Algebraic subgroups of the plane Cremona group over a perfect field

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Julia Schneider ◽  
Susanna Zimmermann
Keyword(s):  
Rational Points ◽  
Perfect Field ◽  
Algebraic Subgroup ◽  
Cremona Group ◽  
Birational Map

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in a maximal algebraic subgroup of the plane Cremona group. We classify the maximal groups, and their subgroups of rational points, up to conjugacy by a birational map.

Bound for the order for p-elementary subgroups in the plane cremona group over a perfect field

2012 ◽  
Vol 56 (2) ◽  
pp. 509-513
Author(s):  
A. L. Fomin
Keyword(s):  
Sharp Bound ◽  
Perfect Field ◽  
Cremona Group

AbstractWe obtain a sharp bound for p-elementary subgroups in the Cremona group Cr2(k) over an arbitrary perfect field k.

RELATIONS IN THE TWO-DIMENSIONAL CREMONA GROUP OVER A PERFECT FIELD

1994 ◽  
Vol 42 (3) ◽  
pp. 427-478 ◽  
Author(s):  
V A Iskovskikh ◽  
F K Kabdykairov ◽  
S L Tregub
Keyword(s):  
Two Dimensional ◽  
Perfect Field ◽  
Cremona Group

scholarly journals Minimal rational threefolds II

1988 ◽  
Vol 110 ◽  
pp. 15-80 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Algebraic Operation ◽  
View Point ◽  
Algebraic Subgroup ◽  
Birational Morphism ◽  
Cremona Group ◽  
Algebraic Operations

The Enriques-Fano classification ([E.F], [F]) of the maximal connected algebraic subgroups of the three variable Cremona group, despite of its group theoretic feature, seems to be the most significant result on the rational threefolds so far known. In this paper as in [MU] we interpret the Enriques-Fano classification from a geometric view point, namely the geometry of minimal rational threefolds. We explained in [MU] the link between the two objects; the maximal algebraic subgroups and the minimal rational threefolds. Let (G, X) be a maximal algebraic subgroup of three variable Cremona group. We denote by ℓ(G, X) the set of all the algebraic operations (G, Y) such that Y is non-singular and projective and such that (G, Y) is isomorphic to (G, X) as law chunks of algebraic operation: namely (G, Y) is birationally equivalent to (G, X). Then we define an order in ℓ(G, X): for (G, Z), (G, W) ∊ ℓ(G, X), (G, Z)>(G, W) if there exists an G-equivariant birational morphism of Z onto W.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

scholarly journals Ordinary varieties with trivial canonical bundle are not uniruled

2021 ◽  
Author(s):  
Zsolt Patakfalvi ◽  
Maciej Zdanowicz
Keyword(s):  
Canonical Bundle ◽  
Perfect Field ◽  
Tangent Bundles

AbstractWe prove that smooth, projective, K-trivial, weakly ordinary varieties over a perfect field of characteristic $$p>0$$ p > 0 are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer’s results, implies that varieties of the above type have strongly semistable tangent bundles with respect to every polarization.

scholarly journals p-Elementary subgroups of the Cremona group

2007 ◽  
Vol 314 (2) ◽  
pp. 553-564 ◽  
Author(s):  
Arnaud Beauville
Keyword(s):  
Cremona Group
Sign in / Sign up

Export Citation Format

Share Document