Bounding the degrees of generators of a homogeneous dimension 2 toric ideal

• Autores: Hugh Thomas
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 53, Fasc. 3, 2002, págs. 277-286
• Idioma: inglés
• Títulos paralelos:
  • Limitación de grado de los generadores de un ideal tórico homogeneo de dimensión 2
• Enlaces
  • Texto completo
• Resumen

  • Let $I$ be the toric ideal defined by a $2\times n$ matrix of integers, $$\mathcal{A}= \left(\begin{array}{cccc} 1&1&\ldots&1\\ a_1&a_2&\ldots&a_n\end{array}\right) $$ with $a_1 < a_2 < \ldots < a_n$. We give a combinatorial proof that I is generated by elements of degree at most the sum of the two largest differences $a_i-a_{i-1}$. The novelty is in the method of proof: the result has already been shown by L'vovsky using cohomological arguments.

• Referencias bibliográficas
  • E. Briales, A. Campillo, C. Marijuán, and P. Pisón, Minimal systems of generators for ideals of semigroups, J. Pure Appl. Algebra, 124 (1998),...
  • E. Briales, A. Campillo, and P. Pisón, On the equations defining toric projective varieties, Geometric and combinatorial aspects of commutative...
  • A. Campillo and Ph. Giménez, Graphes arithmétiques et syzygies, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 313–316.
  • A. Campillo and Ph. Giménez, Syzygies of affine toric varieties, J. Algebra 225 (2000), 142–161.
  • A. Campillo and C. Marijuán, Higher order relations for a numerical semigroup, Sém. Théor. Nombres Bordeaux 3 (1991), 249–260.
  • A. Campillo and P. Pisón, L’idéal d’un semigroupe de type fini, C. R. Acad. Sci. Paris, Sér. I Math. 316 (1993), 1303–1306.
  • L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), 491–506.
  • S. L’vovsky, On inflection points, monomial curves, and hypersurfaces containing projective curves, Math. Ann. 306 (1996), 719–735.
  • B. Sturmfels, Gröbner bases and convex geometry, American Mathematical Society, Providence, R. I., 1996.