Ample line bundles and intersection theory

2006 ◽  
pp. 69-76
Author(s):  
Johan Jong
Keyword(s):  
Intersection Theory ◽  
Line Bundles

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.

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.

scholarly journals Homotopical intersection theory I

2007 ◽  
Vol 11 (2) ◽  
pp. 939-977 ◽  
Author(s):  
John R Klein ◽  
E Bruce Williams
Keyword(s):  
Intersection Theory

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

scholarly journals Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces

2021 ◽  
Vol 8 (1) ◽  
pp. 223-229
Author(s):  
Callum R. Brodie ◽  
Andrei Constantin ◽  
Rehan Deen ◽  
Andre Lukas
Keyword(s):  
Topological Index ◽  
Line Bundles ◽  
Hirzebruch Surfaces ◽  
Del Pezzo

Abstract We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.

Sign in / Sign up

Export Citation Format

Share Document