Osamu Fujino, Wenfei Liu
We show that any union of slc strata of a Fano log pair with semi-log canonical singularities is simply connected. In particular, Fano log pairs with semi-log canonical singularities are simply connected, which confirms a conjecture of the first author.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Rational connectedness of log $Q$-Fano varieties
本文对极小模型程序中的一个猜想给予肯定的回答。证明了dim $n$的log $Q$-Fano簇是有理连通的.我们还研究了正则丛在射影态射下的行为。
On a general Fano hypersurface in projective space, we determine for infinitely many $k$ the minimal degree $e$ of a rational curve through a general collection of $k$ points. In the case of a hypersurface of index 1, our results hold for all $k\geq 1$. In an appendix, M.C. Chang proves an arithmetical result which implies that in the case of index $>1$, the density of the set of curve degrees $e$ covered by our method is approximately $\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.
Separable rational connectedness and stability
在这个简短的说明中，我们证明了在许多情况下，一个簇的失败是由切层的不稳定性引起的（如果没有其他明显的原因）。结果的一个简单应用证明了光滑Fano完全交是可分离有理连通的当且仅当它是可分离非圆的。特别地，一般的这样的Fano完全交是可分离有理连通的。
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 the non-klt locus of singularities of pairs. We show that given a pair $(X,B)$ and a projective morphism $X\to Z$ with connected fibres such that $-(K_X+B)$ is nef over $Z$, the non-klt locus of $(X,B)$ has at most two connected components near each fibre of $X\to Z$. This was conjectured by Hacon and Han. In a different direction we answer a question of Mark Gross on connectedness of the non-klt loci of certain pairs. This is motivated by constructions in Mirror Symmetry.
In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many further unirationality results for special moduli spaces of rational curves on quadric hypersurfaces and del Pezzo surfaces are obtained via explicit birational methods.
Deformations of weak Fano 3-folds with only terminal singularities
本文证明了任何只有终端奇点的$\Bbb Q$ -阶乘弱Fano 3-重都是光滑的.
On Fano threefolds with canonical Gorenstein singularities
我们分类的三维Fano品种与典型的Gorenstein奇点的程度大于64。
We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and the anticanonical degree (-KX)^3 equal 64.
A remark on Fano threefolds with canonical Gorenstein singularities
我们给出了具有典型Gorenstein奇点的Fano三重的一些合理性构造。
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.
In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the uniruling index for a smooth Fano variety of Picard number one to be separably rationally connected. As an application, we prove some weak approximation results for Fano complete intersections in positive charactersitic. For example, we show that weak approximation holds at place of strong potentially good reduction for a Fano complete intersection in $\mathbb…
We classify nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.
Campana rational connectedness and weak approximation
本文证明了复曲线上Campana纤维化在良好约化位置处Campana截面的弱逼近性质，并验证了环状Campana轨道体的更强有理连通性，关键工具为对数几何与稳定对数映射模栈。
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local …
On weak Fano varieties with log canonical singularities
证明了具有对数正则奇点的弱Fano 3-folds的反典范除子是半充分的.此外，我们还考虑了具有对数正则奇点的弱对数Fano对的反对数典范除子的半满性。通过构造几个例子，证明了半满性一般不成立。基于这些例子，我们提出了充分条件，这似乎是最好的可能性，我们证明了在这样的条件下半满。特别地，我们得到了lc中心至多为1维的对数正则弱Fano 4-folds的反典范除子的半满性.我们还研究了具有对数正则奇点的弱对数Fano对的Kleiman-Mori锥。
Birationally rigid Fano hypersurfaces with isolated singularities
证明了射影空间中具有一般位置的孤立奇点的指数为1的一般Fano超曲面是双有理刚性的。因此，它不能被一个有理映射纤维化为一个更小维度的非圆簇。双有理自映射群要么是平凡的，要么是二阶循环的。
On the degree of Fano threefolds with canonical Gorenstein singularities
本文研究具有canonical Gorenstein奇点的Fano三维流形，证明了其度数的严格上界为72。
Singularities of Cox Rings of Fano Varieties
设$X$是$\mathbb{C}$上的光滑完备Fano簇.证明了Cox环$\bigoplus_{L\in\text{Pic}(X)}H^0(X,\mathcal{O}_X(L))$是具有正则奇点的Gorenstein环.
Rational curves on Fano threefolds with Gorenstein terminal singularities
本文研究具有Gorenstein终端奇点的Fano三维簇上的有理曲线，推广了几何Manin猜想至奇异情形，并证明了某些奇异del Pezzo三维簇的该猜想。
Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.
Smoothings of Fano schemes with normal crossing singularities of dimension at most three
研究了具有维数不超过3的正规交叉奇点的Fano簇X的形变理论。我们得到了X的一阶变形层T^1（X）在X的适当对数分解及其奇异轨迹Z中的一个公式，并得到了X光滑存在的显式判据。
On the degree of Fano threefolds with canonical Gorenstein singularities, I
我们考虑具有正则Gorenstein奇点的Fano三重$X$。在进一步假设$X$至少有一个非cDV点的条件下，我们证明了其度的一个精确界：$-K_X^3\le 72$ .
