Q-Fano threefolds of large Fano index, I

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.

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。

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。

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.

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.

Fano threefolds with canonical Gorenstein singularities and big degree

We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and the anticanonical degree (-KX)^3 equal 64.

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.

On G-Fano threefolds

We study Fano threefolds with~terminal singularities admitting a "minimal" action of a finite group. We prove that under certain additional assumptions such a variety does not contain planes. We also obtain an upper bounds of the number of singular points of certain Fano threefolds with~terminal factorial singularities.

实法诺三维簇的一个合理性准则

A rationality criterion for real Fano threefolds

该研究探讨光滑几何有理法诺三维簇实轨迹的连通性，提出并证明了一个实合理性的充分准则，为相关几何对象的合理性分析提供了理论依据。

Toric G-solid Fano threefolds

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

Fano threefolds with sections in $Ω_V^1(1)$

We study Fano threefolds with Picard number one equipped with a holomorphic section in $Ω_V^1(1)$.

Degree of irrationality of Fano threefold hypersurfaces

We study degree of irrationality of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces that have terminal singularities.

A remark on Fano threefolds with canonical Gorenstein singularities

We give some rationality constructions for Fano threefolds with canonical Gorenstein singularities.

Fano三重积分的Landau-Ginzburg模型及例外集合

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。

Calabi-Yau double coverings of Fano-Enriques threefolds

This note is a report on the observation that the Enriques-Fano threefolds with terminal cyclic quotient singularities admit Calabi-Yau threefolds as their double coverings. We calculate the invariants of those Calabi-Yau threefolds when the Picard number is one. It turns out that all of them are new examples.

Fano threefolds and K3 surfaces

We discuss in this note which K3 surfaces appear as anticanonical divisors in a Fano threefold. We prove in particular that a general K3 surface with given Picard lattice P and polarization class h in P is an anticanonical divisor in a Fano threefold if and only if (P,h) is isomorphic to (Pic(V), c_1(V)) for some Fano threefold V, where Pic(V) is equipped with the intersection product (L,M) --> (L.M.c_1(V)).

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.

关于具有零共正则性的光滑Fano三维流形

On smooth Fano threefolds with coregularity zero

本文构造了次数为2、4、6、8的光滑三维Fano完全交的零共正则性实例，并结合已有定理证明所有光滑Fano三维流形族均含零共正则性元素。

3.10家族中Fano三维流形的K-模尼

K-moduli of Fano threefolds in family 3.10

研究3.10家族中光滑Fano三维流形的K-多项稳定极限，揭示其几何结构与稳定性条件。

Rationality of Fano threefolds over non-closed fields

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.

Katzarkov-Kontsevich-Pantev猜想在Fano三维簇上的验证

Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds

本文验证了光滑Fano三维簇的Landau-Ginzburg模型满足Katzarkov-Kontsevich-Pantev猜想。

The zeta functions of the Fano surfaces of cubic threefolds

We give an algorithm to compute the zeta function of the Fano surface of lines of a smooth cubic threefold $F$ into $\mathbb{P}^4$ defined over a finite field. We obtain some examples of Fano surfaces with supersingular reduction.

三次Fano三维簇与14次Fano三维簇的稳定等变双有理等价

Stable equivariant birationalities of cubic and degree 14 Fano threefolds

本研究提出Pfaffian-Grassmannian对应的等变版本，以此构造出三次三维簇与14次Fano三维簇间非平凡的扭稳定等变双有理等价实例，为等变双有理几何提供新案例。

The degree of Q-Fano threefolds

We prove that the degree of Fano threefolds with terminal Q-factorial singularities and Picard number one is at most 125/2 and the bound is sharp.

On the existence of certain weak Fano threefolds of Picard number two

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.

On Fano threefolds with canonical Gorenstein singularities

We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.

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.

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.

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。

Derived category of V_{12} Fano threefolds

A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two exceptional bundles and of the derived category of a curve of genus 7. As an application we show that the Fano surface of X (the surface parameterizing conics on X) is canonically isomorphic to the symmetric square of the associated genus 7 curve.