Fano fivefolds of index two with blow-up structure

Log Fano structures and Cox rings of blow-ups of products of projective spaces

The aim of this paper is twofold. Firstly, we determine which blow-ups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Secondly, we describe generators of the Cox rings of some cases.

三次五维体的库兹涅佐夫分量上的稳定性条件和模空间

Stability conditions and moduli spaces on the Kuznetsov component of cubic fivefolds

研究三次五维体的库兹涅佐夫分量，通过二次曲面纤维化构造塞尔不变的布里奇兰德稳定性条件，证明相关模空间非空等，得到拉格朗日浸入超凯勒簇，还恢复了伊利耶夫 - 马尼维尔的几何构造。

On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces

We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in products of projective spaces. We give a classification result for a special case. In the process of proof, we describe explicitly the structure of nef cones and compute the self intersection numbers of anti-canonical divisors for such weak Fano manifolds.

Fano manifolds and blow-ups of low-dimensional subvarieties

We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the possible cones of curves of these manifolds, and we prove that there is only one such manifold without a fiber type elementary contraction.

Toric varieties whose blow-up at a point is Fano

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a $T$-fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to $\PP^n$ or to the blow-up of $\PP^n$ along a $\PP^{n-2}$. As expected, such results are proved using toric Mori theory due to Reid.

On the existence of good divisors on Fano varieties of coindex 3

We prove that the linear system $|-1/3K_X| on a non-singular Fano fivefold $X$ of index 3 contains an irreducible divisor with only canonical singularities.

A Weak Fano Threefold Arising as a Blowup of a Curve of Genus 5 and Degree 8 on $\mathbb{P}^3$

This article constructs a smooth weak Fano threefold of Picard number two with small anti-canonical morphism that arises as a blowup of a smooth curve of genus 5 and degree 8 in $\mathbb{P}^3$. While the existence of this weak Fano was known as a numerical possibility in \cite{CM13} and constructed in BL12, this paper removes the dependencies on the results in \cite{JPR11} needed in the construction of BL12 and constructs the link in the style of ACM17.

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.

Pseudo-index of Fano manifolds and smooth blow-ups

Suppose $π: X \to Y$ is a smooth blow-up along a submanifold $Z$ of $Y$ between complex Fano manifolds $X$ and $Y$ of pseudo-indices $i_X$ and $i_Y$ respectively (recall that $i_X$ is defined by $i_X := \min \{-K_X \cdot C | C {is a rational curve of} X\}$). We prove that $i_X \leq i_Y$ if $2\dim (Z)< \dim(Y) + i_Y -1$ and show that this result is optimal by classifying the ``boundary'' cases. As expected, these results are obtained by studying rational curves on $X$ and $Y$.

Toward the classification of higher-dimensional toric Fano varieties

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite succession…

On the Griffiths Groups of Fano Manifolds of Calabi-Yau Hodge Type

A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a n…

Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

We characterise smooth curves in P^3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) $4d-30 \le g\le 14$ or $(g,d) = (19,12)$, (ii) there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C. This generalises the classical similar situation for the blow-up of points in P^2. We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.

The blow-up of $\mathbb{P}^4$ at 8 points and its Fano model, via vector bundles on a del Pezzo surface

Building on the work of Mukai, we explore the birational geometry of the moduli spaces M_{S,L} of semistable rank two torsion-free sheaves, with c_1=-K_S and c_2=2, on a polarized degree one del Pezzo surface (S,L); this is related to the birational geometry of the blow-up X of P^4 in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space Y=M_{S,-K_S}, which is a remarkable family of smooth Fano 4-folds. In particular we describe the relevant cones of divisors of Y,…

弱Fano三维流形作为超二次曲面在P^4中沿曲线的吹胀

Weak Fano threefolds arising as the blowup of a hyperquadric in $\mathbb{P}^4$ along a curve

本文研究了超二次曲面在P^4中沿特定曲线的吹胀所得弱Fano三维流形的条件和性质，揭示了其几何可实现性及构造方法。

Fano manifolds which are not slope stable along curves

We show that a Fano manifold (X,-K_X) is not slope stable with respect to a smooth curve Z if and only if (X,Z) is isomorphic to one of (projective space, line), (product of projective line and projective space, fiber of second projection) or (blow up of projective space along linear subspace of codimension two, nontrivial fiber of blow up).

