. Fulton Introduction to Toric Varieties, Ann. of Math. Studies, No 131, Princeton Univ. Press, 1993 | MR | Zbl

[GL] M. Green; R. Lazarsfeld A simple proof of Petri's Theorem on canonical curves, Geometry of Today, Giornate di Geometria (Roma, 1984), Volume vol. 60 (1985), pp. 129-142 | Zbl

[I] S. Iitaka Commutative rings, Kiso Sugaku Algebra (Japanese), vol. 4, Iwanami Shoten, Tokyo, 1977 | MR

[K1] R.J. Koelman The number of moduli of families of curves on toric surfaces (1991) (thesis, Katholieke Universiteit te Nijmengen)

[K2] R.J. Koelman Generators for the ideal of a projectively embedded toric surfaces, Tohoku Math. J, Volume 45 (1993), pp. 385-392 | DOI | MR | Zbl

[K3] R.J. Koelman A criterion for the ideal of a projectively embedded toric surfaces to be generated by quadrics, Beiträger zur Algebra und Geometrie, Volume 34 (1993), pp. 57-62 | MR | Zbl

[M] D. Mumford Varieties defined by quadric equations, Questions on Algebraic Varieties (Corso CIME) (1969), pp. 30-100 | Zbl

[NO] K. Nakagawa; S. Ogata On generators of ideals defining projective toric varieties, Manuscripta Math, Volume 108 (2002), pp. 33-42 | DOI | MR | Zbl

[Od] T. Oda Convex Bodies and Algebraic Geometry, Ergebnisse der Math, 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988 | MR | Zbl

[Og] S. Ogata Quadratic generation of ideals defining projective toric varieties, Kodai Math. J, Volume 26 (2003), pp. 137-146 | DOI | MR | Zbl

[S1] B. Sturmfels Gröbner bases and Convex Polytopes, University Lecture Series, Vol. 8, American Mathematics Society, Providence, RI, 1995 | MR | Zbl

[S2] B. Sturmfels Equations defining toric varieties, Algebraic Geometry (Santa Cruz, 1995) (Proc. Sympos. Pure Math), Volume 62 (1997), pp. 437-449 | Zbl

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