Line bundles on arithmetic surfaces and intersection theory

1996 ◽  
Vol 91 (1) ◽  
pp. 103-119
Author(s):  
Jörg Jahnel
Keyword(s):  
Intersection Theory ◽  
Line Bundles ◽  
Arithmetic Surfaces

scholarly journals ON THE VOEVODSKY MOTIVE OF THE MODULI STACK OF VECTOR BUNDLES ON A CURVE

2020 ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Vector Bundles ◽  
Intersection Theory ◽  
Line Bundles ◽  
Projective Curve ◽  
Classifying Space ◽  
Smooth Projective Curve ◽  
Quot Schemes ◽  
Homotopy Colimit ◽  
Moduli Stack ◽  
Fixed Rank

Abstract We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

Positive Line Bundles on Arithmetic Surfaces

1992 ◽  
Vol 136 (3) ◽  
pp. 569 ◽  
Author(s):  
Shouwu Zhang
Keyword(s):  
Line Bundles ◽  
Arithmetic Surfaces ◽  
Positive Line

Teissier's problem on inequalities of nef divisors

2015 ◽  
Vol 14 (09) ◽  
pp. 1540002 ◽  
Author(s):  
Steven Dale Cutkosky
Keyword(s):  
Characteristic Zero ◽  
Intersection Theory ◽  
Line Bundles ◽  
Closed Field ◽  
Arbitrary Field ◽  
Riemann Manifold ◽  
Complete Proof ◽  
Complete Variety ◽  
Direct System ◽  
Nef Divisors

Teissier has proven remarkable inequalities [Formula: see text] for intersection numbers si = (ℒi ⋅ ℳd-i) of a pair of nef line bundles ℒ, ℳ on a d-dimensional complete algebraic variety over a field. He asks if two nef and big line bundles are numerically proportional if the inequalities are all equalities. In this paper, we show that this is true in the most general possible situation, for nef and big line bundles on a proper irreducible scheme over an arbitrary field k. Boucksom, Favre and Jonsson have recently established this result on a complete variety X over an algebraically closed field of characteristic zero. Their proof involves an ingenious extension of the intersection theory on a variety to its Zariski Riemann Manifold. This extension requires the existence of a direct system of nonsingular varieties dominating X. We make use of a simpler intersection theory which does not require resolution of singularities, and extend volume to an arbitrary field and prove its continuous differentiability, extending results of Boucksom, Favre and Jonsson, and of Lazarsfeld and Mustaţă. A goal in this paper is to provide a manuscript which will be accessible to many readers. As such, subtle topological arguments which are required to give a complete proof in [S. Bouksom et al., J. Algebraic Geometry18 (2009) 279–308] have been written out in this manuscript, in the context of our intersection theory, and over arbitrary varieties.

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