Convex-normal (Pairs of) Polytopes

Haase, Christian ; Hofmann, Jan

Canadian mathematical bulletin, Tome 60 (2017) no. 3, pp. 510-521

Résumé

In 2012, Gubeladze (Adv. Math. 2012) introduced the notion of $k$ -convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence between $k$ - and $(k+1)$ -convex-normality (for $k\ge 3$ ) and improve the bound to $2d(d+1)$ . In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $\text{Q}$ , where the normal fan of $P$ is a refinement of the normal fan of $\text{Q}$ , if every edge ${{e}_{P}}$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) ${{e}_{\text{Q}}}$ of $\text{Q}$ , then $(P+\text{Q})\cap {{\mathbb{Z}}^{d}}=(P\cap {{\mathbb{Z}}^{d}})+(\text{Q}\cap {{\mathbb{Z}}^{d}})$ .

Détail

DOI : 10.4153/CMB-2016-057-0

Mots-clés : 52B20, 14M25, 90C10, integer decomposition property, integrally closed, projectively normal, lattice polytopes

Bibliographie

[BG99] [BG99] Bruns, W. and Gubeladze, J., Normality and covering properties ofaffine semigroups. J. Reine Angew. Math. 510(1999), 161–178. http://dx.doi.Org/10.1515/crll.1999.044 Google Scholar

[BGT97] [BGT97] Bruns, W., Gubeladze, J., and Trung, N. G., Normalpolytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485(1997), 123–160. Google Scholar

[EW91] [EW91] Ewald, G. and Wessels, U., On the ampleness of invertible sheaves in complete projective toric varieties. Results Math. 19(1991), no. 3-4, 275–278. http://dx.doi.Org/10.1007/BF03323286 Google Scholar

[Gubl2] [Gubl2] Gubeladze, J., Convex normality of rational polytopes with long edges. Adv. Math. 230(2012), no. 1,372-389. http://dx.doi.Org/10.1016/j.aim.2011.12.003 Google Scholar

[HHM07] [HHM07] Haase, C., Hibi, T., and Maclagan, D., Mini-workshop: Projective normality of smooth toric varieties, abstracts from the mini-workshop: held August 12–18, 2007. Oberwolfach Reports 4(2007), 2283-2319. Google Scholar

[KS03] [KS03] Kantor, J.-M. and Sarkaria, K. S., On primitive subdivisions of an elementary tetrahedron. Pacific J. Math. 211(2013), 123–155. http://dx.doi.Org/10.2140/pjm.2003.211.123 Google Scholar

[LTZ93] [LTZ93] Liu, J. Y., Trotter, L. E., Jr., and Ziegler, G. M., On the height of the minimal Hilbert basis. Results Math. 23(1993), no. 3-4, 374–376. http://dx.doi.Org/! 0.1007/BF03322309 Google Scholar