Geodesic
Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors
Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 562-572

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

In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety $X$ over $\mathbf{k}\,=\,\mathbb{C}$ . In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field $\mathbf{k}$ . Secondly we give an isomorphism between the group of Cartier $b$ -divisors on the birational class of $X$ and the Grothendieck group of the semigroup of subspaces of rational functions on $X$ . The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier $b$ -divisors and their intersection theory.
DOI : 10.4153/CMB-2013-039-6
Mots-clés : 14C20, 14Exx, intersection number, Cartier divisor, Cartier b-divisor, Grothendieck group 
Kaveh, Kiumars; Khovanskii, A. G. Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors. Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 562-572. doi: 10.4153/CMB-2013-039-6
@article{10_4153_CMB_2013_039_6,
     author = {Kaveh, Kiumars and Khovanskii, A. G.},
     title = {Note on the {Grothendieck} {Group} of {Subspaces} of {Rational} {Functions} and {Shokurov's} {Cartier} b-divisors},
     journal = {Canadian mathematical bulletin},
     pages = {562--572},
     year = {2014},
     volume = {57},
     number = {3},
     doi = {10.4153/CMB-2013-039-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2013-039-6/}
}
TY  - JOUR
AU  - Kaveh, Kiumars
AU  - Khovanskii, A. G.
TI  - Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors
JO  - Canadian mathematical bulletin
PY  - 2014
SP  - 562
EP  - 572
VL  - 57
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2013-039-6/
DO  - 10.4153/CMB-2013-039-6
ID  - 10_4153_CMB_2013_039_6
ER  - 
%0 Journal Article
%A Kaveh, Kiumars
%A Khovanskii, A. G.
%T Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors
%J Canadian mathematical bulletin
%D 2014
%P 562-572
%V 57
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2013-039-6/
%R 10.4153/CMB-2013-039-6
%F 10_4153_CMB_2013_039_6

[Fulton98] [Fulton98] Fulton, W., Intersection theory. Second ed. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, A Series of Modern Surveys in Mathematics, 2, Springer-Verlag, Berlin, 1998. Google Scholar

[Hartshorne77] [Hartshorne77] Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar

[Iskovskikh03] [Iskovskikh03] Iskovskikh, V. A., b-divisors and Shokurov functional algebras. (Russian) Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 8–20; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), 4–15. Google Scholar

[K-K10] [K-K10] Kaveh, K. and Khovanskii, A. G., Mixed volume and an extension of intersection theory of divisors. Mosc. Math. J. 10 (2010), no. 2, 343–375, 479. Google Scholar

[K-K12] [K-K12] Kaveh, K. and Khovanskii, A. G., Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. of Math. 176 (2012), no. 2, 925–978. Google Scholar | DOI

[L04] [L04] Lazarsfeld, R., Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, Folge, A Series of Modern Surveys in Mathematics, 48, Springer-Verlag, Berlin, 2004. Google Scholar

[M74] [M74] Manin, Yu. I., Cubic forms: algebra, geometry, arithmetic. North-Holland Mathematical Library, 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974. Google Scholar

[S-Z60] [S-Z60] Samuel, P. and Zariski, O., Commutative algebra. Vol. II. Reprint of the 1960 ed., Graduate Texts in Mathematics, 29, Springer, New York, 1976. Google Scholar

[S03] [S03] Shokurov, V. V., Prelimiting flips. Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 82–219; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), 75–213. Google Scholar

Cité par Sources :