no. 7

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 : https://doi.org/10.5802/aif.2005
Classification : 14M25,  14J40,  52B20
Mots clés : variétés toriques, polytopes convexes, générateurs d'idéaux 
@article{AIF_2003__53_7_2243_0,
     author = {Ogata, Shoetsu},
     title = {On projective toric varieties whose defining ideals have minimal generators of the highest degree},
     journal = {Annales de l'Institut Fourier},
     pages = {2243--2255},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     doi = {10.5802/aif.2005},
     mrnumber = {2044172},
     zbl = {1069.14057},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2005/}
}
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. https://aif.centre-mersenne.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, Tome 485 (1997), pp. 123-160 | MR 1442191 | Zbl 0866.20050

[ES] D. Eisenbud; B. Sturmfels Binomial ideals, Duke Math. J, Tome 84 (1996), pp. 1-45 | Article | MR 1394747 | Zbl 0873.13021

[EW] G. Ewald; U. Wessels On the ampleness of invertible sheaves in complete projective toric varieties, Results in Mathematics, Tome 19 (1991), pp. 275-278 | MR 1100674 | Zbl 0739.14031

[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 0353.14011

[Fl] W. Fulton Introduction to Toric Varieties, Ann. of Math. Studies, Tome No 131, Princeton Univ. Press, 1993 | MR 1234037 | Zbl 0813.14039

[GL] M. Green; R. Lazarsfeld A simple proof of Petri's Theorem on canonical curves, Geometry of Today, Giornate di Geometria (Roma, 1984), Tome vol. 60 (1985), pp. 129-142 | Zbl 0577.14018

[I] S. Iitaka Commutative rings, Kiso Sugaku Algebra (Japanese), Tome vol. 4, Iwanami Shoten, Tokyo, 1977 | MR 569688

[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, Tome 45 (1993), pp. 385-392 | Article | MR 1231563 | Zbl 0809.14042

[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, Tome 34 (1993), pp. 57-62 | MR 1239278 | Zbl 0781.14025

[M] D. Mumford Varieties defined by quadric equations, Questions on Algebraic Varieties (Corso CIME) (1969), pp. 30-100 | Zbl 0198.25801

[NO] K. Nakagawa; S. Ogata On generators of ideals defining projective toric varieties, Manuscripta Math, Tome 108 (2002), pp. 33-42 | Article | MR 1912946 | Zbl 0997.14014

[Od] T. Oda Convex Bodies and Algebraic Geometry, Ergebnisse der Math, Tome 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988 | MR 922894 | Zbl 0628.52002

[Og] S. Ogata Quadratic generation of ideals defining projective toric varieties, Kodai Math. J, Tome 26 (2003), pp. 137-146 | Article | MR 1993670 | Zbl 02076054

[S1] B. Sturmfels Gröbner bases and Convex Polytopes, University Lecture Series, Tome Vol. 8, American Mathematics Society, Providence, RI, 1995 | MR 1363949 | Zbl 0856.13020

[S2] B. Sturmfels Equations defining toric varieties, Algebraic Geometry (Santa Cruz, 1995) (Proc. Sympos. Pure Math) Tome 62 (1997), pp. 437-449 | Zbl 0914.14022