Manin's conjecture for two quartic del Pezzo surfaces with 3 \mathbfA1and \mathbfA1+ \mathbfA2singularity types

2012 ◽  
Vol 151 (2) ◽  
pp. 109-163 ◽  
Author(s):  
Pierre Le Boudec
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
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 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.

scholarly journals GENERALISED DIVISOR SUMS OF BINARY FORMS OVER NUMBER FIELDS

2017 ◽  
Vol 19 (1) ◽  
pp. 137-173 ◽  
Author(s):  
Christopher Frei ◽  
Efthymios Sofos
Keyword(s):  
Dirichlet Character ◽  
Number Fields ◽  
Del Pezzo Surfaces ◽  
Binary Forms ◽  
Subsequent Work ◽  
Order Of Magnitude ◽  
Del Pezzo ◽  
Divisor Sums ◽  
Manin’S Conjecture ◽  
Sparse Set

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

scholarly journals Counting imaginary quadratic points via universal torsors, II

2014 ◽  
Vol 156 (3) ◽  
pp. 383-407 ◽  
Author(s):  
ULRICH DERENTHAL ◽  
CHRISTOPHER FREI
Keyword(s):  
Number Fields ◽  
Del Pezzo Surfaces ◽  
Quadratic Number ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Del Pezzo ◽  
Manin’S Conjecture

AbstractWe prove Manin's conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.

scholarly journals Counting imaginary quadratic points via universal torsors

2014 ◽  
Vol 150 (10) ◽  
pp. 1631-1678 ◽  
Author(s):  
Ulrich Derenthal ◽  
Christopher Frei
Keyword(s):  
Arbitrary Number ◽  
Number Fields ◽  
Quadratic Fields ◽  
Del Pezzo Surfaces ◽  
Fano Varieties ◽  
Imaginary Quadratic Fields ◽  
Del Pezzo Surface ◽  
Singularity Type ◽  
Del Pezzo ◽  
Manin’S Conjecture

AbstractA conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields$K$for the quartic del Pezzo surface$S$of singularity type${\boldsymbol{A}}_{3}$with five lines given in${\mathbb{P}}_{K}^{4}$by the equations${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

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