Toric Orbifold
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In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a toric stack is a stacky generalization of a
toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be ...
. More precisely, a toric stack is obtained by replacing in the construction of a toric variety a step of taking
GIT quotient In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of ...
s with that of taking
quotient stack In algebraic geometry, a quotient stack is a stack (mathematics), stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a Scheme (mathematics), scheme or a algebraic variety, variety by a Group (mathematics), group ...
s. Consequently, a toric variety is a coarse approximation of a toric stack. A toric orbifold is an example of a toric stack.


See also

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Stanley–Reisner ring In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial ring, polynomial algebra over a field (algebra), field by a square-free monomial ideal, monomial ideal (ring theory), ideal. Such ideals are described more geomet ...


References

* * * Algebraic geometry {{algebraic-geometry-stub