Computing Fano 3-folds of index >=3

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.

Deformations of weak Fano 3-folds with only terminal singularities

In this article, we prove that any $\Bbb Q$-factorial weak Fano 3-fold with only terminal singularities has a smoothing.

Remarks on abelian surfaces in nonsingular toric Fano 4-folds

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.

Classification of Toric 2-Fano 4-folds

In this notes we classify toric Fano 4-folds having positive second Chern Character.

Birationally superrigid Fano 3-folds of codimension 4

We determine birational superrigidity for a quasi-smooth prime Fano 3-fold of codimension 4 with no projection centers. In particular we prove birational superrigidity for Fano 3-folds of codimension 4 with no projection centers which are recently constructed by Coughlan and Ducat. We also pose some questions and a conjecture regarding the classification of birationally superrigid Fano 3-folds.

Fano n-folds with nef tangent bundle and Picard number greater than n-5

We prove that Fano n-folds with nef tangent bundle and Picard number greater than n-5 are rational homogeneous manifolds.

Hodge numbers and deformations of Fano 3-folds

We calculate the Hodge numbers of quasismooth Fano 3-folds whose total anti-canonical embedding has small codimension, and relate these to the number of deformations.

On Fano indicies of Q-Fano 3-folds

This paper considers Q-Fano 3-folds X with ρ=1. The aim is to determine the maximal Fano index f of X. We prove that f<= 19, and that in case of equality, the Hilbert series of X equals that of weighted projective space PP(3,4,5,7). From the previous version, we restrict the list of all possibilities of X to the case f>8. It will appear in a forthcoming paper.

Nef and big divisors on toric weak Fano 3-folds

We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that all ample line bundles on a nonsingular toric weak Fano 3-fold are normally generated. We also show that an ample line bundle whose adjoint bundle has non-zero global sections on a Gorensein toric Fano 3-fold is normally generated. In our proof we do not use full classifications of Fano polytopes.

Sarkisov links for index 1 Fano 3-folds in codimension 4

We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.

Picard秩3且度为20的Fano三维簇的K-稳定性

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。

Constructing Q-Fano 3-folds à la Prokhorov & Reid

We generalise a construction by Prokhorov & Reid of two families of Q-Fano 3-folds of index 2 to obtain five more families of Q-Fano 3-folds; four of index 2 and one of index 3. Two of the families constructed have the same Hilbert series and we study these cases in more detail.

法诺三维加权完全交的K-稳定性与双有理刚性

K-stability and birational rigidity of Fano 3-fold weighted complete intersections

本研究证明了双有理刚性的良构拟光滑法诺三维加权完全交具有K-稳定性。

Fano 5-folds with nef tangent bundles and Picard numbers greater than one

We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.

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

Alpha invariants of birationally bi-rigid Fano 3-folds I

We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold Kähler-Einstein metric and are K-stable. As an application, we give examples of super-rigid affine Fano 4-folds.

Numerical obstructions to abelian surfaces in toric Fano 4-fold

This paper is withdrawn because it is superseded by arXiv:0710.4284

Fano 3-folds and double covers by half elephants

We construct a deformation family for each of the 34 Hilbert series of index 2 Fano 3-folds. In 18 cases we construct two different families, distinguished by the topology of their general members.

Picard秩为3且次数为22的Fano 3-fold的K-稳定性

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。

A note on the Picard number of singular Fano 3-folds

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 deformations of Q-Fano threefolds

We study the deformation theory of a Q-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a Q-Fano 3-fold is smooth. Second, we show that every Q-Fano 3-fold with only "ordinary" terminal singularities is Q-smoothable, that is, it can be deformed to a Q-Fano 3-fold with only quotient singularities. Finally, we prove Q-smoothability of a Q-Fano 3-fold assuming the existence of a Du Val anticanonical element. As an application, we get the genus bound for primary Q-Fano 3-folds with Du Val anticanonical elements.

On the birational geometry of toric Fano 4-folds

We announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the 3-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.

