Pedro Núñez
Fano fibrations arise naturally in the birational classification of algebraic varieties. We show that these morphisms always induce a semiorthogonal decomposition on the derived category of the fibred space, extending classic results such as Orlov's projective bundle formula to the non-flat and singular case.
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We define, basing upon semiorthogonal decompositions of $\Db(X)$, categorical representability of a projective variety $X$ and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of $X$ by providing examples and stating open questions.
Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations
本文研究了正交格拉姆斯曼纤维的导出范畴的半正交分解，证明了在一定条件下，导出范畴嵌入到相对正交格拉姆斯曼的导出范畴中，并在特定情况下计算了残余范畴。
We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond.
We construct $S$-linear semiorthogonal decompositions of derived categories of smooth Fano threefold fibrations $X/S$ with relative Picard rank $1$ and rational geometric fibers and discuss how the structure of components of these decompositions is related to rationality properties of $X/S$.
We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $ε$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $ε$-Fano type.
A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at leas…
In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has "good" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geometry such as Fano varieties, Mori fibre spaces, flipping and divisorial contractions, crepant models, germs of singulari…
We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.
This article treats smooth weak Fano 3-folds having an extremal ray of type D. Smooth weak Fano 3-folds with an extremal ray of type D except of degree 6 are classified into 47 deformation types.
We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this kernel to its convolution with the kernel of the Serre functor. We investigate some basic properties of Hochschild homology and cohomology of admissible subcategories. In particular, we check that the Hoch…
We prove that a fibration X \to \Bbb P_1, the general fiber of which is a smooth Fano threefold, is rationally connected. The proof is based on a generalization of Tsen's classical theorem: a fibration X/C over a curve the general fiber of which is a Fano complete intersection in a product of weighted projective spaces has a section.
By Jahnke-Peternell-Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exists 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.
We study the group of relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstrass and Fano or anti-Fano fibrations we are able to describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determ…
Stein degree on non-Fano type fibrations
研究非Fano型log Calabi-Yau纤维上垂直除子的Stein度无界性。
On existence of bounded relative-global complements for Fano fibrations
本文证明了固定维数且具 $ε$-lc 奇点的 Fano 纤维化存在有界相对全局补，并在基为一维时进一步证明了有界相对全局 klt 补的存在性。
Smoothness criterion for Fano orbifolds and bases of Lagrangian fibrations
该研究建立了具商奇点的法诺簇的光滑性准则，并将其应用于证明紧超凯勒流形的拉格朗日纤维化若基具商奇点则基为射影空间。
We refine the classification of weak Fano threefolds with sextic del Pezzo fibrations by considering the Hodge numbers of them. By the refined classification result, such threefolds are classified into 17 cases. The main result of this paper is to show that there exists an example for each of 17 cases, except for 2 cases.
In this paper, we study the structure of Fano fibrations of varieties admitting an int-amplified endomorphism. We prove that if a normal $\mathbb{Q}$-factorial klt projective variety $X$ has an int-amplified endomorphism, then there exists an étale in codimension one finite morphism $\widetilde{X} \to X$ such that $\widetilde{X}$ is of Fano type over its albanese variety. As a corollary, if we further assume that $X$ is smooth and rationally connected, then $X$ is of Fano type.
Singularities on Fano fibrations and beyond
本文研究Fano型等纤维化的奇点问题，证明了McKernan、Shokurov及作者提出的相关猜想，推导出有理连通簇猜想的变体，给出多个代数几何问题的应用，并证明了控制纤维化纤维重数的关键一般性结果。
The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the Curve Exclusion Theorem, which first appeared in the author's unpublished PhD thesis [Ry02] and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.
On multiplicities of fibers of Fano fibrations
本文将birational几何中的多个猜想，包括Shokurov关于log Calabi-Yau纤维化基的奇点猜想和有理连通Calabi-Yau品种的有界猜想，归结为Fano纤维化过曲线的纤维乘积猜想。
We show boundedness of $3$-folds of $ε$-Fano type with Mori fibration structures. The proof is based on the birational boundedness result in our previous work arXiv:1509.08722 combining with arguments in Kawamata \cite{K} and Kollár--Miyaoka--Mori--Takagi \cite{KMMT}.
