Stably rational surfaces over a quasi-finite field

2019 ◽  
Vol 83 (3) ◽  
pp. 113-126
Author(s):  
Jean-Louis Colliot-Thelene
Keyword(s):  
Finite Field ◽  
Rational Surface ◽  
Cyclic Extension ◽  
Rational Surfaces

Let $k$ be a field and $X$ a smooth, projective, stably $k$-rational surface. If $X$ is split by a cyclic extension, for instance if the field $k$ is finite or more generally quasi-finite, then the surface $X$ is $k$-rational. Bibliography: 22 titles.

Stably rational surfaces over a quasi-finite field

2019 ◽  
Vol 83 (3) ◽  
pp. 521-533
Author(s):  
J.-L. Colliot-Thélène
Keyword(s):  
Finite Field ◽  
Rational Surfaces

Differential Graded Quivers of Smooth Rational Surfaces

2016 ◽  
Vol 60 (4) ◽  
pp. 859-876
Author(s):  
Agnieszka Bodzenta
Keyword(s):  
Minimal Model ◽  
Rational Surface ◽  
Line Bundles ◽  
Toric Surfaces ◽  
Rational Surfaces ◽  
Exceptional Collection ◽  
Dg Algebras

AbstractLetXbe a smooth rational surface. We calculate a differential graded (DG) quiver of a full exceptional collection of line bundles onXobtained by an augmentation from a strong exceptional collection on the minimal model ofX. In particular, we calculate canonical DG algebras of smooth toric surfaces.

scholarly journals Strange Duality on Rational Surfaces II: Higher-Rank Cases

2018 ◽  
Vol 2020 (10) ◽  
pp. 3153-3200 ◽  
Author(s):  
Yao Yuan
Keyword(s):  
Exact Sequence ◽  
Moduli Space ◽  
Chern Class ◽  
Rational Surface ◽  
Rational Surfaces ◽  
One Dimensional ◽  
Hirzebruch Surfaces ◽  
Duality Map ◽  
Higher Rank ◽  
Strange Duality

Abstract We study Le Potier’s strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ that involves the moduli space of rank $r$ sheaves with trivial 1st Chern class and 2nd Chern class $n$, and the moduli space of one-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show there is an exact sequence relating the map $SD_{c_r^r,L}$ to $SD_{c^{r-1}_{r},L}$ and $SD_{c_r^r,L\otimes K_X}$ for all $r\geq 1$ under some conditions on $X$ and $L$ that applies to a large number of cases on $\mathbb{P}^2$ or Hirzebruch surfaces. Also on $\mathbb{P}^2$ we show that for any $r>0$, $SD_{c^r_r,dH}$ is an isomorphism for $d=1,2$, injective for $d=3,$ and moreover $SD_{c_3^3,rH}$ and $SD_{c_3^2,rH}$ are injective. At the end we prove that the map $SD_{c_n^2,L}$ ($n\geq 2$) is an isomorphism for $X=\mathbb{P}^2$ or Fano rational-ruled surfaces and $g_L=3$, and hence so is $SD_{c_3^3,L}$ as a corollary of our main result.

Classification of Algebraic Surfaces with Sectional Genus less than or Equal to Six. II: Ruled Surfaces with dim ϕKx⊗L(x) = l

1986 ◽  
Vol 38 (5) ◽  
pp. 1110-1121 ◽  
Author(s):  
Elvira Laura Livorni
Keyword(s):  
Line Bundle ◽  
Hyperplane Section ◽  
Ample Line Bundle ◽  
Rational Surface ◽  
Algebraic Surfaces ◽  
Sectional Genus ◽  
Rational Surfaces ◽  
Image Position ◽  
Smooth Hyperplane Section

Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].If g = 2 ≠ hl,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.Since X is ruled, h2,0(X) = 0 andsee [4] and [12, p. 390].

scholarly journals Rationality of moduli spaces of vector bundles on rational surfaces

2002 ◽  
Vol 165 ◽  
pp. 43-69 ◽  
Author(s):  
Laura Costa ◽  
Rosa M. Miro-Ŕoig
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Rational Surface ◽  
Rational Surfaces ◽  
Stable Vector Bundles

Let X be a smooth rational surface. In this paper, we prove the rationality of the moduli space MX,L(2; c1; c2) of rank two L-stable vector bundles E on X with det (E) = c1 ∈ Pic(X) and c2(E) = c2 ≫ 0.