Rational chain connectedness in the complex analytic setting
本文通过最小模型程序扩展了有理链连通性定理，证明了复分析kawamata log终端奇点的解析性纤维具有有理链连通性。
Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities. This is shown as a corollary to the characterization of a singularity that is obtained as a $K3^{[2]}$-type contraction and has a unique symplectic resolution.
Smoothings of Fano varieties with normal crossing singularities
本文给出了定义在特征为零的代数闭域上的具有正规交叉奇点的Fano簇X是光滑的判别准则。与原始版本的不同之处在于，本文利用对数结构和变形理论证明了X可被一个光滑簇光滑的充要条件是T^1（X）=O_D，其中D是X的奇异轨迹。
Deformations of Q-Calabi-Yau 3-folds and Q-Fano 3-folds of Fano index 1
本文证明了任何只具有普通终端奇点的Q-Calabi-Yau 3-fold和任何只具有终端奇点的Fano指数为1的Q-Fano 3-fold都有Q-smoothing。
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。
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.
Singularities on the base of a Fano type fibration
设$f\colon X\to Z$是Mori纤维空间. McKernan证明了$Z$的奇点是以$X$的奇点为界的。Shokurov将这一点推广到对：让$(X,B)$是一个klt对，$f\colon X\to Z$是一个收缩，使得$K_X+B\sim_\R 0/Z$和$f$的一般纤维都是Fano型变种;纤维空间的附加产生一个判别因子$B_Z$和一个关于$Z$的模因子$M_Z$。证明了$(Z,B_Z+M_Z)$的奇点是以$(X,B)$的奇点为界的。当$(F,\Supp B_F)$属于一个有界族时，证明了Shokurov猜想，其中$F$是$f$和$K_F+B_F=(K_X+B)|_F$的一般纤维.
Global smoothing of singular Fano and Calabi-Yau varieties
本文研究了Fano和Calabi-Yau品种的全局平滑问题，证明了在特定奇点条件下，这些品种可以通过变形获得更光滑的结构，并推广了已有结论。
We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous singularities of multiplicity asymptotically bounded by twice the square root of the dimension of the hypersurface, (2) hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree sufficiently close to the expected degree.
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.
We show that any Fano fivefold with canonical Gorenstein singularities has an effective anticanonical divisor. Moreover,if a general element of the anticanonical system is reduced, then it has canonical singularities. We also prove nonvanishing of anticanonical system in the case of log canonical Gorenstein singularities.
A type of the Lefschetz hyperplane section theorem on \Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities
利用退化方法证明了一类Picard数为1且具有$1/2(1,1,1)$ -奇点的Q-Fano 3-折叠上的Lefschetz超平面截口定理.作为副产品，我们得到了一个新的例子，Calabi-Yau 3-fold $X$的皮卡德数为1，其不变量是$$(H_X^3, c_2 (X) \cdot H_X, {e} (X)) = (8, 44, -88),$$，其中$H_X$，$e(X)$和$c_2(X)$分别是$\Pic(X)$的充分生成元，拓扑欧拉特征数和第二陈类$X$。
Rational connectedness modulo the Non-nef locus
众所周知，光滑射影Fano簇是有理连通的.最近Zhang（以及后来Hacon和McKernan作为他们关于Shokurov RC猜想的工作的特例）证明了同样的结论对一个klt对$(X,\D)$成立，使得$-(K_X+\D)$是大的和nef的。我们在这里证明了上述结果的一个自然推广，通过放弃nefness假设。也就是说，我们证明了一个klt对$(X,\D)$，使得$-(K_X+\D)$是大的是合理的模非nef轨迹的$-(K_X+\D)$。这个结果是一个更一般的结构定理的任意对$(X,\D)$与$-(K_X+\D)$ pseff的后果。
We determine restrictions on the singularity content of a Fano polygon, or equivalently of certain orbifold del Pezzo surfaces. We establish bounds on the maximum number of 1/R(1,1) singularities in the basket of residual singularities. In particular, there are no Fano polygons without T-singularities and with a basket given by (i) {k x 1/R(1,1)} where k is a positive integer and R>4, or (ii) {1/R1(1,1), 1/R2(1,1), 1/R3(1,1)}.
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.
We study log canonical thresholds (also called global log canonical threshold or $α$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $ε$, the set of Fano varieties of dimension $d$ with $ε$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a…
Kawamata-Miyaoka-type inequality for $\mathbb Q$-Fano varieties with canonical singularities II: Terminal $\mathbb Q$-Fano threefolds
本文证明了 Fano 指数至少为 $3$ 的终端 $\mathbb Q$-Fano 三维簇的最优 Kawamata-Miyaoka 型不等式，并应用得到所有终端 $\mathbb Q$-Fano 三维簇满足 $c_1(X)^3 < 3c_2(X)c_1(X)$。
We study deformations of a pair of a $\mathbb{Q}$-Fano $3$-fold $X$ with only terminal singularities and its elephant $D \in |{-}K_X|$. We prove that, if there exists $D \in |{-}K_X|$ with only isolated singularities, the pair $(X,D)$ can be deformed to a pair of a $\mathbb{Q}$-Fano $3$-fold with only quotient singularities and a Du Val elephant. When there are only non-normal elephants, we reduce the existence problem of such a deformation to a local problem around the singular locus of the elephant. We also give several examples of $\mathbb{Q}$-Fano $3$-folds without Du Val elephants.
