On the Brauer group of Enriques surfaces

On the Brauer group of Enriques surfaces
Arnaud Beauville
Universite de Nice
Madrid, December 2009
Arnaud Beauville On the Brauer group of Enriques surfacesBr(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
S smooth projective variety overC
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Why is it interesting?
The Brauer group
Arnaud Beauville On the Brauer group of Enriques surfacesBr(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Why is it interesting?
The Brauer group
S smooth projective variety overC
Arnaud Beauville On the Brauer group of Enriques surfacesBr(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
2 H (S;O )= torshol
Why is it interesting?
The Brauer group
S smooth projective variety overC
2Br(S) := H (S;G )met
Arnaud Beauville On the Brauer group of Enriques surfacesBr(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
Why is it interesting?
The Brauer group
S smooth projective variety overC
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Arnaud Beauville On the Brauer group of Enriques surfacesBr(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
The Brauer group
S smooth projective variety overC
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Why is it interesting?
Arnaud Beauville On the Brauer group of Enriques surfacesBr(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
The Brauer group
S smooth projective variety overC
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Why is it interesting?
Br(K ) important in number theory and algebra.
Arnaud Beauville On the Brauer group of Enriques surfacesnBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
The Brauer group
S smooth projective variety overC
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Why is it interesting?
Br(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
Arnaud Beauville On the Brauer group of Enriques surfaces2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
For S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
The Brauer group
S smooth projective variety overC
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Why is it interesting?
Br(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
Arnaud Beauville On the Brauer group of Enriques surfacesFor S over a number eld, Br(S) provides the Brauer-Manin
obstruction to the existence of rational points.
The Brauer group
S smooth projective variety overC
2 2 Br(S) := H (S;G ) H (S;O )=m torset hol
Why is it interesting?
Br(K ) important in number theory and algebra.
Br(S) birational invariant { detects non-rational varieties
(Artin-Mumford).
nBr(S) classi es Severi-Brauer brations (= P -bundles) over S
2Br(S) an abelian category Coh (S) of \-twisted
coherent sheaves".
Arnaud Beauville On the Brauer group of Enriques surfaces