Geodesic
Divisors on Varieties Over a Real Closed Field
Canadian mathematical bulletin, Tome 35 (1992) no. 4, pp. 503-509

Voir la notice de l'article provenant de la source Cambridge University Press

Let X be a projective nonsingular variety over a real closed field R such that the set X(R) of R-rational points of X is nonempty. Let ClR(X) = Cl(X)/Γ(X), where Cl(X) is the group of classes of linearly equivalent divisors on X and Γ(X) is the subgroup of Cl(X) consisting of the classes of divisors whose restriction to some neighborhood of X(R) in X is linearly equivalent to 0. It is proved that the group ClR(X) is isomorphic to (Z/2)s for some non-negative integer s. Moreover, an upper bound on s is given in terms of the Z/2-dimension of the group cohomology modules of Gal(C/R), where , with values in the Néron-Severi group and the Picard variety of Xc = X xR C.
DOI : 10.4153/CMB-1992-066-3
Mots-clés : 14C20, 14P05 
Kucharz, W. Divisors on Varieties Over a Real Closed Field. Canadian mathematical bulletin, Tome 35 (1992) no. 4, pp. 503-509. doi: 10.4153/CMB-1992-066-3
@article{10_4153_CMB_1992_066_3,
     author = {Kucharz, W.},
     title = {Divisors on {Varieties} {Over} a {Real} {Closed} {Field}},
     journal = {Canadian mathematical bulletin},
     pages = {503--509},
     year = {1992},
     volume = {35},
     number = {4},
     doi = {10.4153/CMB-1992-066-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1992-066-3/}
}
TY  - JOUR
AU  - Kucharz, W.
TI  - Divisors on Varieties Over a Real Closed Field
JO  - Canadian mathematical bulletin
PY  - 1992
SP  - 503
EP  - 509
VL  - 35
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1992-066-3/
DO  - 10.4153/CMB-1992-066-3
ID  - 10_4153_CMB_1992_066_3
ER  - 
%0 Journal Article
%A Kucharz, W.
%T Divisors on Varieties Over a Real Closed Field
%J Canadian mathematical bulletin
%D 1992
%P 503-509
%V 35
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1992-066-3/
%R 10.4153/CMB-1992-066-3
%F 10_4153_CMB_1992_066_3

[1] 1. Becker, E., Valuations and real places in the theory of formally real fields. In: Géométrie Algébrique Réelle et Formes Quadratiques, Lecture Notes in Math. 959, Springer, 1982 1–40. Google Scholar

[2] 2. Bochnak, J., Coste, M. and Roy, M.-F., Géométrie Algébrique Réelle, Ergebnisse der Math., 12, Berlin, Heidelberg, New York, Springer Verlag, 1987. Google Scholar

[3] 3. Bröcker, L., Reelle Divisoren, Arch, der Math. 35(1980), 140–143. Google Scholar

[4] 4. Brumfiel, G. W., Partially Ordered Rings and Semi-algebraic Geometry, Cambridge Univ. Press, Cambridge, 1979. Google Scholar

[5] 5. Delfs, H. and Knebusch, M., Semi-algebraic topology over a real closed field I, II, Math. Z. 177(1981), 107–129; Math. Z. 178(1981), 175–213. Google Scholar

[6] 6. Delfs, H., On the homology of algebraic varieties over real closed fields, J. Reine Angew. Math. 335(1982), 122–163. Google Scholar

[7] 7. Evans, E. G., Projective modules as fiber bundles, Proc. Amer. Math. Soc. 27(1971), 623–626. Google Scholar

[8] 8. Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79(1964), 109–326. Google Scholar

[9] 9. Knebusch, M., On algebraic curves over real closed fields I, Math Z. 150(1976), 49–70. Google Scholar

[10] 10. Lang, S. and A. Néron, Rational points of abelian varieties over function fields, Amer. J. Math. 81(1959), 95–118. Google Scholar

[11] 11. Serre, J.-P., Groups Algébriques et Corps de Classes, Hermann, Paris, 1959. Google Scholar

[12] 12. Silhol, R., Real Algebraic Surfaces, Lecture Notes in Math. 1392, Springer, 1989. Google Scholar

[13] 13. Swan, R. G., Topological examples of projective modules, Trans. Amer. Math. Soc. 230(1977), 201–234. Google Scholar

Cité par Sources :