Families of rationally connected varieties

By T. Graber, J. Harris, J. Starr. 2001

546 citations recorded by the source corpus.

Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X. This is clearly a birationally invariant property. When X is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset Γ ⊂ X, there is a map g : P → X whose image contains Γ and such that gTX is an ample bundle.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/714f620d9e4933b920c85a02c7493d397a7547d4

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Citations in the corpus

Varieties over a finite field with trivial Chow group of 0-cycles have a rational point (2002, background): “3 can be compared to the main theorems of [12], resp.”
Weak approximation over function fields (2004, background): “The Hasse principle says that X has an F -rational point provided it has a rational point over each completion of F .”
Families of rationally simply connected varieties over surfaces and torsors for semisimple groups (2008, background): “For example, trying to use a property such as (1) above to prove (P) one runs into the problem of not having a result like [ GHS03 ] for families of ind-schemes.”
Threefold thresholds (2003, background): “By the main Theorem of [10], B is not uniruled.”
Dynamics of automorphisms on projective complex manifolds (2008, unknown): “Further, Z is non-uniruled by Graber-Harris-Starr [14] (1.”
Building blocks of étale endomorphisms of complex projective manifolds (2009, background): “If X is a uniruled nonsingular projective variety, then there exists uniquely up to birational equivalence, a rational fiber space π : X ···→Y over a nonsingular projective variety Y satisfying the following conditions, from [10, 22, 40].”
On Fano foliations (2011, unknown): “By [23], codimT (S) ≥ 2.”
Jordan property for Cremona groups (2012, background): “Rationally connected varieties enjoy the following important property (see [12, Corollary 1.3] for the proof over C ; the case of an arbitrary ﬁeld of characteristic 0 follows by the usual Lefschetz principle).”
A decomposition theorem for projective manifolds with nef anticanonical bundle (2017, background): “ leaf, we know by [Hör07, Cor.2.11] that there exists a smooth morphism ϕ: X→ Y such that TX/Y = F. Since −KX is nef and ϕis smooth, we know that −KY is nef [Deb01, Cor.3.15]. Since Y is not uniruled [GHS03], this proves that KY ≡ 0. We claim that ϕis locally trivial. By the Fischer-Grauert theorem ”
Rationally connected manifolds and semipositivity of the Ricci curvature (2012, background): “nce Xis uniruled by [BDPP]. We consider the quotient with maximal rationally connected ﬁbers (rational quotient or MRC ﬁbration, see [Cam92], [KMM92]) f: X > W to a smooth projective variety W. By [GHS01], Wis not uniruled, otherwise we could lift the ruling to Xand the ﬁbers of fwould not be max”
On the universal $CH_0$ group of cubic hypersurfaces (2014, background): “The morphism ΦZ has then rationally connected general fiber overC, and thus by [13], there is a lift σ : C → B such that ΦZ◦σ = j.”