scholarly journals The split torsor method for Manin’s conjecture

2020 ◽  
Vol 373 (12) ◽  
pp. 8485-8524
Author(s):  
Ulrich Derenthal ◽  
Marta Pieropan
Keyword(s):  
Manin’S Conjecture

scholarly journals Manin’s conjecture for quartic Del Pezzo surfaces with a conic fibration

2011 ◽  
Vol 160 (1) ◽  
pp. 1-69 ◽  
Author(s):  
R. De la Bretèche ◽  
T. D. Browning
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Manin’S Conjecture

scholarly journals La conjecture de Manin pour une famille de variétés en dimension supérieure

2018 ◽  
Vol 166 (3) ◽  
pp. 433-486 ◽  
Author(s):  
KEVIN DESTAGNOL
Keyword(s):  
Infinite Family ◽  
Strong Form ◽  
Higher Dimension ◽  
Crepant Resolution ◽  
Projective Varieties ◽  
Formula Group ◽  
Image Position ◽  
Manin’S Conjecture

AbstractInspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$in ℙℚ2n−1for alln⩾ 3. This paper comes with an Appendix by Per Salberger constructing a crepant resolution of the above varieties.

scholarly journals Manin's conjecture for certain biprojective hypersurfaces

2016 ◽  
Vol 2016 (714) ◽  
Author(s):  
Damaris Schindler
Keyword(s):  
Circle Method ◽  
Height Function ◽  
Singular Locus ◽  
Rational Points ◽  
Large Dimension ◽  
Side Length ◽  
Asymptotic Formulas ◽  
Complete Intersections ◽  
Integer Points ◽  
Manin’S Conjecture

AbstractUsing the circle method, we count integer points on complete intersections in biprojective space in boxes of different side length, provided the number of variables is large enough depending on the degree of the defining equations and certain loci related to the singular locus. Having established these asymptotics we deduce asymptotic formulas for rational points on such varieties with respect to the anticanonical height function. In particular, we establish a conjecture of Manin for certain smooth hypersurfaces in biprojective space of sufficiently large dimension.

scholarly journals Manin's conjecture for a cubic surface with 2A2 + A1 singularity type

2012 ◽  
Vol 153 (3) ◽  
pp. 419-455 ◽  
Author(s):  
PIERRE LE BOUDEC
Keyword(s):  
Arithmetic Progressions ◽  
Divisor Function ◽  
Cubic Surface ◽  
Singularity Type ◽  
Manin’S Conjecture ◽  
A1 Singularity

AbstractWe establish Manin's conjecture for a cubic surface split over ℚ and whose singularity type is 2A2 + A1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath–Brown) and draws on the work of Deligne.

scholarly journals Equivariant Compactifications of Two-Dimensional Algebraic Groups

2014 ◽  
Vol 58 (1) ◽  
pp. 149-168 ◽  
Author(s):  
Ulrich Derenthal ◽  
Daniel Loughran
Keyword(s):  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Direct Products ◽  
Two Dimensional ◽  
Del Pezzo ◽  
Manin’S Conjecture ◽  
Transitive Actions

AbstractWe classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .

scholarly journals On Manin's conjecture for singular del Pezzo surfaces of degree four, II

2007 ◽  
Vol 143 (3) ◽  
pp. 579-605 ◽  
Author(s):  
R. DE LA BRETÈCHE ◽  
T. D. BROWNING
Keyword(s):  
Zeta Function ◽  
Half Plane ◽  
Open Subset ◽  
Height Function ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Del Pezzo ◽  
Manin’S Conjecture ◽  
Height Zeta Function

AbstractThis paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four$X \subset \bfP^4$. In fact, ifU⊂Xis the open subset formed by deleting the lines fromX, andHis the usual projective height function on$\bfP^4(\Q)$, then the height zeta function$ \sum_{x \in U(\Q)}{H(x)^{-s}} $is analytically continued to the half-plane ℜe(s) > 17/20.

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