Christopher D Hacon, James McKernan
We prove a conjecture of V. V. Shokurov which in particular implies that the fibers of a resolution of a variety with divisorial log terminal singularities are rationally chain connected.
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Let $(X,Δ)$ be a log pair over $S$, such that $-(K_X+Δ)$ is nef over $S$. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of $(X,Δ)$ with any fiber $X_s$ has at most two connected components. We prove this conjecture in dimension $\leq 4$ and in arbitrary dimension assuming the termination of klt flips.
In a previous paper the authors develop an intersection theory for subspaces of rational functions on an algebraic variety X over complex numbers. In this note, we first extend this intersection theory to an arbitrary algebraically closed ground field. Secondly we give an isomorphism between the group of b-divisors on the birational class of X and the Grothendieck group of the semigroup of subspaces of rational functions on X. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative approach to b-divisors and their intersection theory.
It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang (and later Hacon and McKernan as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair $(X,\D)$ such that $-(K_X+\D)$ is big and nef. We prove here a natural generalization of the above result by dropping the nefness assumption. Namely we show that a klt pair $(X,\D)$ such that $-(K_X+\D)$ is big is rationally connected modulo the non-nef locus of $-(K_X+\D)$. This result is a consequence of a more general structure theorem for arbitrary pair…
We use the Shokurov connectedness principle and the Corti inequality to prove the birational superrigidity of a nodal hypersurface in $\mathbb{P}^{5}$ of degree 5.
We discuss the strong rational connectedness of smooth rationally connected surfaces. We prove in lots of cases, including the smooth locus of a log del Pezzo surface, the rational connectedness indeed implies the strong rational conectedness. This confirms a conjecture due to Hassett and Tschinkel in \cite{ht08}.
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.
In this paper we prove two results about the rational chain connectedness for klt threefolds with anti-big canonical divisors in the relative setting.
The notion of 'slope rational connectedness' is introduced in the context of smooth orbifold pairs. The main result parallels the characterization of the rational connectedness of projective manifolds in terms of either the non-existence of holomorphic covariant tensors, or of absence of fibrations onto manifolds with pseudo-effective canonical bundle, or of existence of movable classes for which the minimal slope of the tangent bundle is positive. We then use this result to construct the `rational quotient map' in orbifold category.
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。
We prove birational superrigidity of generic Fano fiber spaces $V/{\mathbb P}^1$, the fibers of which are Fano complete intersections of index 1 and dimension $M$ in ${\mathbb P}^{M+k}$, provided that $M\geq 2k+1$. The proof combines the traditional quadratic techniques of the method of maximal singularities with the linear techniques based on the connectedness principle of Shokurov and Koll\' ar. Certain related results are also considered.
In this paper, we give an affirmative answer to a conjecture in the Minimal Model Program. We prove that log $Q$-Fano varieties of dim $n$ are rationally connected. We also study the behavior of the canonical bundles under projective morphisms.
In this paper, the technique of foliations in characteristic $p$ is used to investigate the difference between rational connectedness and separable rational connectedness in positive characteristic. The notion of being freely rationally connected is defined; a variety is freely rationally connected if a general pair of points can be connected by a free rational curve. It is proved that a freely rationally connected variety admits a finite purely inseparable morphism to a separably rationally connected variety. As an application, a generalized Graber-Harris-Starr type theorem in positive charac…
In this paper, we show that projective globally $F$-regular threefolds, defined over an algebraically closed field of characteristic $p\geq 11$, are rationally chain connected.
Campana separable rational connectedness of toric orbifold
本文证明了光滑非klt轨道折叠具有可分离Campana有理连通性，并展示了在正特征下，某些奇异加权射影空间不满足此性质。
In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no other obvious reasons). A simple application of the results proves that a smooth Fano complete intersection is separably rationally connected if and only if it is separably uniruled. In particular, a general such Fano complete intersection is separably rationally connected.
Let X be any generalized flag variety with Picard group of rank one. Given a degree d, consider the Gromov-Witten variety of rational curves of degree d in X that meet three general points. We prove that, if this Gromov-Witten variety is rationally connected for all large degrees d, then the structure constants of the small quantum K-theory ring of X vanish for large degrees.
