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.
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}.
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.
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.
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.
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.
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.
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 $\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.
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
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Michigan State University（密歇根州立大学）；Peking University（北京大学）
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…
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…
We review the present state of the problem, for each degree $d$ and genus $g$, is the Hilbert scheme of locally Cohen--Macaulay curves in ${\mathbb P}^3$ connected?
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.
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, …
Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties, including combinatorial type and degree. We also talk about the moduli space, a geometric object that parameterizes all possible types of abstract or planar tropical curves subject to certain conditions. Our research focuses on the moduli spaces of planar tropical curves of genus one, arbitrary degree d and any number of marked, unbounded edges. We prove that these …
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…
$\mathbb{A}^1$-connectedness of moduli of semistable bundles and symplectic bundles on a curve
Harish-Chandra Research Institute（Harish-Chandra 研究所）；Homi Bhabha National Institute（霍米·巴哈国家研究所）；School of Mathematical Sciences, National Institute of Science Education and Research（国家科学教育与研究学院数学科学学院）
It is a result of Gruson and Peskine that the invariants of a set points in $\ptwo$ in general position are connected. Associated to a space curve there are sequences of invariants which generalize the invariants of points in $\ptwo$. The main result of this paper is to show that the invariants of reduced, irreducible, non-degenerate curves in $\pthree$ also satisfy a connectedness property. This result greatly restricts the kinds of Borel-fixed monomial ideals which can occur as generic initial ideals of such curves and thus gives us more control over their Hilbert functions.
A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in characteristic $p$. We prove the connectedness of the moduli spaces of $\mathbb{A}^1$-curves with given degree, classify their irreducible components, and exhibit the inseparable $\mathbb{A}^1$-connectedness via the $\mathbb{A}^1$-curves parameterized by each irreducible component. The key to these results is the strangeness of all $\mathbb{A}^1$-curves. As an …
$\mathbb{A}^1$-connectedness of moduli of vector bundles on a stacky curve
Indian Institute of Science Education and Research Tirupati（印度科学教育研究所蒂鲁帕提分校）；National Institute of Science Education and Research, HBNI（印度国家科学教育研究中心，HBNI）
We show that limit linear series spaces for chains of curves are reduced. Using new advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification at up to two points. We find that in general, these spaces may not be connected even when they have positive dimension, but we prove a criterion for connectedness which generalizes the theorem previously proved by Fulton and Lazarsfeld in the case without imposed ramification.
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).
We study some examples of Bridgeland-Douglas stability conditions on triangulated categories. From one side we give a complete description of the stability manifolds for smooth projective curves of positive genus. From the other side we study stability conditions on triangulated categories generated by an exceptional collection. In the case of the projective line this leads to the connectedness and simply-connectedness of the stability manifold.
Let $(X,D)$ be a pair where $X$ is a projective variety. We study in detail how the behavior of rational curves on $X$ as well as the positivity of $-(K_X+D)$ and $D$ influence the behavior of rational curves on $D$. In particular we give criteria for uniruledness and rational connectedness of components of $D$.
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. …
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…
By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface, a projective homogeneous space admits a rational point if the elementary obstruction vanishes.
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.
We study the moduli space $\mathcal{AS}_{g}$ of Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of positive characteristic $p$. The moduli space is partitioned by irreducible strata, where each stratum parameterizes Artin-Schreier curves whose ramification divisors have the same coefficients. We construct deformations of these curves to study the relations between those strata. As an application, when $p=3$, we prove that $\mathcal{AS}_{g}$ is connected for all possible $g$. When $p>3$, it turns out that $\mathcal{AS}_{g}$ is connected for sufficiently large value of …
Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the separable rational connectedness of a large class of varieties, including general Fano complete intersections of hypersurfaces of degree at least three in flag varieties, in arbitrary characteristic. In addition, we give a new way of computing the normal bundle of certain rational curves in products of varieties in terms of their restricted tangent bundles and nor…
Rajiv Gandhi Institute of Petroleum Technology（拉吉夫·甘地石油技术研究所）；Indian Institute of Technology (Indian School of Mines) IIT (ISM) Dhanbad（印度理工学院（印度矿业学院）丹巴德分校）；Indian Institute of Technology(BHU), Varanasi（印度理工学院（贝拿勒斯 Hindu 大学），瓦拉纳西）
A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all p…
We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We show that for every $k \geq 1$ there are ${\rm gcd}(r, °L)$ unobstructed components. In addition, if $k$ is divisible by $r_1(r-r_1)(g-1)$ for $1\le r_1\le r-1$, there is an additional obstructed component of the expected dimension for each such $r_1$. We construct families of obstructed components and show that their generic point is not the generic vector…
Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is non-singular, then this rational map is a morphism. As a consequence, we obtain that, if the surface is rational and we fix a divisor class containing a non-singular rational curve, then the scheme parametrizing rational curves on this class is irreducible. Further, if the class has non-negative self-intersection, then the scheme of rational curves has expecte…
Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.
We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain polyhedral cones which we describe explicitly. Not all homology classes in the cone correspond to rational curves, but all classes sufficiently far from the walls of the cone do. The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.