光滑法诺超曲面的爆破、其双有理几何与除子稳定性

Blowups of smooth Fano hypersurfaces, their birational geometry and divisorial stability

针对光滑法诺超曲面$X$沿子超曲面$Γ$的爆破$Y$，构造性证明$Y$是Mori梦想空间，描述其Mori chamber分解与双有理模型，分类使$Y$为法诺流形的$X$和$Γ$并启动K稳定性研究。

Anticanonical geometry of the blow-up of $\mathbb{P}^4$ in $8$ points and its Fano model

Building on the work of Casagrande-Codogni-Fanelli, we develop our study on the birational geometry of the Fano fourfold $Y=M_{S,-K_S}$ which is the moduli space of semi-stable rank-two torsion-free sheaves with $c_1=-K_S$ and $c_2=2$ on a polarised degree-one del Pezzo surface $(S,-K_S)$. Based on the relation between $Y$ and the blow-up of $\mathbb{P}^4$ in $8$ points, we describe completely the base scheme of the anticanonical system $|{-}K_Y|$. We also prove that the Bertini involution $ι_Y$ of $Y$, induced by the Bertini involution $ι_S$ of $S$, preserves every member in $|{-}K_Y|$. In pa…

Fano manifolds of index n-2 and the cone conjecture

The Morrison--Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair have rational, polyhedral fundamental domains. In a previous paper, we proved the conjecture for certain blowups of Fano manifolds of index n-1. In this paper, we consider the Morrison--Kawamata conjecture for blowups of Fano manifolds of index n-2.

Fano symmetric varieties with low rank

The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are Fano. When $G$ is semisimple we classify also the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are only quasi-Fano. Moreover, we classify the Fano symmetric $G$-varieties of rank 3 obtainable from a wonderful variety by a sequence of blow-ups along $G$-stable varieties. Finally, we classify the Fano symmetric varieties of arbitrary rank …

Spherical blow-ups of Grassmannians and Mori Dream Spaces

In this paper we classify weak Fano varieties that can be obtained by blowing-up general points in prime Fano varieties. We also classify spherical blow-ups of Grassmannians in general points, and we compute their effective cone. These blow-ups are, in particular, Mori dream spaces. Furthermore, we compute the stable base locus decomposition of the blow-up of a Grassmannian in one point, and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the Grassmannian at the blown-up point, and by the rational normal curves in the Grassmannian passing throu…

Toric Fano varieties with divisorial contractions to curves

In this paper, we obtain a complete classification of smooth toric Fano varieties equipped with extremal contractions which contract divisors to curves for any dimension. As an application, we obtain a complete classification of smooth projective toric varieties which can be equivariantly blown-up to Fano along curves.

Toric Fano varieties and birational morphisms

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leqρ_X-ρ_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $ρ_X-ρ_D>0$ (with some additional hypotheses if $ρ_X-ρ_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $ρ_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension …

Some boundedness results for Fano-like Moishezon manifolds

We prove finiteness of the number of smooth blow-downs on Fano manifolds and boundedness results for the geometry of non projective Fano-like manifolds. Our proofs use properness of Hilbert schemes and Mori theory.

Variétés projectives complexes dont l'éclatée en un point est de Fano

We classify complex projective manifolds $X$ for which there exists a point $a$ such that the blow-up of $X$ at $a$ is Fano.

Stable Ulrich Bundles on Fano Threefolds with Picard Number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of $\mathbb{P}^{3}$, $Q$ (smooth quadric in $\mathbb{P}^{4}$), $V_{3}$ (smooth cubic in $\mathbb{P}^{4}$) or $V_{4}$ (complete intersection of two quadrics in $\mathbb{P}^{5}$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in $\mathbb{P}^{3}$. Also, we prove that there exist stable rank two Ulrich bundles with $c_{1}=3H$ on a generic member o…

On $K$-stability of $\mathbb{P}^3$ blown up along the disjoint union of a twisted cubic curve and a line

We find all K-polystable smooth Fano threefolds that can be obtained as blowup of projective space along the disjoint union of a twisted cubic curve and a line.

关于射影丛爆破的K-稳定性

On the K-stability of blow-ups of projective bundles

研究法诺簇上射影线丛爆破的K-稳定性，证明当除子B与2L成比例时，爆破的K-半稳定和K-多稳定等价于对数法诺对的相应性质，并给出系数；当比例非2时爆破K-不稳定。

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.