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。
We prove the connectedness of the moduli space of maps (of fixed genus and homology class) to the homogeneous space G/P by degeneration via the maximal torus action. In the genus 0 case, the irreducibility of the moduli of maps is a direct consequence of connectedness. An analysis of a related Bialynicki-Birula stratification of the map space yields a rationality result: the (coarse) moduli space of genus 0 maps to G/P is a rational variety. The rationality argument depends essentially upon rationality results for quotients of SL2 representations proven by Katsylo and Bogomolov.
Connectedness principle for $3$-folds in characteristic $p>5$
本文在正特征代数几何中研究 Shokurov–Kollár 连通性原理，证明其对特征 $p>5$ 的三维簇成立，并刻画非 klt 轨迹不连通的情形。
On the connectedness principle and dual complexes for generalized pairs
本文证明 Shokurov–Kollár 连通性原理，刻画非 klt 轨迹纤维的不连通情形，并推广至广义配对，进而用于广义对数 Calabi–Yau 配对的对偶复形研究。
In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there exists a rational curve $f:\mathbb{P}^1\to X$ passing through $P_1,...,P_r$, disjoint from $S\setminus \{P_1,...,P_r\}$ (see Main Theorem). As a corollary, we prove that the smooth loci of complete toric varieties are strongly rationally connected.
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 $\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.
Unirationality is the same thing as Rational Connectedness in Characteristic Zero
本文证明特征零域上光滑射影变体的单有理性、有理连通性与有理链连通性等价，通过引入MU(X)并结合MRC纤维丛与归纳论证完成证明。
Rationally connectedness and O'Grady's generalized Franchetta conjecture for K3 surfaces
本文利用Bergeron--Li的同调结果，证明了在万有K3曲面有理连通时（含亏格22），O'Grady广义Franchetta猜想成立。
We define, for smooth projective orbifold pairs $(X,D)$ notions of `slope Rational connectedness', and of orbifold `slope Rational quotient' . These notions extend to this larger context the classical notions of rationally connected manifold and `rational quotient' (sometimes called `MRC fibration'). Our notions and proofs work entirely in characteristic zero, and are based on the consideration of foliations with minimal positive slope with respect to some suitable movable class. The existence of covering or connecting families of `orbifold rational curves' is indeed presently unknown in the o…
On rational chain connectedness of globally +-regular varieties
本文研究了全局$+$-正则簇的有理链连通性，证明了其在特定条件下的连通性，并引入了强全局$+$-正则的概念。
On the equivalence between the effective adjunction conjectures of Prokhorov-Shokurov and of Li
本文证明了Prokhorov-Shokurov与Li的有效伴随猜想等价，其核心是通过构建一致有理多胞体并利用代数可积叶状极小模型纲领的最新进展。
Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of the map $f$ have Picard number 1 and a further smoothness assumption is imposed. The main application is when $X \subset \PP^n$ is a smooth complete intersection of type $(d_1, ..., d_c)$ and $\sum d_i^2 \leq n$. In this case, we take $W$ to be the space of pointed lines contained in $X$ and the positive answer to the question implies that $X$ contains very t…
Campana Rational Connectedness and Weak Approximation of Del Pezzo Orbifolds
本文证明低次及具不可约边界的高次del Pezzo轨形族是强Campana无理性覆盖的，从而推出Campana有理连通性，为弱逼近定理提供二维非环面例子，并在附录中分类了次数6、7、8的情形。
Separable rational connectedness and $k$-plane sections of hypersurfaces
该研究证明了在特定条件下，超曲面具有可分离的有理连通性，并探讨了k-平面截面与线性截面空间的关系。
In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi-Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi-Yau or hyperkähler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least two contain a rational curve, …
We show that an $n$-dimensional compact Kähler manifold $X$ admitting a non-degenerate meromorphic map $f:{\bf C}^n\to X$ of order $ρ_f<2$ is rationally connected.
