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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pefav

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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
Universit´e de Nice
Ramanan 70, Miraﬂores, June 2008
Arnaud Beauville The theta map for principal bundles on curves/
/
M moduli space of (semi-stable) principal G-bundles on CG
Normal projective variety, with mild singularities. But : embedding
nin P not explicit.
Pic(M ) =Z[L] ,L = determinant bundle. Theta map:G
∗_ _ _θ :M |L| ,G
0rational map deﬁned by the sections ofL , (|L| =P(H (M ,L))).G
Theme of the talk : What can we say about that map?
Introduction
1. G = SL(r)
2. G = SO(r)
3. G = Sp(2r)
Introduction
C curve of genus g ≥ 2 G simple algebraic group
Arnaud Beauville The theta map for principal bundles on curves/
/
Normal projective variety, with mild singularities. But : embedding
nin P not explicit.
Pic(M ) =Z[L] ,L = determinant bundle. Theta map:G
∗_ _ _θ :M |L| ,G
0rational map deﬁned by the sections ofL , (|L| =P(H (M ,L))).G
Theme of the talk : What can we say about that map?
Introduction
1. G = SL(r)
2. G = SO(r)
3. G = Sp(2r)
Introduction
C curve of genus g ≥ 2 G simple algebraic group
M moduli space of (semi-stable) principal G-bundles on CG
Arnaud Beauville The theta map for principal bundles on curves/
/
But : embedding
nin P not explicit.
Pic(M ) =Z[L] ,L = determinant bundle. Theta map:G
∗_ _ _θ :M |L| ,G
0rational map deﬁned by the sections ofL , (|L| =P(H (M ,L))).G
Theme of the talk : What can we say about that map?
Introduction
1. G = SL(r)
2. G = SO(r)
3. G = Sp(2r)
Introduction
C curve of genus g ≥ 2 G simple algebraic group
M moduli space of (semi-stable) principal G-bundles on CG
Normal projective variety, with mild singularities.
Arnaud Beauville The theta map for principal bundles on curves/
/
Pic(M ) =Z[L] ,L = determinant bundle. Theta map:G
∗_ _ _θ :M |L| ,G
0rational map deﬁned by the sections ofL , (|L| =P(H (M ,L))).G
Theme of the talk : What can we say about that map?
Introduction
1. G = SL(r)
2. G = SO(r)
3. G = Sp(2r)
Introduction
C curve of genus g ≥ 2 G simple algebraic group
M moduli space of (semi-stable) principal G-bundles on CG
Normal projective variety, with mild singularities. But : embedding
nin P not explicit.
Arnaud Beauville The theta map for principal bundles on curves/
/
Theta map:
∗_ _ _θ :M |L| ,G
0rational map deﬁned by the sections ofL , (|L| =P(H (M ,L))).G
Theme of the talk : What can we say about that map?
Introduction
1. G = SL(r)
2. G = SO(r)
3. G = Sp(2r)
Introduction
C curve of genus g ≥ 2 G simple algebraic group
M moduli space of (semi-stable) principal G-bundles on CG
Normal projective variety, with mild singularities. But : embedding
nin P not explicit.
Pic(M ) =Z[L] ,L = determinant bundle.G
Arnaud Beauville The theta map for principal bundles on curves/
/
Theme of the talk : What can we say about that map?
Introduction
1. G = SL(r)
2. G = SO(r)
3. G = Sp(2r)
Introduction
C curve of genus g ≥ 2 G simple algebraic group
M moduli space of (semi-stable) principal G-bundles on CG
Normal projective variety, with mild singularities. But : embedding
nin P not explicit.
Pic(M ) =Z[L] ,L = determinant bundle. Theta map:G
∗_ _ _θ :M |L| ,G
0rational map deﬁned by the sections ofL , (|L| =P(H (M ,L))).G
Arnaud Beauville The theta map for principal bundles on curves/
/
Introduction
1. G = SL(r)
2. G = SO(r)
3. G = Sp(2r)
Introduction
C curve of genus g ≥ 2 G simple algebraic group
M moduli space of (semi-stable) principal G-bundles on CG
Normal projective variety, with mild singularities. But : embedding
nin P not explicit.
Pic(M ) =Z[L] ,L = determinant bundle. Theta map:G
∗_ _ _θ :M |L| ,G
0rational map deﬁned by the sections ofL , (|L| =P(H (M ,L))).G
Theme of the talk : What can we say about that map?
Arnaud Beauville The theta map for principal bundles on curves