Derived Categories of Toric Fano 3-Folds via the Frobenius Morphism

In a work of Costa and Miró-Roig state the following conjecture: Every smooth complete toric Fano variety has a full strongly exceptional collection of line bundles. The goal of this article is to prove it for toric Fano 3-folds.

Numerical obstructions to abelian surfaces in toric Fano 4-folds

Some of the 124 toric Fano 4-folds contain abelian surfaces but most do not: in a few cases it is not known whether they do or not. By elementary methods, with a little computer help, we exclude some more possibilities.

Birationally rigid Pfaffian Fano 3-folds

We classify birationally rigid orbifold Fano 3-folds of index one defined by $5 \times 5$ Pfaffians. We give a sharp criterion for birational rigidity of these families based on the type of singularities that the varieties admit. Various conjectures are born out of our study, highlighting a possible approach to the classification of terminal Fano 3-folds. The birationally rigid cases are the first known rigid examples of Fanos that are not (weighted) complete intersection.

Deformations of weak $\mathbb{Q}$-Fano 3-folds

We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak $\mathbb{Q}$-Fano $3$-fold. We also treat a stacky proof of the unobstructedness of deformations of a $\mathbb{Q}$-Fano $3$-fold.

关于奇异双曲Fano 3折面的K稳定性

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。

Fano 3-folds from homogeneous vector bundles over Grassmannians

We rework the Mori-Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

Terminal toric Fano three-folds with certain numerical conditions

We completely classify the Q-factorial terminal toric Fano three-folds such that the sum of the squared torus invariant prime divisors is non-negative.

A characterization of some Fano 4-folds through conic fibrations

We find a characterization for Fano 4-folds $X$ with Lefschetz defect $δ_{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $ρ_{X}-ρ_{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $δ_{X}=3$.

Fano compactifications of contractible affine 3-folds with trivial log canonical divisors

T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this article, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.

Fano 3-folds of index 2

We study Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X) = 1, Q-factorial terminal singularities and -K_X = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties X,A and deduce both the nonvanishing of H^0(X,-K_X) and the sharp bound (-K_X)^3 >= 8/165. We list families that can be realised in codimension up to 4.

On toric geometry and K-stability of Fano varieties

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.

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.

满足条件(A)的光滑Fano 3-簇

Smooth Fano 3-folds satisfying Condition (A)

本研究针对满足条件(A)（自同构群的所有有限阿贝尔子群均有不动点）的光滑Fano 3-簇，完成了对应的分类工作，为代数几何领域相关簇的研究提供了分类结果。

Generalized Bogomolov-Gieseker type inequalities on Fano 3-folds

We modify the conjectural Bogomolov-Gieseker type inequality introduced by Bayer, Macri and Toda to construct a family of geometric Bridgeland stability conditions on smooth projective 3-folds. We give an equivalent conjecture which needs to check these inequalities for a small class of tilt stable objects. We extend some of the techniques in Li's work for Fano 3-folds of Picard rank one to establish our modified Bogomolov-Gieseker type inequality conjecture for general Fano 3-folds.

光滑 Fano 3-folds 的 K-模空间中一维分支

One-dimensional components in the K-moduli of smooth Fano 3-folds

本文通过在 4 个特定形变族中识别 K-多稳定极限，完成了可光滑化 Fano 3-folds 的 K-模空间中一维分支的分类。

On the minimal length of extremal rays for Fano 4-folds

The minimum of intersection numbers of the anti-canonical divisor with rational curves on a Fano manifold is called pseudo-index. It is expected that the intersection number of anti-canonical divisor attains to the minimum on an extremal ray, i.e. there exists an extremal rational curve whose intersection number with the anti-canonical divisor equals the pseudo-index. In this note, we prove this for smooth Fano 4-folds having birational contractions.

双有理超刚性Fano三维加权超曲面的K-稳定性

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

该研究证明指数为1的拟光滑Fano三维加权超曲面的α不变量≥1/2，结合已有双有理超刚性与K-稳定性的关系结论，推导出这类双有理超刚性曲面的K-稳定性。

Birational Mori fiber structures of Q-Fano 3-fold weighted complete intersections, II

In this paper we study 14 families among 85 families of anticanonically embedded Q-Fano threefolds weighted complete intersections of codimension 2 and show that every quasismooth member is birationally birigid, that is, it is birational to another Q-Fano threefold but not birational to any other Mori fiber space.

