7,545 research outputs found
Bunches of cones in the divisor class group -- A new combinatorial language for toric varieties
As an alternative to the description of a toric variety by a fan in the
lattice of one parameter subgroups, we present a new language in terms of what
we call bunches -- these are certain collections of cones in the vector space
of rational divisor classes. The correspondence between these bunches and fans
is based on classical Gale duality. The new combinatorial language allows a
much more natural description of geometric phenomena around divisors of toric
varieties than the usual method by fans does. For example, the numerically
effective cone and the ample cone of a toric variety can be read off
immediately from its bunch. Moreover, the language of bunches appears to be
useful for classification problems.Comment: Minor changes, to appear in Int. Math. Res. No
Three lectures on Cox rings
Notes of an introductory course given at the conference "Torsors: Theory and
Applications" in Edinburgh, January 2011.Comment: Minor corrections, 37 page
Homogeneous coordinates for algebraic varieties
We associate to every divisorial (e.g. smooth) variety with only constant
invertible global functions and finitely generated Picard group a
-graded homogeneous coordinate ring. This generalizes the usual
homogeneous coordinate ring of the projective space and constructions of Cox
and Kajiwara for smooth and divisorial toric varieties. We show that the
homogeneous coordinate ring defines in fact a fully faithful functor. For
normal complex varieties with only constant global functions, we even
obtain an equivalence of categories. Finally, the homogeneous coordinate ring
of a locally factorial complete irreducible variety with free finitely
generated Picard group turns out to be a Krull ring admitting unique
factorization.Comment: 30 page
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