Unirationality is the Same as Rational Connectedness in Characteristic Zero
本文通过构造类似 MRC 纤维化的 MU 纤维化，证明了特征零域上光滑簇的单有理性、有理连通性与有理链连通性等价，并指出 MRC 商非单有理覆盖。
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。
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…
Naive $\mathbb{A}^1$-connectedness of retract rational varieties
本文在无限域情形下改进收缩有理簇的 $\mathbb{A}^1$-连通性结果，证明光滑紧致收缩有理簇具有朴素 $\mathbb{A}^1$-连通性。
We prove the irredcibility (and the rational connectedness) of the moduli spaces of (free) morphisms from a projective line to a successive blowing-up of a product of projective spaces if a suitable numerical condition on morphisms is provided (and also if all centers of the blowing-up are rationally connected).
Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$, Galois with group G, such that F is a regular extension of k (i.e. k is algebraically closed in F). Moreover, one may arrange that a given k-place of k(t) be totally split in F. Harbater proved this theorem for k an arbitrary complete valued field. Rather formal arguments ([10, §4.5]; \S2 hereafter) then imply that the theorem holds over any `large' field k. …
Rational connectedness for groups of proper projective similitudes
本文研究了特征不为2的域上二次型$φ$的真射影相似群${\bf PSim}^+(φ)$的有理连通性，给出了基于$φ$结构性质的充分条件，并构造了非有理连通的新二次型例子。
On rational connectedness and parametrization of finite Galois extensions
该研究围绕有理数域的$G$-伽罗瓦扩张的$R$-等价关系展开，确定了基本群族$G$的$R$-等价类，在$G$无通用扩张的情况下实现了对应伽罗瓦扩张的参数化。
Vanishing theorems and rational connectedness on holomorphic tensor fields
本文通过建立统一RC k-正的全纯向量丛的消散定理，证明了具有正k-瑞奇曲率的紧复流形的全纯切丛是统一RC k-正的，并应用该结果证明某些全纯张量场空间的平凡性及紧凯勒流形的有理连通性。
Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness
本文研究了积分点的Log Manin猜想及其与$\mathbb A^1$-连通性的联系，证明了Campana有理曲线和分裂托里除数上的Log Manin猜想，并结合Cox环描述和Batyrev的启发式计数方法给出几何解释。
In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of $k$-positive Ricci curvature and prove that for a projective $n$-dimensional Kähler manifold of $k$-positive Ricci curvature the MRC fibration has generic fibers of dimension at least $n-k+1$. We also establish an analogous result for projec…
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。
In this paper, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles and prove that, if $E$ is an RC-positive vector bundle over a compact complex manifold $X$, then for any vector bundle $A$, there exists a positive integer $c_A=c(A,E)$ such that $$H^0(X,\mathrm{Sym}^{\otimes \ell}E^*\otimes A^{\otimes k})=0$$ for $\ell\geq c_A(k+1)$ and $k\geq 0$. Moreover, we obtain that, on a compact Kähler manifold $X$, if $Λ^p T_X$ is RC-positive for every $1\leq p\leq \dim X$, then $X$ is projective and rationally connected. As applications, we show that if a compact Kähler ma…
We prove in this note a stabilized version of a conjecture on $\A^1$-connectedness. For the stabilized version of this conjecture, we introduce the notion of stable $\A^1$-connectedness, which is can be seen as the stabilization of $\A^1$-connectedness. The notion of stable $\A^1$-connectedness is in connection to the existence of zero cycles of degree one rather than rational points.
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 introduce and develop the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with trivial Bohr compactification, that is the class of minimally almost periodic groups. As an application we generalize some res…
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.
On global ACC for foliated threefolds
本文证明了叶状三重流形全局ACC的有理系数情形，通过引入广义叶状四元组建立任意维叶状典范丛公式，并将Shokurov全局指标猜想推广到叶状三元组。
A Hermitian metric $ω$ on a complex manifold is called SKT or pluriclosed if $dd^cω=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kähler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from rational connectedness of the twistor space, due to F. Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein.
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space of b-classes of higher codimension cycles, and endow this space with various Banach norms. Building on these constructions, we design a natural extension to higher dimensions of the Picard-Manin space introduced by Cantat and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the Hodge index theorem, and a surprising compactness result in…
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