Gorenstein Fano Research Articles

On Gorenstein Fano Threefolds with an Action of a Two-Dimensional Torus

We classify the non-toric, Q-factorial, Gorenstein, log terminal Fano threefolds of Picard number one that admit an effective action of a two-dimensional algebraic torus.

K-stability of Gorenstein Fano group compactifications with rank two

We give a classification of Gorenstein Fano bi-equivariant compactifications of semi-simple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler–Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kähler–Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the Kähler–Ricci flow is of type II.

Small toric resolutions of toric varieties of string polytopes with small indices

Let G be a semisimple algebraic group over [Formula: see text]. For a reduced word [Formula: see text] of the longest element in the Weyl group of G and a dominant integral weight [Formula: see text], one can construct the string polytope [Formula: see text], whose lattice points encode the character of the irreducible representation [Formula: see text]. The string polytope [Formula: see text] is singular in general and combinatorics of string polytopes heavily depends on the choice of [Formula: see text]. In this paper, we study combinatorics of string polytopes when [Formula: see text], and present a sufficient condition on [Formula: see text] such that the toric variety [Formula: see text] of the string polytope [Formula: see text] has a small toric resolution. Indeed, when [Formula: see text] has small indices and [Formula: see text] is regular, we explicitly construct a small toric resolution of the toric variety [Formula: see text] using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when [Formula: see text]. As a byproduct, we show that if [Formula: see text] has small indices then [Formula: see text] is integral for any dominant integral weight [Formula: see text], which in particular implies that the anticanonical limit toric variety [Formula: see text] of a partial flag variety [Formula: see text] is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold [Formula: see text] and obtain a formula of the disk potential of the Lagrangian torus fibration on [Formula: see text] obtained from a flat toric degeneration of [Formula: see text] to the toric variety [Formula: see text].

On the hyperplane conjecture for periods of Calabi–Yau hypersurfaces in $\mathbf{P}^n$

In [HLY1], Hosono, Lian, and Yau gave a conjectural characterization of the set of solutions to certain Gelfand–Kapranov–Zelevinsky hypergeometric equations which are realized as periods of Calabi–Yau hypersurfaces in a Gorenstein Fano toric variety $X$. We prove this conjecture in the case where $X$ is a complex projective space.

Gorenstein property for phylogenetic trivalent trees

We study the Gorenstein property for phylogenetic group-based models. We prove that the varieties associated to a trivalent tree and any of the groups Z3 and Z2×Z2 are Gorenstein Fano varieties, extending the results of Buczyńska and Wiśniewski for the group Z2.

The Minkowski Property and Reflexivity of Marked Poset Polytopes

We study the Minkowski property and reflexivity of marked poset polytopes. Both are relevant to the study of toric varieties associated to marked poset polytopes: the Minkowski property can be used to obtain generators of coordinate rings, while reflexive polytopes correspond to Gorenstein–Fano toric varieties.

Polyhedral realizations of crystal bases and convex-geometric Demazure operators

The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano.

Some examples of non-smoothable Gorenstein Fano toric threefolds

We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show examples of singular Gorenstein Fano toric threefolds which have compound Du Val, hence smoothable, singularities but are not smoothable.

Volume, Facets and Dual Polytopes of Twinned Chain Polytopes

Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with $|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q) \subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$ and $Q$ is the lattice polytope $\Gamma(\mathcal{C}(P),\mathcal{C}(Q)) \subset \mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup (-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes.

Classifying Fano complexity-one T-varieties via divisorial polytopes

The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and S\"u{\ss} to a correspondence between Gorenstein Fano complexity-one $T$-varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger's classification of Gorenstein del Pezzo $\mathbb{K}^*$-surfaces. Furthermore, we show that any three-dimensional Fano divisorial polytope is equivalent to one involving only eight functions.

Calabi\u2013Yau Volumes and Reflexive Polytopes

We study various geometrical quantities for Calabi–Yau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the Sasaki–Einstein base of the corresponding Calabi–Yau cone are calculated. By doing so, we conjecture new bounds for the Sasaki–Einstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence.

Reverse lexicographic squarefree initial ideals and Gorenstein Fano polytopes

Via the theory of reverse lexicographic squarefree initial ideals of toric ideals, we give a new class of Gorenstein Fano polytopes (reflexive polytopes) arising from a pair of stable set polytopes of perfect graphs.

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.

Facets and volume of Gorenstein Fano polytopes

AbstractIt is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension . Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.

Quadratic Gröbner bases of twinned order polytopes

Let P and Q be finite partially ordered sets on [d]={1,…,d}, and O(P)⊂Rd and O(Q)⊂Rd their order polytopes. The twinned order polytope of P and Q is the convex polytope Δ(P,−Q)⊂Rd which is the convex hull of O(P)∪(−O(Q)). It follows that the origin of Rd belongs to the interior of Δ(P,−Q) if and only if P and Q possess a common linear extension. It will be proved that, when the origin of Rd belongs to the interior of Δ(P,−Q), the toric ideal of Δ(P,−Q) possesses a quadratic Gröbner basis with respect to a reverse lexicographic order for which the variable corresponding to the origin is the smallest. Thus in particular if P and Q possess a common linear extension, then the twinned order polytope Δ(P,−Q) is a normal Gorenstein Fano polytope.

Centrally symmetric configurations of order polytopes

It is shown that the toric ideal of the centrally symmetric configuration of the order polytope of a finite partially ordered set possesses a squarefree quadratic initial ideal. It then follows that the convex polytope arising from the centrally symmetric configuration of an order polytope is a normal Gorenstein Fano polytope.

English

We give a characterizaion of Gorenstein toric Fano $n$-folds with index $n-1$, which is called Gorenstein toric Del Pezzo $n$-folds, among toric varieties. In practice, we obtain a condition for a lattice $n$-polytope to be a Gorenstein Fano polytope. In our proof, we do not use Batyrev-Juny's classification of Gorenstein Fano polytopes.

On the Picard Number of Singular Fano Varieties

Let X be a Q-factorial Gorenstein Fano variety. Suppose that the singularities of X are canonical and that the locus where they are non-terminal has dimension zero. Let D be a prime divisor of X. We show that rho_X - rho_D < 9 (where rho is the Picard number). Moreover, if rho_X - rho_D > 3, there exists a finite morphism from X to S x Y, where S is a surface with rho_S at most 9. As an application we prove that, if X has dimension 3, then rho_X is at most 10.

A classification of terminal quartic 3-folds and applications to rationality questions

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not $${\mathbb{Q}}$$ -factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program on X, a small $${\mathbb{Q}}$$ -factorialization of Y. In this case, the generators of Cl Y/ Pic Y are “topological traces” of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces $${Y_4 \subset \mathbb{P}^4}$$ with rk Cl Y = 2 and show that when rk Cl Y ≥ 6, Y is always rational.

Roots of Ehrhart polynomials arising from graphs

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart polynomials of polytopes of dimension $D$ satisfy $-D \le \Re(\alpha) \le D - 1$, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order $d$ lie in the circle $|z+\tfrac{d}{4}| \le \tfrac{d}{4}$ or are negative integers, and (2) a Gorenstein Fano polytope of dimension $D$ has the roots of its Ehrhart polynomial in the narrower strip $-\tfrac{D}{2} \leq \Re(\alpha) \leq \tfrac{D}{2}-1$. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.