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

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Yang Cao
Keyword(s):  
Projective Space ◽  
Characteristic Zero ◽  
Sufficient Condition ◽  
Cohomology Groups ◽  
Del Pezzo Surface ◽  
The Third ◽  
Galois Structure ◽  
Unramified Cohomology ◽  
Del Pezzo ◽  
Open Question

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

Geometry related to the key del Pezzo surface and the associated mapping of plane cubics

1987 ◽  
Vol 107 (1-2) ◽  
pp. 75-86
Author(s):  
W. L. Edge
Keyword(s):  
Projective Space ◽  
Pezzo Surface ◽  
Del Pezzo Surface ◽  
New Approach ◽  
Del Pezzo

SynopsisWhen the points of a projective space [9] map ternary cubics the maps of perfect cubes are the points of a del Pezzo surface F. Several manifolds linked with F are listed and the cubics mapped on these are noted. A new approach leads to the first of Aronhold's two invariants.

scholarly journals On log del Pezzo surfaces in large characteristic

2017 ◽  
Vol 153 (4) ◽  
pp. 820-850 ◽  
Author(s):  
Paolo Cascini ◽  
Hiromu Tanaka ◽  
Jakub Witaszek
Keyword(s):  
Characteristic Zero ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Vanishing Theorem ◽  
Del Pezzo Surface ◽  
Algebraically Closed Field ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally $F$-regular or it admits a log resolution which lifts to characteristic zero. As a consequence, we prove the Kawamata–Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.

scholarly journals Affine cones over cubic surfaces are flexible in codimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Perepechko
Keyword(s):  
Automorphism Group ◽  
Open Subset ◽  
Pezzo Surface ◽  
Ample Divisor ◽  
Transitive Action ◽  
Del Pezzo Surface ◽  
Codimension One ◽  
Cubic Surfaces ◽  
Special Automorphism ◽  
Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

scholarly journals Degree three unramified cohomology groups

2016 ◽  
Vol 458 ◽  
pp. 120-133 ◽  
Author(s):  
Akinari Hoshi ◽  
Ming-chang Kang ◽  
Aiichi Yamasaki
Keyword(s):  
Cohomology Groups ◽  
Unramified Cohomology

Noncommutative G-semihereditary rings

2018 ◽  
Vol 17 (01) ◽  
pp. 1850014 ◽  
Author(s):  
Jian Wang ◽  
Yunxia Li ◽  
Jiangsheng Hu
Keyword(s):  
Associative Ring ◽  
Sufficient Condition ◽  
Projective Modules ◽  
The Third ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Direct Limits ◽  
Gorenstein Flat ◽  
Finitely Presented ◽  
Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

scholarly journals Troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique sur un corps de fonctions d'une variable

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Alena Pirutka
Keyword(s):  
Function Field ◽  
Cohomology Group ◽  
Chow Group ◽  
Hodge Conjecture ◽  
Complex Curve ◽  
The Third ◽  
Generic Fibre ◽  
Nous Montrons ◽  
Unramified Cohomology ◽  
Cubic Threefold

En combinant une m\'ethode de C. Voisin avec la descente galoisienne sur le groupe de Chow en codimension $2$, nous montrons que le troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique lisse d\'efini sur le corps des fonctions d'une courbe complexe est nul. Ceci implique que la conjecture de Hodge enti\`ere pour les classes de degr\'e 4 vaut pour les vari\'et\'es projectives et lisses de dimension 4 fibr\'ees en solides cubiques au-dessus d'une courbe, sans restriction sur les fibres singuli\`eres. --------------- We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension $2$ Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree $4$ classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres. Comment: in French

scholarly journals Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

Polynomial rings over NLI rings need not be NLI

2015 ◽  
Vol 52 (1) ◽  
pp. 129-133 ◽  
Author(s):  
Weixing Chen
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Sufficient Condition ◽  
Open Question

It is proved that there exists an NI ring R over which the polynomial ring R[x] is not an NLI ring. This answers an open question of Qu and Wei (Stud. Sci. Math. Hung., 51(2), 2014) in the negative. Moreover a sufficient condition of R[x] to be an NLI ring is included for an NLI ring R.

scholarly journals Local BPS Invariants: Enumerative Aspects and Wall-Crossing

2018 ◽  
Vol 2020 (17) ◽  
pp. 5450-5475 ◽  
Author(s):  
Jinwon Choi ◽  
Michel van Garrel ◽  
Sheldon Katz ◽  
Nobuyoshi Takahashi
Keyword(s):  
Projective Space ◽  
Euler Characteristic ◽  
Moduli Spaces ◽  
Del Pezzo Surfaces ◽  
Arithmetic Genus ◽  
One Dimensional ◽  
Wall Crossing ◽  
Bps Invariants ◽  
Del Pezzo ◽  
Dimensional Projective Space

Abstract We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincaré polynomials of the moduli spaces for the curve classes $\beta $ having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of $((-K_S).\beta -1)$-dimensional projective space. This conjecture motivates the upcoming work on log BPS numbers [8].

Sign in / Sign up

Export Citation Format

Share Document