Towards the problem of rationality of conic bundles

Geometrically rational real conic bundles and very transitive actions

Compositio Mathematica ◽

10.1112/s0010437x10004835 ◽

2010 ◽

Vol 147 (1) ◽

pp. 161-187 ◽

Cited By ~ 6

Author(s):

Jérémy Blanc ◽

Frédéric Mangolte

Keyword(s):

Automorphism Groups ◽

Algebraic Surfaces ◽

Algebraic Models ◽

Transitive Automorphism ◽

Real Locus ◽

Real Algebraic Surfaces ◽

Conic Bundles ◽

Transitive Actions ◽

Insight Into

AbstractIn this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

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BIRATIONAL AUTOMORPHISMS OF CONIC BUNDLES

Mathematics of the USSR-Izvestiya ◽

10.1070/im1981v017n01abeh001326 ◽

1981 ◽

Vol 17 (1) ◽

pp. 177-202 ◽

Cited By ~ 17

Author(s):

V G Sarkisov

Keyword(s):

Conic Bundles ◽

Birational Automorphisms

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COUNTING MULTISECTIONS IN CONIC BUNDLES OVER A CURVE DEFINED OVER 𝔽q

International Journal of Number Theory ◽

10.1142/s179304211100485x ◽

2011 ◽

Vol 07 (06) ◽

pp. 1663-1680

Author(s):

SEYFI TÜRKELLI

Keyword(s):

Function Field ◽

Algebraic Numbers ◽

Conic Bundle ◽

Conic Bundles ◽

Special Case

For a given conic bundle X over a curve C defined over 𝔽q, we count irreducible branch covers of C in X of degree d and height e ≫ 1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field 𝔽q(C).

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STABILITY OF CONIC BUNDLES: WITH AN APPENDIX BY MUNDET I RIERA

International Journal of Mathematics ◽

10.1142/s0129167x00000507 ◽

2000 ◽

Vol 11 (08) ◽

pp. 1027-1055 ◽

Cited By ~ 6

Author(s):

TOMÁS L. GÓMEZ ◽

IGNACIO SOLS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Conic Bundle ◽

Conic Bundles ◽

Definition Of

Roughly speaking, a conic bundle is a surface, fibered over a curve, such that the fibers are conics (not necessarily smooth). We define stability for conic bundles and construct a moduli space. We prove that (after fixing some invariants) these moduli spaces are irreducible (under some conditions). Conic bundles can be thought of as generalizations of orthogonal bundles on curves. We show that in this particular case our definition of stability agrees with the definition of stability for orthogonal bundles. Finally, in an appendix by I. Mundet i Riera, a Hitchin-Kobayashi correspondence is stated for conic bundles.

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