A polar dual to the momentum of toric Fano manifolds

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.

Quadratic Gröbner bases arising from partially ordered sets

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.

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

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.