A remark on Fano threefolds with canonical Gorenstein singularities

Gorenstein Fano threefolds with base points in the anticanonical system

We classify all Gorenstein Fano threefolds with at worst canonical singularities for which the anticanonical system has a nonempty base locus.

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

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.

非Gorenstein的极小Fano三维簇的反canonical次数

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。

Rationality of Fano threefolds with terminal Gorenstein singularities, I

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

On classification of Q-Fano 3-folds with Gorenstein index 2 and Fano index 1/2

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 $ord f(Z,U)=1$.

Rationality of Fano threefolds with terminal Gorenstein singularities, II

We classify nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.

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.

Gorensteinness and iteration of Cox rings for Fano type varieties

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.

Gorenstein formats, canonical and Calabi-Yau threefolds

We extend the known classification of threefolds of general type that are complete intersections to various classes of non-complete intersections, and find other classes of polarised varieties, including Calabi-Yau threefolds with canonical singularities, that are not complete intersections. Our methods apply more generally to construct orbifolds described by equations in given Gorenstein formats.

具有Gorenstein终端奇点的Fano三维簇上的有理曲线

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。

关于具有canonical Gorenstein奇点的Fano三维流形的度数

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。

Rationality of $\mathbb{Q}$-Fano threefolds of large Fano index

We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.

Smooth Fano four folds in Gorenstein formats

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.

Threefold extremal curve germs with one non-Gorenstein point

An extremal curve germ is the analytic germ of a threefold with terminal singularities along a reduced complete curve admitting a contraction whose fibers have dimension at most one. The aim of the present paper is to review the results concerning those contractions whose central fiber is irreducible and contains only one non-Gorenstein point.

Fano threefolds of large Fano index and large degree

We classify Q-Fano threefolds of Fano index > 2 and big degree.

On the degree of Fano threefolds with canonical Gorenstein singularities, I

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: $-K_X^3\le 72$.

On the rational Enriques-Fano threefolds discovered by Fano

It was Fano who first classified Enriques-Fano threefolds. However his arguments appear to contain several gaps. In this paper, we will verify some of his assertions through the use of modern techniques.

On the Rationality of Fano-Enriques Threefolds

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.

Low codimension Fano--Enriques threefolds

In this paper we study Fano threefolds with a torsion divisor (Fano--Enriques). Due to this torsion divisor, they can be described as quotients of Fano threefolds by a finite abelian group action. We start from lists of Fano threefolds by Reid, Fletcher and Altınok and check which of them admit such an action with a Fano--Enriques quotient.

Q-Fano threefolds of large Fano index, I

We study Q-Fano threefolds of large Fano index. In particular, we prove that the maximum of Fano index is attained for the weighted projective space P(3,4,5,7).

Singularities of pluri-fundamental divisors on Gorenstein Fano varieties of coindex $4$

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.

Small toric degenerations of Fano threefolds

We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points.

On geometry of Fano threefold hypersurfaces

We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one.

Hyperbolicity of the moduli of certain Fano threefolds

We prove the Shafarevich conjecture for Fano threefolds of Picard rank 1, index 1 and degree 4.

Singular Fano threefolds of genus 12

We study singular Fano threefolds of type $V_{22}$.

关于大Fano指数的Q-Fano三维流形的理性质，I

On the birational geometry of $\mathbf{Q}$-Fano threefolds of large Fano index, I

本文研究Fano指数≥2的Q-Fano三维流形的理性问题，探讨其几何性质和分类方法。

On stable rationality of Fano threefolds and del Pezzo fibrations

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities

The main propose of this paper is to show that Bridgeland's moduli space of perverse point sheaves for certain flopping contractions gives the flops, and the Fourier-Mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories.

正特征下的Fano三重曲面IV

Fano threefolds in positive characteristic IV

本文在正特征下对Fano三重曲面进行了分类，延续了前几部分的研究工作。

Picard 数为一且具有 $\mathbb{K}^*$ 作用、轨道商为 $\mathbb{P}_2$ 的 Gorenstein Fano 三维簇

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。

K-stable Fano threefolds of rank 2 and degree 30

We find all K-stable smooth Fano threefolds in the family No. 2.22.

Conic bundle structures on Q-Fano threefolds

We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure.

Rationality of Fano threefolds of degree 18 over nonclosed fields

We study unirationality and rationality of Fano threefolds of degree 18 over nonclosed fields.

