Hiromichi Takagi
We obtain a classification of a Q-factorial Q-Fano 3-fold $X$ with the following properties: the Picard number of $X$ is 1; the Gorenstein index of $X$ is 2; the Fano index of $X$ is 1/2; $h^0 (-K_X) \geq 4$; there exists an index 2 point $P$ such that $(X,P)\simeq (\{xy +f(z^2,u)=0\} / \Bbb Z_2 (1,1,1,0), o)$ with $ord f(Z,U)=1$.
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We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.
On Gorenstein Fano toric complete intersections
本文分类了假加权射影空间中的Q-因子Gorenstein Fano非退化完全交三维流形。
We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and the anticanonical degree (-KX)^3 equal 64.
We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary $d$-dimensional $\mathbb{Q}$-factorial Gorenstein spherical Fano variety is bounded by $2d$. This paper also contains an overview of the description of the natural representative of the anticanonical divisor class of a spherical va…
We give some rationality constructions for Fano threefolds with canonical Gorenstein singularities.
We classify all Gorenstein Fano threefolds with at worst canonical singularities for which the anticanonical system has a nonempty base locus.
We classify the non-toric, $\mathbb Q$-factorial, Gorenstein, log terminal Fano threefolds of Picard number one that admit an effective action of a two-dimensional algebraic torus.
Degrees of non-Gorenstein canonical Fano threefolds with Picard number one
研究了Picard数为一的非Gorenstein Q-极小canonical Fano三维簇的反canonical次数上界，得出最佳上界为200/3。
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 nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.
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.
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 varieties of Fano type and Kawamata log terminal quasicones, iteration of Cox rings is finite with factorial master Cox ring. Moreover, we prove a relative version of Cox ring iteration for almost principal solvable $G$-bundles.
On the degree of Fano threefolds with canonical Gorenstein singularities
本文研究具有canonical Gorenstein奇点的Fano三维流形，证明了其度数的严格上界为72。
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 anti-canonical embedding in some weighted projective space. The constructed families have relatively smaller anti-canonical degrees than most other known families of smooth Fano 4-folds.
Rational curves on Fano threefolds with Gorenstein terminal singularities
本文研究具有Gorenstein终端奇点的Fano三维簇上的有理曲线，推广了几何Manin猜想至奇异情形，并证明了某些奇异del Pezzo三维簇的该猜想。
We consider Fano threefolds $X$ with canonical Gorenstein singularities. Under additional assumption that $X$ has at least one non-cDV point we prove a sharp bound of the degree: $-K_X^3\le 72$.
Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.
Gorenstein Fano threefolds of Picard number one with a $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$
本文分类了一类具有 $\mathbb{K}^*$ 作用和最大轨道商 $\mathbb{P}_2$ 的非环、$\mathbb{Q}$-因子、Gorenstein、Picard 数为一的 Fano 三维簇。
This paper studies the defect of terminal Gorenstein Fano 3 folds. I determine a bound on the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 that do not contain a plane. I give a general bound for quartic 3-folds and indicate how to study the defect of terminal Gorenstein Fano 3-folds with Picard rank 1 that contain a plane.
Smoothing Gorenstein toric Fano 3-folds
该研究针对三维自反多面体提出可容许闵可夫斯基分解数据的组合概念，证明其可确定对应Gorenstein环面Fano 3-簇的光滑化，并给出计算该光滑化贝蒂数的有效方法。
Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic
该研究在特征p>5的不可数代数闭域上，通过引入Gorenstein指数、补与Kodaira型零化条件，部分证明了三维簇的BAB猜想，同时给出了射影三维簇上nef大可逆层Seshadri常数的一致下界。
We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where the key varieties are constructed by extending partially the Sarkisov…
We give a characterization of Gorenstein toric Fano varieties of dimension $n$ with index $n$ among toric varieties. As an application, we give a strong version of Fujita's freeness conjecture and also give a simple proof of Fujita's very ampleness conjecture on Gorenstein toric varieties.
Anticanonical Volumes of Terminal and Canonical Gorenstein Fano Fourfolds
本文证明终端和典范 Gorenstein Fano 四折叠的反典范体积上界，并给出尖锐界及特殊情形下的改进。
We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.
We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.
On a combinatorial description of the Gorenstein index for varieties with torus action
本文证明复杂度一的环面作用 Fano 簇及一般排列簇的 Gorenstein 指数可由反典范复形的格距离读出，并据此界定近齐次 Fano 三维簇的定义数据。
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study the rationality questions for most of these varieties.
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.
We show that any Fano fivefold with canonical Gorenstein singularities has an effective anticanonical divisor. Moreover,if a general element of the anticanonical system is reduced, then it has canonical singularities. We also prove nonvanishing of anticanonical system in the case of log canonical Gorenstein singularities.