从素Fano三维流形沿两条线得到的初等链接

Elementary links from prime Fano threefolds along two lines

本文研究了素Fano三维流形沿两条不相交直线的吹胀操作，建立了与Fano三维流形的初等链接，并探讨了不同亏格下的几何构造与反向链接。

射影空间在环面不变点处爆破的反翻转

Anti-flips of the blow-ups of the projective spaces at torus invariant points

本文通过反翻转显式构造光滑环面法诺簇，使其同构于射影空间在余维一环面不变点处的爆破。

A note on the smooth blow-ups of P(1,1,1,k) in torus-invariant subvarieties

This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) } for any positive integer k, building on the work of Batyrev and Watanabe-Watanabe. This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore we identify birational relationships between entries of the classification. For a fixed value k>4, there are exactly two such Fano 3-folds linked by a blow-up in a torus-invariant line.

Fano manifolds with long extremal rays

Let X be a Fano manifold of pseudoindex i_X whose Picard number is at least two and let R be an extremal ray of X with exceptional locus Exc(R). We prove an inequality which bounds the length of R in terms of i_X and of the dimension of Exc(R) and we investigate the border cases. In particular we classify Fano manifolds X of pseudoindex i_X obtained blowing up a smooth variety Y along a smooth subvariety T such that dim T < i_X.

非交换沙漏 I：关于通过导出范畴对 Gorenstein 指数为 2 的 Q-Fano 3-fold 进行分类

Non-commutative hourglasses I: On classification of the Q-Fano 3-folds Gorenstein index 2 via Derived category

本文通过给出广义加权爆破的导出范畴公式，研究了Takagi簇在Sarkisov链接下的导出范畴变化行为，并利用非交换投影给出了例外收集及其几何意义。

关于沿五次椭圆曲线吹胀的 $\mathbb{P}^3$ 的 K-稳定性

On K-stability of $\mathbb{P}^3$ blown up along a quintic elliptic curve

本文研究沿五次光滑椭圆曲线吹胀三维射影空间所得光滑 Fano 三维流形的 K-稳定性，刻画了此类流形的稳定性性质。

关于秩2弱Fano三维流形分类的更新

An Update on the Classification of Rank 2 Weak Fano Threefolds

本文更新了皮卡德数二且小反典范映射的光滑弱Fano三维流形分类，通过爆破特定曲线构造几何实例，将开放情形从14个减少到4个。

Instanton bundles on two Fano threefolds of index $1$

We deal with instanton bundles on the product ${\mathbb P}^1\times{\mathbb P}^2$ and the blow up of ${\mathbb P}^3$ along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.

关于渐近对数Fano簇的丰富角体

On the body of ample angles of asymptotically log Fano varieties

本文将二维渐近对数Fano对的分类问题归结为特定吹胀上的一般性条件问题，并证明了任意维数下其丰富角体始终为有理多面体。

关于Fano最后Fano的K-稳定性

On K-stability of Fano's last Fanos

研究通过沿圆锥曲线爆破$\mathbb{P}^5$中两个二次曲面的光滑完全交得到的Picard秩2、次数22的光滑Fano三维流形的K-稳定性，并描述其自同构群。

Automorphisms of blowups of threefolds being Fano or having Picard number $1$

Let $X_0$ be a smooth projective threefold which is Fano or which has Picard number $1$. Let $π:X\rightarrow X_0$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow X_0$, on which any automorphism is of zero entropy. The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology class…

Fano箭图模空间的量子上同调

Quantum Cohomology of a Fano Quiver Moduli Space

本文计算了作为精细箭图模空间的指标为3的本原Fano 6-维簇$Y$的量子上同调环，推导出Schubert型子簇的量子Chevalley公式，并验证了Dubrovin猜想。

Sarkisov Links with centres space curves on smooth cubic surfaces

We construct and study Sarkisov links obtained by blowing up smooth space curves lying on smooth cubic surfaces. We restrict our attention to the case where the blowup is not weak Fano. Together with the results of arXiv:1106.3716 which cover the weak Fano case, we provide a classification of all such curves. This is achieved by computing all curves which satisfy certain necessary criteria on their multisecant curves and then constructing the Sarkisov link step by step.

