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Introduction G SL r G SO r G Sp 2r

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Introduction 1. G = SL(r) 2. G = SO(r) 3. G = Sp(2r) The theta map for principal bundles on curves Arnaud Beauville Universite de Nice Ramanan 70, Miraflores, June 2008 Arnaud Beauville The theta map for principal bundles on curves

  • normal projective

  • theta map

  • algebraic group

  • universite de nice

  • determinant bundle


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The
theta
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
map
for
principal
bundles
Arnaud Beauville
Universit´edeNice
on
Ramanan 70, Miraflores, June 2008
Arnaud Beauville
curves
The theta map for principal bundles on curves
Introduction
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
Ccurve of genusg2
Gsimple algebraic group
Arnaud Beauville
The theta map for principal bundles on curves
Introduction
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
Ccurve of genusg2
Gsimple algebraic group
MGmoduli space of (semi-stable) principalG-bundles onC
Arnaud Beauville
The theta map for principal bundles on curves
Introduction
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
Ccurve of genusg2
Gsimple algebraic group
MGmoduli space of (semi-stable) principalG-bundles onC
Normal projective variety, with mild singularities.
Arnaud Beauville
The theta map for principal bundles on curves
Introduction
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
Ccurve of genusg2
MG
Gsimple algebraic group
moduli space of (semi-stable) principalG-bundles onC
Normal projective variety, with mild singularities.But :embedding inPnnot explicit.
Arnaud Beauville
The theta map for principal bundles on curves
Introduction
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
Ccurve of genusg2
MG
Gsimple algebraic group
moduli space of (semi-stable) principalG-bundles onC
Normal projective variety, with mild singularities.But :embedding inPnnot explicit.
Pic(MG) =Z[L] ,L= determinant bundle.
Arnaud Beauville
The theta map for principal bundles on curves
Introduction
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
Ccurve of genusg2
MG
Gsimple algebraic group
moduli space of (semi-stable) principalG-bundles onC
Normal projective variety, with mild singularities.But :embedding inPnnot explicit.
Pic(MG) =Z[L] ,L map: Theta= determinant bundle.
θ:MG
_ _ _
|L|
rational map defined by the sections ofL, (|L|=P(H0(MG
Arnaud Beauville
L))).
The theta map for principal bundles on curves
Introduction
Introduction 1.G=SL(r) 2.G=SO(r) 3.G=Sp(2r)
Ccurve of genusg2
MG
Gsimple algebraic group
moduli space of (semi-stable) principalG-bundles onC
Normal projective variety, with mild singularities.But :embedding inPnnot explicit.
Pic(MG) =Z[L] ,L Theta= determinant bundle. map:
θ:MG
_ _ _
|L|
rational map defined by the sections ofL, (|L|=P(H0(MG
Theme of the talk :What can we say about that map?
Arnaud Beauville
The
L))).
theta map for principal bundles on curves