scholarly journals Generalized Tsen's theorem and rationally connected Fano fibrations

2002 ◽  
Vol 193 (10) ◽  
pp. 1443-1468 ◽  
Author(s):  
F Campana ◽  
Thomas Peternell ◽  
A V Pukhlikov
Keyword(s):  
Rationally Connected ◽  
Fano Fibrations

scholarly journals KODAIRA DIMENSION OF SUBVARIETIES II

2006 ◽  
Vol 17 (05) ◽  
pp. 619-631 ◽  
Author(s):  
THOMAS PETERNELL
Keyword(s):  
Normal Bundle ◽  
General Type ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Fundamental Groups ◽  
Weak Version ◽  
Joint Paper ◽  
Rationally Connected

This paper continues the study of non-general type subvarieties begun in a joint paper with Schneider and Sommese [14]. We prove uniruledness of a projective manifold containing a submanifold not of general type whose normal bundle has positivity properties and study moreover the rational quotient. We also relate the fundamental groups and a prove a cohomological criterion for a manifold to be rationally connected (weak version of a conjecture of Mumford).

scholarly journals Reflexive differential forms on singular spaces. Geometry and cohomology

2014 ◽  
Vol 2014 (697) ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Thomas Peternell
Keyword(s):  
Differential Forms ◽  
Extension Theorem ◽  
Singular Variety ◽  
Singular Spaces ◽  
Complex Spaces ◽  
Rationally Connected ◽  
Smooth Locus ◽  
Recent Extension ◽  
Rationally Connected Varieties

AbstractBased on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials.First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira–Akizuki–Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

scholarly journals On birational boundedness of Fano fibrations

2018 ◽  
Vol 140 (5) ◽  
pp. 1253-1276 ◽  
Author(s):  
Chen Jiang
Keyword(s):  
Fano Fibrations

scholarly journals Rationally connected rational double covers of primitive Fano varieties

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Aleksandr V. Pukhlikov
Keyword(s):  
Galois Group ◽  
Rational Maps ◽  
Fano Varieties ◽  
Rationally Connected ◽  
Double Covers

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers $X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties. Comment: the final journal version

scholarly journals FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY

2020 ◽  
Vol 8 ◽  
Author(s):  
NATHAN CHEN ◽  
DAVID STAPLETON
Keyword(s):  
Minimal Degree ◽  
Rational Map ◽  
Rationally Connected ◽  
General Complex ◽  
Special Fibers ◽  
Fano Index ◽  
Low Dimensional ◽  
The Way ◽  
Rationally Connected Varieties

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index  $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index  $e$ and dimension n is bounded from below by a constant times  $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

scholarly journals Normal bundles of rational curves on complete intersections

2019 ◽  
Vol 21 (02) ◽  
pp. 1850011 ◽  
Author(s):  
Izzet Coskun ◽  
Eric Riedl
Keyword(s):  
Complete Intersection ◽  
Normal Bundle ◽  
Rational Curves ◽  
Complete Intersections ◽  
Type Formula ◽  
Normal Bundles ◽  
Rationally Connected

Let [Formula: see text] be a general Fano complete intersection of type [Formula: see text]. If at least one [Formula: see text] is greater than [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. If all [Formula: see text] are [Formula: see text] and [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. As an application, we prove a stronger version of the theorem of Tian [27], Chen and Zhu [4] that [Formula: see text] is separably rationally connected by exhibiting very free rational curves in [Formula: see text] of optimal degrees.

scholarly journals Nef anti-canonical divisors and rationally connected fibrations

2019 ◽  
Vol 155 (7) ◽  
pp. 1444-1456
Author(s):  
Sho Ejiri ◽  
Yoshinori Gongyo
Keyword(s):  
Projective Variety ◽  
Kodaira Dimension ◽  
Canonical Divisor ◽  
Complex Projective Variety ◽  
Rationally Connected ◽  
Complex Projective

We study the Iitaka–Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka–Kodaira dimension of the anti-canonical divisor.

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