A complete classification is presented of elliptic and K3 fibrations birational to certain mildly singular complex Fano 3-folds. Detailed proofs are given for one example case, namely that of a general hypersurface X of degree 30 in weighted P^4 with weights 1,4,5,6,15; but our methods apply more generally. For constructing birational maps from X to elliptic and K3 fibrations we use Kawamata blowups and Mori theory to compute anticanonical rings; to exclude other possible fibrations we make a close examination of the strictly canonical singularities of (X,(1/n)H), where H is the linear system …
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 $δ_{X}=3$.
Stably semiorthogonally indecomposable varieties
提出非交换稳定半正交不可分解(NSSI)簇的概念，证明到阿贝尔簇有有限态射的簇是NSSI，并应用于证明某些簇中没有幻影子范畴。
In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations $(X,B)\to Z$ under mild conditions. We show that if $D$ is a component of $B$ with coefficient $\ge t>0$ which is contracted to a point on $Z$, then $D$ is birational to $\mathbb{P}^1\times C$ where $C$ is a smooth projective curve with gonality bounded depending only on $t$. Moreover, if $t>\frac{1}{2}$, then genus of $C$ is bounded depending only on $t$.
The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to the complete flag manifold $G/B$ where $G$ is a semi-simple Lie algebraic group and $B$ is a Borel subgroup of $G$.
Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes
本文确定了由Fano多面体导出的环面$K3$超曲面的Néron-Severi格，通过引入特定椭圆纤维化并证明其生成元，从而证明了Dolgachev猜想。
On Sun-Zhang's theory of Fano fibrations -- weighted volumes, moduli and bubbling Fano fibrations
本文从代数几何角度重审 Sun-Zhang 的一般 Fano 纤维化理论，并提出加权体积计算方法及“冒泡”构造的新结果与猜想。
Let X be a smooth, complex Fano variety. For every prime divisor D in X, we set c(D):=dim ker(r:H^2(X,R)->H^2(D,R)), where r is the natural restriction map, and we define an invariant of X as c_X:=max{c(D)|D is a prime divisor in X}. In a previous paper we showed that c_X<9, and that if c_X>2, then either X is a product, or X has a flat fibration in Del Pezzo surfaces. In this paper we study the case c_X=2. We show that up to a birational modification given by a sequence of flips, X has a conic bundle structure, or an equidimensional fibration in Del Pezzo surfaces. We also show a weaker prope…
Boundedness of klt complements on Fano fibrations over surfaces
本文证明 Birkar–Shokurov 关于 ε-lc 对局部有效 Cartier 除子补的猜想等价于 Shokurov 的 Fano 纤维化 klt 补有界性猜想，并在二维情形加以证实，从而推出曲面上 Fano 纤维化的 klt 补有界性。
We continue to study birational geometry of Fano fibrations $π\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of sufficient twistedness over the base, we prove birational rigidity (in particular, it means that there are no other structures of a fibration into rationally connected varieties) and compute their groups of birational self-maps. We considerably improve the principal components of the method of maximal singularities, in the first place, the technique of counting multip…
We prove that a general Fano fibration $π\colon V\to {\mathbb P}^1$, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on $V$ there are no other structures of a rationally connected fibration. The proof is based on the method of maximal singularities.
