Arithmetic of Del Pezzo surfaces

Várilly-Alvarado, Anthony

doi:10.1007/978-1-4614-6482-2_12

Cited by 28 publications

(14 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The goal of this section is to review some geometric facts about a del Pezzo surface of any degree. For more details see [29]. We then prove some special facts about a del Pezzo surface of degree 4.…”

Section: Geometry Of a Del Pezzo Surface Of Degreementioning

confidence: 92%

“…proof of Theorem 2.2.2. This proof can be found in [29,Theorem 1.6]; we restate it for the reader's convenience. We consider Y a minimal model of X s and the corresponding birational morphism f : X s Ñ Y .…”

Section: Geometry Of a Del Pezzo Surface Of Degreementioning

confidence: 99%

“…[29, Theorem 1.6] Let X be a del Pezzo surface of degree d over a field k. Then X s is isomorphic to the blowup of P2 k s at 9´d points in general position where 0 ď d ď 8, or to P 1 k sˆP 1 k s . In the latter case d " 8.Proposition 2.2.3.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Vanishing of the Brauer group of a del Pezzo surface of degree 4

Riman¹

2020

Journal of Number Theory

View full text Add to dashboard Cite

The Vanishing of the Brauer Group of a del Pezzo Surface of Degree 4 Manar RimanChair of the Supervisory Committee: Associate Professor Bianca Viray Mathematics A del Pezzo surface X of degree 4 over a field k can be thought of as the smooth complete intersection of 2 quadrics in P 4 . Arithmetic geometers are interested in computing the quotient of its Brauer group Br X{ Br 0 X, where Br 0 X " impBr k Ñ Br Xq. Several algorithms have been implemented to compute this quotient; see [2], [30]. In [30], the algorithm relies on a specific arithmetic input related to the solvability of certain quadrics associated to the pencil determined by X. We explicitly construct a del Pezzo surface X of degree 4 over a field k such that H 1 pk, Pic Xq » Z{2Z while Br X{ Br k is trivial. This proves that the algorithm to compute the Brauer group in [30] cannot be generalized in some cases.

show abstract

Section: Geometry Of a Del Pezzo Surface Of Degreementioning

confidence: 92%

Section: Geometry Of a Del Pezzo Surface Of Degreementioning

confidence: 99%

See 1 more Smart Citation

Vanishing of the Brauer group of a del Pezzo surface of degree 4

Riman¹

2020

Journal of Number Theory

View full text Add to dashboard Cite

show abstract

“…. , x 5 in general position [Va,Theorem 1.6]. In the blown-up model, the 16 lines are given as follows, cf.…”

Section: The Picard Group and Its Automorphism Groupmentioning

confidence: 99%

On the algebraic Brauer classes on open degree four del Pezzo surfaces

Jahnel¹,

Schindler

2019

Journal of Number Theory

View full text Add to dashboard Cite

We study the algebraic Brauer classes on open del Pezzo surfaces of degree 4. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree 4. We show that the 2-torsion part is generated by classes of two different types. Moreover, there are two types of 4-torsion classes. For each type, we discuss methods for the evaluation of such a class at a rational point over a p-adic field. *

show abstract

“…D'après (1.2), on a H 1 (k, Sym 2 Pic(Xk)) = 0. Par le théorème 4.7 et sa preuve, il suffit de montrer que l'on a X(k) ∅ ou que le morphisme (Sym 2 T * ) Γ k ∪ − → CH 2 (Xk) Z dans le théorème 4.7 (2) est surjectif.Notons d le degré de X.Si d = 5 ou 7, par les travaux de Enriques, Châtelet,Manin, Section 4] ou[VA, Théorème 2.1]), on a X(k) ∅.…”

unclassified

Troisi\`eme groupe de cohomologie non ramifi\'ee des torseurs universels sur les surfaces rationnelles

Cao¹

2018

Épijournal De Géométrie Algébrique

View full text Add to dashboard Cite

Let $k$ a field of characteristic zero. Let $X$ be a smooth, projective, geometrically rational $k$-surface. Let $\mathcal{T}$ be a universal torsor over $X$ with a $k$-point et $\mathcal{T}^c$ a smooth compactification of $\mathcal{T}$. There is an open question: is $\mathcal{T}^c$ $k$-birationally equivalent to a projective space? We know that the unramified cohomology groups of degree 1 and 2 of $\mathcal{T}$ and $\mathcal{T}^c$ are reduced to their constant part. For the analogue of the third cohomology groups, we give a sufficient condition using the Galois structure of the geometrical Picard group of $X$. This enables us to show that $H^{3}_{nr}(\mathcal{T}^{c},\mathbb{Q}/\mathbb{Z}(2))/H^3(k,\mathbb{Q}/\mathbb{Z}(2))$ vanishes if $X$ is a generalised Ch\^atelet surface and that this group is reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at least 2. Comment: 27 page, in French

show abstract

Arithmetic of Del Pezzo surfaces

Cited by 28 publications

References 35 publications

Vanishing of the Brauer group of a del Pezzo surface of degree 4

Vanishing of the Brauer group of a del Pezzo surface of degree 4

On the algebraic Brauer classes on open degree four del Pezzo surfaces

Troisi\`eme groupe de cohomologie non ramifi\'ee des torseurs universels sur les surfaces rationnelles

Contact Info

Product

Resources

About