Existence of complements for foliations
本文证明了klt环境簇上代数可积log-Fano叶状结构存在$\mathbb{Q}$-补，并研究了其伴随反转与连通性原理。
We prove the $W\mathcal{O}$-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic $p>5$. As a consequence, any klt Fano threefold over a finite field has a rational point.
Boundedness of Fano threefolds with log-terminal singularities of given index
证明了具有对数终端奇点的Fano三重方程的有界性定理。这是我们早期结果的改进，在那里我们还要求簇是Q阶乘的，Picard数为1。本文的创新之处在于：1.利用Alexeev极小模型程序，在适当的边界条件下，求出水平极值收缩。2.使用Kollár的有效基点自由度定理。3.利用Kawamata关于极值曲线长度的结果，在适当的边界条件下，避免了某些情况下曲线的粘连。
We study Kawamata log terminal singularities of full rank, i.e., $n$-dimensional klt singularities containing a large finite abelian group of rank $n$ in its regional fundamental group. The main result of this article is that klt singularities of full rank degenerate to cones over log crepant equivalent toric quotient varieties. To establish the main theorem, we reduce the proof to the study of Fano type varieties with large finite automorphisms of full rank. We prove that such Fano type varieties are log crepant equivalent toric. Furthermore, any such Fano variety of dimension $n$ contains an…
K-polystability and reduced uniform K-stability of log Fano cone singularities
研究证明log Fano锥奇点在特定条件下K-多项稳定性与特殊测试配置的K-多项稳定性等价，并建立其减少均匀K稳定性等价于K-多项稳定性。
We investigate some coboundary map associated to a $3$-dimensional terminal singularity which is important in the study of deformations of singular $3$-folds. We prove that this map vanishes only for quotient singularities and a $A_{1,2}/4$-singularity, that is, a terminal singularity analytically isomorphic to a $\mathbb{Z}_4$-quotient of the singularity $ (x^2+y^2 +z^3+u^2=0)$. As an application, we prove that a $\mathbb{Q}$-Fano $3$-fold with terminal singularities can be deformed to one with only quotient singularities and $A_{1,2}/4$-singularities. We also treat the $\mathbb{Q}$-smoothabi…
Hyperelliptic and trigonal Fano threefolds
我们分类的Fano 3-folds与典型Gorenstein奇点的反正则线性系没有基点，但不给嵌入，我们分类的反正则嵌入的Fano 3-folds与典型Gorenstein奇点不是二次曲面的交集。我们还研究了大多数这些品种的合理性问题。
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。
A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half…
Fano-Mori contractions of high length on projective varieties with terminal singularities
设X是一个具有终端奇点的射影簇，L是X上的一个充分Cartier因子.证明了：如果f是一个与极值射线$ R \subset \bar {NE(X)}$相关联的双有理压缩，使得R.（K_X+（n-2）L）<0，则f是光滑点的加权blow-up。然后，我们分类与极值射线R相关联的除数收缩，使得R。（K_X+rL）<0，其中r是非负有理数，并且f的纤维具有小于或等于r+1的维数。
Characterization of varieties of Fano type via singularities of Cox rings
证明了全局$F$ -正则型的Mori梦空间都是Fano型的.作为应用，我们给出了Fano型簇的Cox环奇异性的刻画.
We prove that the pull-back of a quasi-log scheme by a smooth quasi-projective morphism has a natural quasi-log structure. We treat an application to log Fano pairs. This paper also contains a proof of the simple connectedness of log Fano pairs with only log canonical singularities by Kento Fujita.
We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure.
We classify all $\mathbb{Q}$-factorial Fano intrinsic quadrics of dimension three and Picard number one having at most canonical singularities.
We find an explicit upper bound for the anticanonical volume of Fano 4-folds with canonical singularities.
Singularities of Landau-Ginzburg models for complete intersections and derived categories
本文在数值层面验证了镜像对称中关于Fano完全交的Landau-Ginzburg模型纤维奇点与例外集合对应的预期。
In this paper we extend to the singular setting the theory of Fano foliations developed in our previous paper. A Q-Fano foliation on a complex projective variety X is a foliation F whose anti-canonical class is an ample Q-Cartier divisor. In the spirit of Kobayashi-Ochiai Theorem, we prove that under some conditions the index i of a Q-Fano foliation is bounded by the rank r of F, and classify the cases in which i=r. Next we consider Q-Fano foliations F for which i=r-1. These are called del Pezzo foliations. We classify codimension 1 del Pezzo foliations on mildly singular varieties.
Kawamata--Miyaoka type inequality for $\mathbb{Q}$-Fano varieties with canonical singularities
针对余维二光滑、皮卡数为1的$n$维正规$\u211a$-因子典范奇点射影$\u211a$-Fano簇，证明其满足Kawamata--Miyaoka型不等式，并对带终端奇点的三维簇得到更强结果。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。