C. Casagrande, P. Jahnke, I. Radloff
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.
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We classify all smooth Calabi-Yau threefolds of Picard number two that have a general hypersurface Cox ring.
We prove the birational boundedness of Q-Fano varieties with Picard number one and dimension $n$ for every $n$. We do not use Mori theory.
We prove that Fano n-folds with nef tangent bundle and Picard number greater than n-5 are rational homogeneous manifolds.
It is conjectured that a Fano manifold of Picard number 1 which is not a projective space admits no endomorphisms of degree bigger than 1. Beauville confirmed this for hypersurfaces of projective space. We study this problem for hypersurfaces of an arbitrary Fano manifold of Picard number 1.
K-stability of Fano 3-folds of Picard rank 3 and degree 20
本文证明了满足明确普遍性条件的Picard秩3且度为20的光滑Fano三维簇的K-稳定性。
On locally factorial Fano fourfolds of Picard number two
本文分类了具有超曲面Cox环且允许三维环面有效作用的Picard数为2的局部析因Fano四重折叠。
We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.
We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.
In this paper, we classify Fano fourfolds whose the second exterior power of tangent bundles are numerically effective with Picard number greater than one.
This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number 1 with the exception of 12 numerical cases.
We classify Q-Fano threefolds of Fano index > 2 and big degree.
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。
We classify smooth Fano manifolds X with the Picard number $ρ_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.
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。
K-stability of Fano 3-folds of Picard rank 3 and degree 22
本文证明了满足非常显式一般性条件的Picard秩为3且次数为22的光滑Fano 3-fold的K-稳定性。
Using a construction due to C. Casagrande and further developed by the author, we prove that the Picard number of a non-smooth Fano 3-fold with isolated factorial canonical singularities, is at most 6.
On smooth Fano fourfolds of Picard number two
本文分类了具有一般超曲面Cox环且Picard数为2的光滑Fano四维簇。
Algebraic hyperbolicity of surfaces in Fano threefolds with Picard number one
本文研究皮卡数为1的一般法诺三重折叠中极一般曲面的代数双曲性，除加权超曲面中的曲面外，对其代数双曲性完成了完整分类。
One-nodal Fano threefolds with Picard number one
分类了Picard数为1的光滑Fano三维簇的所有1-节点退化，并揭示了其与更高Picard秩的光滑Fano三维簇及完全交簇投影之间的几何关系。
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.
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.
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.
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).
We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional assumptions.
Landau-Ginzburg models for Fano threefolds of Picard rank one and exceptional collections
本文研究Fano三重积分Picard秩为一的Landau-Ginzburg模型，探讨其与例外集合的关系，并验证Homological Mirror Symmetry的预测。
In this paper, we investigate the minimal length of chains of minimal rational curves needed to join two general points on a Fano manifold of Picard number 1. In particular, we give a sharp bound of the length by a fundamental argument. As an application, we compute the length for Fano manifolds of dimension < 8.
We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points.
We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one.
We prove the Shafarevich conjecture for Fano threefolds of Picard rank 1, index 1 and degree 4.
We study singular Fano threefolds of type $V_{22}$.
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。
We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.
We prove that the automorphism group of a Calabi-Yau threefold with Picard number three is either finite, or isomorphic to the infinite cyclic group up to finite kernel and cokernel.
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。
We find all K-stable smooth Fano threefolds in the family No. 2.22.
We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure.
We study unirationality and rationality of Fano threefolds of degree 18 over nonclosed fields.
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。
We prove that all smooth Fano threefolds of rank 4 and degree 24 are K-stable.
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.
In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.
We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X \ge (\dim X +3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.
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.
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。
We classify smooth Fano threefolds with infinite automorphism groups.
We study smoothings of Fano threefolds. We prove that the Picard number remains constant in the case of terminal Gorenstein singularities.
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。
We prove the conjectural Bogomolov-Gieseker type inequality for tilt slope stable objects on each Fano threefold X of Picard number one. Based on the previous works on Bridgeland stability conditions, this induces an open subset of geometric stability conditions on D^b(X). We also get a new stronger bound for the Chern characters of slope semistable sheaves on X.
We describe properties of the Kähler cone of general Calabi-Yau-threefolds with Picard number $ρ(X)=2$ and prove the rationality of the Kähler cone, if $X$ is a Calabi-Yau-hypersurface in a ${\mathbb P}^2$-bundle over ${\mathbb P}^2$ and $c_3(X)\le -54$. Without the latter assumption we prove the positivity of $c_2(X)$.
Actions on the Picard group of smooth Fano threefolds
该研究对光滑法诺三维簇自同构群在其皮卡群上作用的可能像分类，同时探究了这类簇族皮卡群的第一群上同调，属代数几何领域的相关工作。
We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.
We compute global log canonical thresholds of some smooth Fano threefolds.
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.
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 .
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.
The Picard number of a Fano manifold X obtained by blowing up a curve in a smooth projective variety is known to be at most 5, in any dimension greater than or equal to 4. We show that the Picard number attains to the maximal if and only if X is the blow-up of the projective space whose center consists of two points, the strict transform of the line joining them and a linear subspace or a quadric of codimension 2. This result is obtained as a consequence of a classification of special types of Fano manifolds.
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.
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。
We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
On K-moduli of Fano threefolds with degree 28 and Picard rank 4
本文研究次数28、Picard秩4的Fano三维簇的K-模，通过构造奇异环面K-多稳定Fano三维簇的代数形变，证明对应光滑Fano三维簇的K-模不可约分支为有理曲面。
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