关于大Fano指数的Q-Fano三维流形的理性几何

On the birational geometry of $\mathbf{Q}$-Fano threefolds of large Fano index, II

本文研究Fano指数≥3的Q-Fano三维流形的理性问题，延续前文研究。

K-stability of Fano threefolds of rank 4 and degree 24

We prove that all smooth Fano threefolds of rank 4 and degree 24 are K-stable.

Fano threefolds with noncyclic torsion in the divisor class group

In this note we study Fano threefolds with noncyclic torsion in the divisor class group. Since they can all be obtained as quotients of Fano threefolds, we get also all examples that can be obtained as quotients of low codimension Fanos in the weighted projective space.

On the singularities and on the projective normality of some Enriques-Fano threefolds

In order to find useful information to complete the classification of Enriques-Fano threefolds, we will computationally study the singularities of some known Enriques-Fano threefolds of genus 6, 7, 8, 9, 10, 13 and 17. We will also deduce the projective normality of these threefolds.

正特征中的Fano三维流形 III

Fano threefolds in positive characteristic III

本文分类了正特征下Picard数至少为二以及Picard秩为二的原始Fano三维流形。

Fano threefolds with infinite automorphism groups

We classify smooth Fano threefolds with infinite automorphism groups.

Terminal Fano threefolds and their smoothings

We study smoothings of Fano threefolds. We prove that the Picard number remains constant in the case of terminal Gorenstein singularities.

Fano三维簇的等变单有理性

Equivariant unirationality of Fano threefolds

研究有限群在Fano三维簇上作用的单有理性，探讨其等变性质。

On the number of singular points of factorial terminal Fano threefolds

We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.

Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre Demailly

We compute global log canonical thresholds of some smooth Fano threefolds.

Picard 秩为一的 Fano 三维流形上的秩二弱 Fano 丛

Rank two Weak Fano bundles on Fano threefolds of Picard rank one

本文对 Picard 秩为一的 Fano 三维流形上的秩二向量丛进行分类，确定其射影化为弱 Fano 的情形，并证明各类例子的存在性，同时给出在二次三维流形上的详细消解。

$\mathbb{Q}$-Fano threefolds with three birational Mori fiber structures

In this paper we give first examples of $\mathbb{Q}$-Fano threefolds whose birational Mori fiber structures consist of exactly three $\mathbb{Q}$-Fano threefolds. These examples are constructed as weighted hypersurfaces in a specific weighted projective space. We also observe that the number of birational Mori fiber structures does not behave upper semi-continuously in a family of $\mathbb{Q}$-Fano threefolds.

Non-rationality of the symmetric sextic Fano threefold

We give a simple proof of the non-rationality of the Fano threefold defined by the equations Σx_i = Σx_i^2 = Σx_i^3 = 0 in P^6 .

Tangent scrolls in prime Fano threefolds

In this paper we prove that any smooth prime Fano threefold, different from the Mukai-Umemura threefold, contains a 1-dimensional family of intersecting lines. Combined with a result of the second author (see J. Algebr. Geom. 8:2 (1999), 221-244) this implies that any morphism from a smooth Fano threefold of index 2 to a smooth Fano threefold of index 1 must be constant, which gives an answer in dimension 3 to a question stated by Peternell.

On Q-Fano threefolds of Fano index 2

We show that, for a Q-Fano threefold X of Fano index 2, the inequality dim |-1/2K_X| <= 4 holds with a single well understood family of varieties having dim |-1/2K_X| = 4.

正特征下的105族Fano三维簇

105 families of Fano threefolds in positive characteristic

在正特征代数闭域上，证明光滑Fano三维簇共有105个不可约形变族。

秩为2且次数为28的法诺三维簇的自同构群

Automorphisms of Fano threefolds of rank 2 and degree 28

本研究针对秩2、次数28的光滑法诺三维簇，明确了其自同构群为有限时的具体结构，填补了该类代数簇自同构群研究的对应空白。

Birationally rigid Fano threefold hypersurfaces

We prove that every quasi-smooth hypersurface in the 95 families of weighted Fano threefold hypersurfaces is birationally rigid.

The defect of Fano 3-folds

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.

Toric $G$-Fano Threefolds

We classify toric Fano threefolds having at worst terminal singularities such that a rank of a $G$-invariant part of a class group equals one, where $G$ is a group acting on the variety by automorphisms.

Birational Unboundedness of $\Bbb Q$-Fano threefolds

In this paper, we prove that the family of $\Bbb Q$-Fano threefolds with Picard number one is birationally unbounded.