Arnaud Beauville
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)).
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We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.
We classify Q-Fano threefolds of Fano index > 2 and big degree.
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 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.
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 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 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.
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。
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.
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.
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。
We prove that every quasi-smooth hypersurface in the 95 families of weighted Fano threefold hypersurfaces is birationally rigid.
We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and the anticanonical degree (-KX)^3 equal 64.
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.
In this paper, we prove that the family of $\Bbb Q$-Fano threefolds with Picard number one is birationally unbounded.
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.
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。
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.
We study Fano threefolds with Picard number one equipped with a holomorphic section in $Ω_V^1(1)$.
We study degree of irrationality of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces that have terminal singularities.
We give some rationality constructions for Fano threefolds with canonical Gorenstein singularities.
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。
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.
We classify all Gorenstein Fano threefolds with at worst canonical singularities for which the anticanonical system has a nonempty base locus.
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。
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。
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 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。
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.
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。
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.
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 three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.
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.
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.
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。
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.
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