Islands in three-dimensional steady flows

1991 ◽  
Vol 227 ◽  
pp. 527-542 ◽  
Author(s):  
C. C. Hegna ◽  
A. Bhattacharjee
Keyword(s):  
Boundary Layer ◽  
Classical Theory ◽  
Three Dimensional ◽  
Rational Surface ◽  
Steady Flows ◽  
Rational Surfaces ◽  
Boundary Layer Analysis ◽  
Layer Analysis ◽  
Euler Flows ◽  
Critical Layers

We consider the problem of steady Euler flows in a torus. We show that in the absence of a direction of symmetry the solution for the vorticity contains δ-function singularities at the rational surfaces of the torus. We study the effect of a small but finite viscosity on these singularities. The solutions near a rational surface contain cat's eyes or islands, well known in the classical theory of critical layers. When the islands are small, their widths can be computed by a boundary-layer analysis. We show that the islands at neighbouring rational surfaces generally overlap. Thus, steady toroidal flows exhibit a tendency towards Beltramization.

scholarly journals Birational classification of curves on rational surfaces

2010 ◽  
Vol 199 ◽  
pp. 43-93
Author(s):  
Alberto Calabri ◽  
Ciro Ciliberto
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Rational Surface ◽  
Minimal Degree ◽  
Classification Theorem ◽  
Plane Curves ◽  
Cremona Transformation ◽  
Rational Surfaces

AbstractIn this paper we consider the birational classification of pairs (S, ℒ), withSa rational surface andℒa linear system onS. We give a classification theorem for such pairs, and we determine, for each irreducible plane curveB, itsCremona minimalmodels, that is, those plane curves which are equivalent toBvia a Cremona transformation and have minimal degree under this condition.

scholarly journals Optimal bound for the number of(−1)-curves on extremal rational surfaces

2002 ◽  
Vol 31 (2) ◽  
pp. 123-126
Author(s):  
Mustapha Lahyane
Keyword(s):  
Rational Surface ◽  
Rational Surfaces ◽  
Optimal Bound

We give an optimal bound for the number of(−1)-curves on an extremal rational surfaceXunder the assumption that−KXis numerically effective and having self-intersection zero. We also prove that a nonelliptic extremal rational surface has at most nine(−1)-curves.

Prevention of ECCD triggering explosive bursts in reversed magnetic shear tokamak plasmas for disruption avoidance

2022 ◽  
Author(s):  
Tong Liu ◽  
Zheng-Xiong Wang ◽  
Lai Wei ◽  
Jialei Wang
Keyword(s):  
Control Strategies ◽  
Control Process ◽  
Rational Surface ◽  
Current Drive ◽  
Tokamak Plasmas ◽  
Tearing Mode ◽  
Magnetic Shear ◽  
Rational Surfaces ◽  
Major Disruption ◽  
Neoclassical Tearing Mode

Abstract The explosive burst excited by neoclassical tearing mode (NTM) is one of the possible candidates of disruptive terminations in reversed magnetic shear (RMS) tokamak plasmas. For the purpose of disruption avoidance, numerical investigations have been implemented on the prevention of explosive burst triggered by the ill-advised application of electron cyclotron current drive (ECCD) in RMS configuration. Under the situation of controlling NTMs by ECCD in RMS tokamak plasmas, a threshold in EC driven current has been found. Below the threshold, not only are the NTM islands not effectively suppressed, but also a deleterious explosive burst could be triggered, which might contribute to the major disruption of tokamak plasmas. In order to prevent this ECCD triggering explosive burst, three control strategies have been attempted in this work and two of them have been recognized to be effective. One is to apply differential poloidal plasma rotation in the proximity of outer rational surface during the ECCD control process; The other is to apply two ECCDs to control NTM islands on both rational surfaces at the same time. In the former strategy, the threshold is diminished due to the modification of classical TM index. In the latter strategy, the prevention is accomplished as a consequence of the reduction of the coupling strength between the two rational surfaces via the stabilization of inner islands. Moreover, the physical mechanism behind the excitation of the explosive burst and the control processes by different control strategies have all been discussed in detail.

Pressure driven long wavelength MHD instabilities in an axisymmetric toroidal resistive plasma

2021 ◽  
Vol 64 (1) ◽  
pp. 014001
Author(s):  
J P Graves ◽  
M Coste-Sarguet ◽  
C Wahlberg
Keyword(s):  
Acoustic Waves ◽  
Rational Surface ◽  
Ion Acoustic Waves ◽  
Equilibrium Conditions ◽  
Rational Surfaces ◽  
Long Wavelength ◽  
Mhd Instabilities ◽  
Axisymmetric Equilibrium ◽  
First Time ◽  
Pressure Driven

Abstract A general set of equations that govern global resistive interchange, resistive internal kink and resistive infernal modes in a toroidal axisymmetric equilibrium are systematically derived in detail. Tractable equations are developed such that resistive effects on the fundamental rational surface can be treated together with resistive effects on the rational surfaces of the sidebands. Resistivity introduces coupling of pressure driven toroidal instabilities with ion acoustic waves, while compression introduces flute-like flows and damping of instabilities, enhanced by toroidal effects. It is shown under which equilibrium conditions global interchange, internal kink modes or infernal modes occur. The m = 1 internal kink is derived for the first time from higher order infernal mode equations, and new resistive infernal modes resonant at the q = 1 surface are reduced analytically. Of particular interest are the competing effects of resistive corrections on the rational surfaces of the fundamental harmonic and on the sidebands, which in this paper is investigated for standard profiles developed for the m = 1 internal kink problem.

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