On the anti-canonical geometry of $\mathbb{Q}$-Fano 3-folds

For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map $φ_{-m}$ is birational onto its image for all $m\geq 39$. By a weak $\mathbb{Q}$-Fano 3-fold $X$ we mean a projective one with at worst terminal singularities on which $-K_X$ is $\mathbb{Q}$-Cartier, nef and big. For weak $\mathbb{Q}$-Fano 3-folds, we prove that $φ_{-m}$ is birational onto its image for all $m\geq 97$.

Fano 3-folds, K3 surfaces and graded rings

Explicit birational geometry of 3-folds represents a second phase of Mori theory, going beyond the foundational work of the 1980s. This paper is a tutorial and colloquial introduction to the explicit classification of Fano 3-folds (Q-Fano 3-folds), a subject that we hope is nearing completion. With the intention of remaining accessible to beginners in algebraic geometry, we include examples of elementary calculations of graded rings over curves and K3 surfaces. For us, K3 surfaces have at worst Du Val singularities and are polarised by an ample Weil divisor; they occur as the general elephant …

On terminal Fano 3-folds with 2-torus action, part II

We continue the classification of terminal Fano threefolds with an effective two-torus action. In earlier work we settled the Q-factorial case with Picard number one. Here we treat the larger class of varieties that do not admit any contraction of a prime divisor; these are called combinatorially minimal.

On some Fano 4-folds with Lefschetz defect 3

We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we discuss classification results for 4-dimensional complex smooth projective varieties admitting some particular fiber type contractions.

On the birational geometry for irreducible symplectic 4-folds related to the Fano schemes of lines

There is two group actions on the Fano scheme of lines such that the quotient becomes an irreducible symplectic manifold. We showed that both quotients are birational to the generalized Kummer variety or the 2-points Hilbert scheme of a K3 surface.

Fano 三维簇的多向量场

Polyvector fields for Fano 3-folds

本文计算 Fano 三维簇 Hochschild 上同调的 HKR 分解，以揭示其 Gerstenhaber 代数结构，并推进高 Picard 秩 Fano 三维簇上 Poisson 结构的分类。

Classification of Fano 4-folds with Lefschetz defect 3 and Picard number 5

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve.

Fano 4-folds, flips, and blow-ups of points

In this paper we study smooth, complex Fano 4-folds X with large Picard number rho(X), with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at a point, then rho(X) is at most 12. We give examples of such X with Picard number up to 9. The main theme (and tool) is the study of fixed prime divisors in Fano 4-folds, especially in the case rho(X)>6, in which we give some general results of independent interest.

Vector bundles on Fano 3-folds without intermediate cohomology

We study the vector bundles without intermediate cohomology on Fano threefolds of index two, degree d=3,4,5 and Betti number one. We obtain a complete characterization in the case of rank-two vector bundles. For arbitrary rank, we give all possible Chern classes of such vector bundles, under some general condition.

Constructing Fano 3-folds from cluster varieties of rank 2

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

双有理刚性的Fano 3-重超曲面

Birationally solid Fano 3-fold hypersurfaces

本文基于Picard数为1的Fano簇的双有理刚性定义，完成了拟光滑双有理刚性Fano 3-重加权超曲面的分类。

从有理曲线出发的例外Fano 3折

Exceptional Fano 3-folds from rational curves

本文通过构造特定的加权超曲面，展示了其作为例外Fano 3折在K-模边界中的位置，并证明其允许凯勒-爱因斯坦度量。

Mori dream spaces and birational rigidity of Fano 3-folds

We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.

无有理点del Pezzo曲面与Fano 3-簇的K-稳定性

K-stability of pointless del Pezzo surfaces and Fano 3-folds

该研究探究子域上Fano簇的有理点存在性与几何模型Kähler-Einstein度量的关联，证明无有理点的del Pezzo曲面及多数光滑Fano 3-簇的几何模型容许该度量，得到新例子并可反推部分簇的有理点存在性。