In our previous research, we constructed the affine varieties $Σ_{\mathbb{A}}^{13}$ and $Π_{\mathbb{A}}^{14}$ whose partial projectivizations admit $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations with relative Picard number one. In this paper, we produce prime quasi-smooth $\mathbb{Q}$-Fano 3-folds which are anticanonically embedded of codimension four and belong to 23 (resp.8) classes in the Graded Ring Database [GRDB], as weighted complete intersections in weighted projectivizations of $Σ_{\mathbb{A}}^{13}$ (resp.$Π_{\mathbb{A}}^{14}$ or its cone). We also show that a general member of the a…
We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X-->Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a non-zero, non-big movable divisor on X. Our main result is that if Y is not P^1 or P^2, then the Picard number rho(X) of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X-->Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a produ…
Let $f\colon X\to Z$ be a Mori fibre space. McKernan conjectured that the singularities of $Z$ are bounded in terms of the singularities of $X$. Shokurov generalised this to pairs: let $(X,B)$ be a klt pair and $f\colon X\to Z$ a contraction such that $K_X+B\sim_\R 0/Z$ and that the general fibres of $f$ are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor $B_Z$ and a moduli divisor $M_Z$ on $Z$. it is then conjectured that the singularities of $(Z,B_Z+M_Z)$ are bounded in terms of the singularities of $(X,B)$. We prove Shokurov conjecture when $(F,\Supp B_F)$ b…
We construct a $14$-dimensional affine variety $Σ^{14}_{\mathbb{A}}$ with a $\rm{GL}_3$- and a $(\mathbb{C}^*)^6$-actions. We denote by $Σ^{13}_{\mathbb{A}}$ the affine variety obtained from $Σ^{14}_{\mathbb{A}}$ by setting one specified variable to $1$ (we refer the precise definition to Definition 1.1 of the paper). We show that several weighted projectivizations of $Σ^{13}_{\mathbb{A}}$ and $Σ^{14}_{\mathbb{A}}$ produce, as weighted complete intersections, examples of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to $24$ classes of the graded ring database. Except No.360 …
Boundedness of complements for fibered Fano threefolds in positive characteristic
本文在正特征大特征情形下证明了Fano型纤维化的典范丛公式，并解决了具有纤维化结构的Fano型三维配对的Shokurov补集有界性猜想，特别覆盖了$-(K_X+B)$为nef且非big的情形。
A. Borisov classified into finitely many series the set of isomorphism classes of germs of toric $\Q$-factorial singularities, of fixed dimension and with minimal log discrepancy over the special point bounded from below by a fixed real number. We extend this classification to germs of toric Fano fibrations, possibly not $\Q$-factorial. As an application, we verify in the toric setting a conjecture proposed by V. V. Shokurov on the existence of bounded complements.
Stable Degenerations of log Fano Fibration Germs
本文证明了Sun-Zhang提出的log Fano纤维芽稳定退化猜想，引入了H-不变量，证明存在唯一准单项估值使H-不变量最小，并证明其关联环是有限生成的，诱导出特殊退化到K-semistable极化log Fano纤维芽。
Let $X/\mathbb{P}^1$ be a Mori fibre space with general fibre of Picard rank at least two. We prove that there is a proper closed subset $S\subsetneq X$, invariant by the connected component of the identity ${\rm Aut}^{\circ}(X)$ of the automorphism group of $X$, which is moreover the orbit of a section $s$ and whose intersection with a fibre is an orbit of the subgroup of ${\rm Aut}^{\circ}(X)$ acting trivially on $\mathbb{P}^1$. Such result is a tool to describe equivariant birational maps from $X/\mathbb{P}^1$ to other Mori fibre spaces and therefore finds its applications in the study of c…
We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories…
Singularities on vertical $ε$-log canonical Fano fibrations
本文在更弱的竖直ε-lc假设下，证明了Fano型对数Calabi-Yau纤维化经典范丛公式得到的广义配对是广义δ-lc的，推广了此前证实Shokurov猜想的结果。
It is conjectured that the base varieties of the Iitaka fibrations are bounded when the Iitaka volumes are bounded above. We confirm this conjecture for Iitaka $ε$-lc Fano type fibrations.
Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that the numerical invariants coincide for all 105. Our construction relies on a notion of toric degeneration for affine manifolds with singularities, and the correspondence we obtain between polytopes and Fano manifolds is compatible with that appearing in the work of Coates-Corti-K…
In this paper we study boundedness properties and singularities of log Calabi-Yau fibrations, particularly those admitting Fano type structures. A log Calabi-Yau fibration roughly consists of a pair $(X,B)$ with good singularities and a projective morphism $X\to Z$ such that $K_X+B$ is numerically trivial over $Z$. This class includes many central ingredients of birational geometry such as Calabi-Yau and Fano varieties and also fibre spaces of such varieties, flipping and divisorial contractions, crepant models, germs of singularities, etc.
In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M$^\rm{c}$Kernan) which roughly says that if $(X,B)\to Z$ is an $ε$-lc Fano type log Calabi-Yau fibration, then the singularities of the log base $(Z,B_Z+M_Z)$ are bounded in terms of $ε$ and $\dim X$ where $B_Z,M_Z$ are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if $X\to Z$ is a toric Fano fibration with $X$ being $ε$-lc, then the multiplicities of the fibres over codimension one…
On decompositions for Fano schemes of intersections of two quadrics
本文研究两个二次曲面交上线性子空间的 Fano 概形，提出以相伴超椭圆或叠曲线的对称幂表示的半正交分解猜想，并通过 Hodge 数公式与组合恒等式提供证据。
Kähler-Ricci shrinkers and Fano fibrations
本文用双有理代数几何证明Kähler-Ricci收缩孤立子是拟射影簇，并提出将其存在性与极化Fano纤维化K-稳定性及奇点切流相联系的统一猜想。
We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.
We construct a $15$-dimensional affine variety $Π_{\mathbb{A}}^{15}$ with a ${\rm GL}_{2}$- and $(\mathbb{C}^{*})^{4}$-actions. We denote by $Π_{\mathbb{A}}^{14}$ the affine variety obtained from $Π_{\mathbb{A}}^{15}$ by setting one specified variable to $1$ (we refer the precise definition to Definition 1.1 of the paper). Let $Π_{\mathbb{P}}^{13}$ be several weighted projectivizations of $Π_{\mathbb{A}}^{14}$, and $Π_{\mathbb{P}}^{14}$ the weighted cone over $Π_{\mathbb{P}}^{13}$ with a weight one coordinate added. We show that $Π_{\mathbb{P}}^{13}$ or $Π_{\mathbb{P}}^{14}$ produce, as weight…
Irrationality of degenerations of Fano varieties
本文研究了任意维数ε-lc Fano变种的无理度，证明了在特定条件下，中心纤维可通过有理映射到有界无理度的光滑射影变种，支持了一个关于log Fano纤维化猜想的泛化界限情况。
Fano fibrations and DK conjecture for relative Grassmann flips
本文针对相对Grassmann翻转，构造半正交分解并证明DK猜想，推广了Leung-Xie的结果，并得到Fano四重态的新完全例外集合。
We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.
Optimal bound for singularities on Fano type fibrations of relative dimension one
本文针对相对维数为1的Fano型纤维化，在Shokurov猜想的基础上，给出了任意$0<ε≤1$时$(Z,B_Z+M_Z)$为$δ$-lc的最优$δ$值，完善了该领域的奇点最优界研究。
We study the birational boundedness of special fibers of log Calabi-Yau fibrations and Fano fibrations. We show that for a locally stable family of Fano varieties or polarised log Calabi-Yau pairs over a curve, if the general fiber satisfies some natural boundedness conditions, then every irreducible component of the special fiber is birationally bounded.
The fibering genus of Fano hypersurfaces
本文拓展Kollár的思路，通过退化到正特征情形并利用Tate亏格变化公式，证明存在纤维化亏格可任意大的Fano超曲面，首次在Fano范围内研究簇的纤维化亏格问题。
Degenerations, fibrations and higher rank Landau-Ginzburg models
本文研究了准Fano簇的退化与高阶Landau-Ginzburg模型的镜像关系，通过拼接低阶模型来揭示其镜像性质，并推导出更高余维数的Calabi-Yau纤维化。
Let $V_1$ be the Fano threefold given as a hypersurface of degree 6 in $P(1,1,1,2,3)$ (over a number field $K$). Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.
We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev…
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