Related papers: Gorenstein toric Fano varieties
We present a new class of affine Gorenstein 6-folds obtained by smoothing the 1-dimensional singular locus of a reducible affine toric surface; their existence is established using explicit methods in toric geometry and serial use of…
This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic…
We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.
We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Pl\"ucker polarization) and study the fixed loci.
An inductive approach to classifying toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688…
We list combinatorial criteria of some singularities, which appear in the Minimal Model Program or in the study of (singular) Fano varieties, for spherical varieties. Most of the results of this paper are already known or are quite easy…
We construct families of non-toric $\mathbb{Q}$-factorial terminal Fano ($\mathbb{Q}$-Fano) threefolds of codimension $\geq 20$ corresponding to 54 mutation classes of rigid maximally mutable Laurent polynomials. From the point of view of…
We study toric degenerations arising from Gr\"obner degenerations or the tropicalization of partial flag varieties. We produce a new family of toric degenerations of partial flag varieties whose combinatorics are governed by matching fields…
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds $F(n_1, ..., n_l, n)$. This construction includes our previous mirror construction for complete intersection in Grassmannians…
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to…
We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for…
A dissection of a polygon is obtained by drawing diagonals such that no two diagonals intersect in their interiors. In this paper, we define a toric variety of Schr\"{o}der type as a smooth toric variety associated with a polygon…
We investigate how ring isomorphisms between mirror closed states of compact toric Fano manifolds come from homological mirror symmetry. A natural closed-open relation on B-side is also discussed.
First, we examine the notion of nonrational convex polytope and nonrational fan in the context of toric geometry. We then discuss and interrelate some recent developments in the subject.
The anticanonical complex has been introduced as a natural generalization of the toric Fano polytope and so far has been succesfully used for the study of varieties with a torus action of complexity one. In the present article we enlarge…
We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan…
We study the anti-canonical ring of a projective variety and we characterise varieties of log Fano type depending on the singularities of these models.
We present some results on projective toric varieties which are relevant in Diophantine geometry. We interpret and study several invariants attached to these varieties in geometrical and combinatorial terms. We also give a B\'ezout theorem…
A matroid is Gorenstein if its toric variety is. Hibi, Laso\'n, Matsuda, Micha\l{}ek, and Vodi\v{c}ka provided a full graph-theoretic classification of Gorenstein matroids associated to simple graphs. We extend this classification to…