By Guenard F., Lelievre G.
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Extra resources for Complements d'Analyse. Topologie. Premiere Partie
Example text
2. Let u be an N-cone. (1) There is a one-to-one correspondence between the following sets: (a) The set A(u) := {T c Np. z; z E X u } of T-orbits in X u , (c) the set of non-empty closed irreducible T-invariant subvarieties of x,, (d) the set of open a f i n e toric subvarieties of X u . (2) This correspondence is explicitly given as follows: To a face r 5 CJ, we first associate a “base point” z, := XV(0) E xu 28 where v is an arbitrary lattice vector in the relative interior 7’. x,, - respectively.
4). 8, this description U = UX,” yields an analogous description u’ = ri. Hence, there is a face ri 5 u satisfying dimTi = dimu’. Together with ~i u’ u,this implies that a’ is included in lin(ri) n u = ri, thus 0 proving u’= ri. u c The one-to-one correspondence between faces and affine open toric subvarieties thus established actually is only one aspect of a larger picture: The combinatorial face structure of a cone also corresponds to the orbit structure and the structure of invariant irreducible closed subvarieties.
The open affine "charts" given by the n-cones U, (for J (0,. . ,n} as above) are cyclic quotients Cn/Gj. There is an isomorphism P(a) z Pn/G(a) with G(a) := Cai acting coordinatewise on P,, so in particular, P(1,. . ,1) equals P,. Moreover, the description P, E (C"+')/D given above generalizes t o the weighted projective space if one replaces the diagonal 1-subtorus D c (C*)n+l with D(a) := {t" = ( P o , . , t a n ); t E C*}. 5 ny=o Secondly, there is a similar equivalence of suitable categories (cf.
Complements d'Analyse. Topologie. Premiere Partie by Guenard F., Lelievre G.
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