Introduction G SL r G SO r G Sp 2r
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pages
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- normal projective
- theta map
- algebraic group
- universite de nice
- determinant bundle
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