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local …
Non-commutative hourglasses I: On classification of the Q-Fano 3-folds Gorenstein index 2 via Derived category
本文通过给出广义加权爆破的导出范畴公式，研究了Takagi簇在Sarkisov链接下的导出范畴变化行为，并利用非交换投影给出了例外收集及其几何意义。
Weighted projective spaces admitting $\mathbb Q$-Gorenstein smoothings to Fano threefolds with Picard number one
研究加权射影三维簇允许Q-戈伦斯坦光滑化为Picard数一的光滑Fano三维簇，推导必要数值与局部条件，计算机搜索确定数值候选，构造若干光滑化并证明一个非光滑化例子，给出消失环准则保持Picard数一。
We prove the Conjecture of Catenese--Chen--Zhang: the inequality $K_X^3\geq \frac{4}{3}p_g(X)-\frac{10}{3}$ holds for all projective Gorenstein minimal 3-folds $X$ of general type.
Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial reflexive d-polytope has at most 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, it is known that equality for d even implies uniqueness up to unimodular equivalence. In this paper we completely classify all simplicial reflexive d-polytopes having 3d-1 vertices, corresponding to d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number 2d-1. For d even, there e…
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.
Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\uı and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list …
Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties
该研究推广了Eikelberg仿射依赖理论，证明维数不小于3的极化Gorenstein环面Fano簇对的Picard秩之和的上界，且该上界在极化对为单-单纯对时取到。
Sharp volume and multiplicity bounds for Fano simplices
该研究给出依赖维度和Gorenstein指数的法诺单纯形体积、马勒体积与重数的精确上界，提出高效分类方法并完成多种Gorenstein指数下四维及以下法诺单纯形的分类。
The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß 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-dimensi…
In this article, we prove that any Q-Calabi-Yau 3-fold with only ordinary terminal singularities and any Q-Fano 3-fold of Fano index 1 with only terminal singularities have Q-smoothings.
Let $X$ be a Gorenstein minimal $3$-fold of general type. We prove the optimal inequality: $$K_X^{3}\geq \frac{4}{3}χ(ω_X)-2,$$ where $χ(ω_X)$ is the Euler-Poincar$\acute{\text{e}}$ characteristic of the dualizing sheaf $ø_{X}$.
We completely classify toric weakened Fano 3-folds, that is, smooth toric weak Fano 3-folds which are not Fano but are deformed to smooth Fano 3-folds. There exist exactly 15 toric weakened Fano 3-folds up to isomorphisms.
We propose a refined but natural notion of toric degenerations that respect a given embedding and show that within this framework a Gorenstein Fano variety can only be degenerated to a Gorenstein Fano toric variety if it is embedded via its anticanonical embedding. This also gives a precise criterion for reflexive polytopes to appear, which might be required for applications in mirror symmetry. For the proof of this statement we will study polytopes whose polar dual is a lattice polytope. As a byproduct we generalize a connection between the number of lattice points in a rational convex polyto…
The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite succession…
We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.
We study canonically polarized Gorenstein $3$-folds with at most terminal singularities and satisfying $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$. We characterize the canonical maps of such $3$-folds, describe a structure theorem for the locally factorial ones and completely classify the smooth ones. New examples of canonically polarized smooth $3$-folds with $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$ are constructed. These examples are natural extensions of those constructed by M.~Kobayashi.
In this paper, we classify smooth toric Fano 5-folds of index 2. There exist exactly 10 smooth toric Fano 5-folds of index 2 up to isomorphisms.
On K-stability of singular hyperelliptic Fano 3-folds
本文研究了具有特定奇点和反canonical线性系性质的Fano 3折面的K稳定性问题。
We classify Fano threefolds with only Gorenstein terminal singularities and Picard number greater than 1 satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect to an action of some group $G$.
We prove that Fano 5-folds with nef tangent bundles are rational homogeneous manifolds.
In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study the structure of them, and in particular completely classify smooth toric special weak Fano 4-folds. As a result, we can confirm that almost every smooth toric special weak Fano 4-fold is a weakened Fano manifold, that is, a weak Fano manifold which can be deformed to a Fano manifold.
A Fano-Enriques threefold is a three-dimensional non-Gorenstein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano-Enriques threefolds with terminal cyclic quotient singularities. We investigate their rationality, and also provide an example of a Fano-Enriques threefold, whose pliability is 9, i.e. a Fano-Enriques threefold birationally equivalent to exactly 9 Mori fibre spaces in Sarkisov category.
We find an explicit upper bound for the anticanonical volume of Fano 4-folds with canonical singularities.
In this article, we prove that any $\Bbb Q$-factorial weak Fano 3-fold with only terminal singularities has a smoothing.
We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases. In the Fano case, we prove that the generalized Mukai conjecture holds.
In this paper, we investigate whether the 124 nonsingular toric Fano 4-folds admit totally nondegenerate embeddings from abelian surfaces or not. As a result, we determine the possibilities for such embeddings except for the remaining 21 nonsingular toric Fano 4-folds.
In this notes we classify toric Fano 4-folds having positive second Chern Character.
We use the computer algebra system Magma to study graded rings of Fano 3-folds of index >=3 in terms of their Hilbert series.
We study Q-factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of P^2 x P^2. These lie in codimension 4 in their total anticanonical embedding and have Picard rank 2. They fit into the current state of classification in three different ways. Some families arise as unprojections of degenerations of complete intersections, where the generic unprojection is a known prime Fano 3-fold in codimension 3; these are new, and an analysis of their Gorenstein projections reveals yet other new families. Others represent the "second Tom" unprojection families already…
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