scholarly journals The number of vertices of a Fano polytope

2006 ◽  
Vol 56 (1) ◽  
pp. 121-130 ◽  
Author(s):  
Cinzia Casagrande
Keyword(s):  
Fano Polytope

scholarly journals A polar dual to the momentum of toric Fano manifolds

2021 ◽  
Vol 8 (1) ◽  
pp. 230-246
Author(s):  
Yuji Sano
Keyword(s):  
Fano Manifold ◽  
Fano Manifolds ◽  
Fano Polytope

Abstract We introduce an invariant on the Fano polytope of a toric Fano manifold as a polar dual counterpart to the momentum of its polar dual polytope. Moreover, we prove that if the momentum of the polar dual polytope is equal to zero, then the dual invariant on a Fano polytope vanishes.

scholarly journals Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Hidefumi Ohsugi ◽  
Kazuki Shibata
Keyword(s):  
Nous Avons ◽  
Ehrhart Polynomial ◽  
Ehrhart Polynomials ◽  
Nous Montrons ◽  
International Audience ◽  
Del Pezzo ◽  
Odd Cycles ◽  
Fano Polytope

International audience The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this extended abstract, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials. Les polytopes d'arêtes symètriques de cycles impaires (del Pezzo polytopes) sont connus sous le nom de polytopes de Fano lisses. Dans ce rèsumè ètendu, nous montrons que si la longueur du cycle est 127, alors le polynôme d'Ehrhart a une racine dont la partie rèele est plus grande que la dimension. En consèquence, nous avons un polytope de Fano lisse qui est un contre exemple à deux conjectures sur les racines de polynômes d'Ehrhart.

scholarly journals Quadratic Gröbner bases arising from partially ordered sets

2017 ◽  
Vol 121 (1) ◽  
pp. 19 ◽  
Author(s):  
Takayuki Hibi ◽  
Kazunori Matsuda ◽  
Akiyoshi Tsuchiya
Keyword(s):  
Convex Hull ◽  
Ordered Set ◽  
Convex Polytope ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Toric Ideals ◽  
Partially Ordered ◽  
Initial Ideals ◽  
Order Polytope ◽  
Fano Polytope

The order polytope $\mathcal {O}(P)$ and the chain polytope $\mathcal {C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ which is the convex hull of $\mathcal {O}(P) \cup (-\mathcal {C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.

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