Related papers: $\mathbb{Q}$-Fano threefolds and Laurent inversion
We continue the classification of terminal Fano threefolds with an effective two-torus action. In earlier work we settled the Q-factorial case with Picard number one. Here we treat the larger class of varieties that do not admit any…
The Ap\'ery numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the…
We classify birationally rigid orbifold Fano 3-folds of index one defined by $5 \times 5$ Pfaffians. We give a sharp criterion for birational rigidity of these families based on the type of singularities that the varieties admit. Various…
For every complete toric variety, there exists a projective toric variety which is isomorphic to it in codimension one. In this paper, we show that every smooth non-projective complete toric threefold of Picard number at most five becomes…
Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an…
In our paper [Tak6], we constructed eight families of quasi-smooth prime $\mathbb{Q}$-Fano threefolds, anticanonically embedded in codimension four, using weighted projectivizations of the $14$-dimensional affine variety…
We show that $G$-Fano threefolds are mirror-modular. 1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of $SL_2(\mathbb{R})$. 2. Quantum periods, shifted by an integer constant (eigenvalue of quantum operator on…
We produce a list of 64 families of Fano fourfolds of K3 type, extracted from our database of at least 634 Fano fourfolds constructed as zero loci of general global sections of completely reducible homogeneous vector bundles on products of…
We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.
For any positive integer $k$ and any integer $n$ large enough, we construct a Fano variety $X$ with Picard number $k$ and dimension $n$ such that $((-K_X)^n)^{1/n}$ grows like $n^k/(\log n)^{k-1}$.
This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a…
T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this…
The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete…
We determine the complete list of anticanonically embedded quasi smooth log Fano 3-folds in weighted projective 4-spaces. This implies that the Reid-Fletcher list of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds…
We prove that for any e>0, there exists only finitely many e-log terminal spherical Fano varieties of fixed dimension. We also introduce an invariant of a spherical subgroup H in a reductive group G which measures how nice an equivariant…
We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold K\"{a}hler-Einstein metric and…
We construct some new deformation families of four-dimensional Fano manifolds of index $1$ in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the…
We study global log canonical thresholds on anticanonically embedded quasismooth weighted Fano threefold hypersurfaces having terminal quotient singularities to prove the existence of a Kahler-Einstein metric on most of them, and to produce…
We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak…
A directed edge polytope $\mathcal{A}_G$ is a lattice polytope arising from root system $A_n$ and a finite directed graph $G$. If every directed edge of $G$ belongs to a directed cycle in $G$, then $\mathcal{A}_G$ is terminal and reflexive,…