scholarly journals Nevanlinna-Pick Families and Singular Rational Varieties

2020 ◽  
pp. 129-145
Author(s):  
Kenneth R. Davidson ◽  
Eli Shamovich
Keyword(s):  
Rational Varieties

scholarly journals Bundles on with vanishing lower cohomologies

2020 ◽  
pp. 1-20
Author(s):  
Mengyuan Zhang
Keyword(s):  
Hilbert Function ◽  
Betti Numbers ◽  
Projective Spaces ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Rational Varieties ◽  
Minimal Free Resolutions ◽  
Graded Lattice

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

scholarly journals Theta Groups and Products of Abelian and Rational Varieties

2013 ◽  
Vol 57 (1) ◽  
pp. 299-304 ◽  
Author(s):  
Yuri G. Zarhin
Keyword(s):  
Elliptic Curve ◽  
Projective Line ◽  
Negative Answer ◽  
General Linear ◽  
Linear Groups ◽  
Rational Varieties ◽  
General Linear Groups ◽  
Finite Subgroups ◽  
Birational Automorphisms

AbstractWe prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the groups of birational automorphisms of products of an elliptic curve and the projective line. This gives a negative answer to a question posed by Vladimir L. Popov.

Construction de Variétés de Groupes Exceptionnels Non $k$-Rationnelles

2015 ◽  
Vol 116 (2) ◽  
pp. 182
Author(s):  
Philippe Gille
Keyword(s):  
Weak Approximation ◽  
Exceptional Groups ◽  
Rational Varieties

Our goal is to construct non $k$-rational varieties of exceptional groups. The relevant invariant is the defect of weak approximation.

scholarly journals UNIFORMLY RATIONAL VARIETIES WITH TORUS ACTION

2017 ◽  
Vol 24 (1) ◽  
pp. 149-153
Author(s):  
ALVARO LIENDO ◽  
CHARLIE PETITJEAN
Keyword(s):  
Torus Action ◽  
Rational Varieties

scholarly journals On the ideals of secant varieties to certain rational varieties

2008 ◽  
Vol 319 (5) ◽  
pp. 1913-1931 ◽  
Author(s):  
M.V. Catalisano ◽  
A.V. Geramita ◽  
A. Gimigliano
Keyword(s):  
Secant Varieties ◽  
Rational Varieties

scholarly journals RATIONALITY PROPERTIES OF MANIFOLDS CONTAINING QUASI-LINES

2003 ◽  
Vol 14 (10) ◽  
pp. 1053-1080 ◽  
Author(s):  
PALTIN IONESCU ◽  
DANIEL NAIE
Keyword(s):  
Open Subset ◽  
Basic Question ◽  
Projective Manifold ◽  
Rational Varieties ◽  
Direct Sum ◽  
Formal Geometry ◽  
Rationally Connected ◽  
Birational Invariant ◽  
Small Deformations

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification [Formula: see text] that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of [Formula: see text]. For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: there is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification [Formula: see text] which is strongly-rational, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion [Formula: see text]. As applications we show various instances in which X is determined by [Formula: see text]. We also formulate a basic question about the birational invariance of ẽ(X, Y).

scholarly journals Examples of Non-rational Varieties of Adjoint Groups

1997 ◽  
Vol 193 (2) ◽  
pp. 728-747 ◽  
Author(s):  
Philippe Gille
Keyword(s):  
Rational Varieties

scholarly journals Morphisms between Cremona groups, and characterization of rational varieties

2014 ◽  
Vol 150 (7) ◽  
pp. 1107-1124 ◽  
Author(s):  
Serge Cantat
Keyword(s):  
Projective Variety ◽  
Abstract Group ◽  
Birational Transformations ◽  
Rational Varieties ◽  
Complex Projective Variety ◽  
Abstract Homomorphisms ◽  
Complex Projective

AbstractWe classify all (abstract) homomorphisms from the group$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\sf PGL}_{r+1}(\mathbf{C})$to the group${\sf Bir}(M)$of birational transformations of a complex projective variety$M$, provided that$r\geq \dim _\mathbf{C}(M)$. As a byproduct, we show that: (i)${\sf Bir}(\mathbb{P}^n_\mathbf{C})$is isomorphic, as an abstract group, to${\sf Bir}(\mathbb{P}^m_\mathbf{C})$if and only if$n=m$; and (ii)$M$is rational if and only if${\sf PGL}_{\dim (M)+1}(\mathbf{C})$embeds as a subgroup of${\sf Bir}(M)$.

scholarly journals Tropical representation of Weyl groups associated with certain rational varieties

2009 ◽  
Vol 221 (3) ◽  
pp. 936-954 ◽  
Author(s):  
Teruhisa Tsuda ◽  
Tomoyuki Takenawa
Keyword(s):  
Weyl Groups ◽  
Rational Varieties
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