High-pliability Fano hypersurfaces

We show that five of Reid's Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type $cA_n$ have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of $\mathbb{P}^2 \times \mathbb{P}^2$-type having Picard rank…

正特征下的 Fano 三维簇 II

Fano threefolds in positive characteristic II

本文研究正特征域上指数和 Picard 数均为一且反典范线性系极丰富的光滑 Fano 三维簇，确定其亏格范围并排除沿光滑曲线爆破仍为 Fano 的情形。

On Classification of $\mathbb{Q}$-Fano 3-folds of Gorenstein index 2. III

We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where the key varieties are constructed by extending partially the Sarkisov…

Fano manifolds of index n-1 and the cone conjecture

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair $(X, Δ)$ have finite, rational polyhedral fundamental domains. Let $Z$ be an $n$-dimensional Fano manifold of index $n-1$ such that $-K_Z = (n-1) H$ for an ample divisor $H$. Let $Γ$ be the base locus of a general $(n-1)$-dimensional linear system $V \subset |H|$. In this paper, we verify the Morrison-Kawamata cone conjecture for the blow-up of $Z$ along $Γ$.

A remark on Fano 4-folds having (3,1)-type extremal contractions

Let X be the blow-up of a smooth projective 4-fold Y along a smooth curve C and let E be the exceptional divisor. Assume that X is a Fano manifold and has an elementary extremal contraction $ϕ: X \to Z$ of (3,1)-type such that E is $ϕ$-ample (recall that a contraction map for a 4-fold is called (3,1)-type if the exceptional locus is a divisor and its image is a curve). We show that if the exceptional divisor of $ϕ$ is smooth, then Y is isomorphic to $\mathbb{P}^{4}$ and C is an elliptic curve of degree 4.

Instanton bundles on the blow up of the projective $3$-space at a point

We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $\mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blow up of $\mathbb P^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties: moreover, we also show that they correspond to smooth points of a component of the moduli space.

K-polystability of two smooth Fano threefolds

We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow up of the complete intersection $$ \Big\{x_0x_3+x_1x_4+x_2x_5=x_0^2+ωx_1^2+ω^2x_2^2+\big(x_3^2+ωx_4^2+ω^2x_5^2\big)+\big(x_0x_3+ωx_1x_4+ω^2x_2x_5\big)\Big\}\subset\mathbb{P}^5 $$ in the conic cut out by $x_0=x_1=x_2=0$, where $ω$ is a~primitive cube root of unity.

K-stability of Fano threefolds of rank 2 and degree 14 as double covers

We prove that every smooth Fano threefold from the family No 2.8 is K-stable. Such a Fano threefold is a double cover of the blow-up of $\mathbb{P}^3$ at one point branched along an anti-canonical divisor.

Rank 2 stable sheaves with odd determinant on Fano threefolds of genus 9

By the description due to Mukai and Iliev, a smooth prime Fano threefold X of genus 9 is associated to a surface P(V), ruled over a smooth plane quartic Gamma. We use Kuznetsov's integral functor to study rank-2 stable sheaves on X with odd determinant. For each c_2 \geq 7, we prove that a component of their moduli space M_X(2,1,c_2) is birational to a Brill-Noether locus of bundles on Gamma having enough sections when twisted by V. Moreover we prove that M_X(2,1,7) is isomorphic to the blowing-up of the Picard variety Pic^2(Gamma) along the curve parametrizing lines contained in X.

环面Fano簇的积分与有理Gongyo指数之间的差距

On the gap between the integral and rational Gongyo indices of toric Fano varieties

本文研究环面Fano簇的有理与积分Gongyo指数之差，证明维数下界并确定等号情形，构造无穷正差距族，并证明所有正有理数均可作为差距出现。

Fano manifolds with Lefschetz defect 3

Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the case where delta(X)=3. We show that there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. Then we apply the structure theorem to Fano 4-fo…

On the Fano variety of linear spaces contained in two odd-dimensional quadrics

In this paper we describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m-1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P^{2m+2}, for m>0. We show that there are exactly 2^{2m+2} distinct isomorphisms in codimension one between G and the blow-up of P^{2m} at 2m+3 general points, parametrized by the 2^{2m+2} distinct m-planes contained in Z, and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of G, as well as their dual cones of curves. Finally, we determin…

Birational geometry of singular Fano double spaces of index two

In this paper we describe the birational geometry of Fano double spaces $V\stackrelσ{\to}{\mathbb P}^{M+1}$ of index 2 and dimension $\geqslant 8$ with at mostquadratic singularities of rank $\geqslant 8$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected fibre space over a base of dimension $\geqslant 2$, that every birational map $χ\colon V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb P}^1$ induces an isomorphism $V^+\cong V'$ of the blow up $V^+$ of the variety $V$ along $σ^{…