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In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = [ Y / G ] {\displaystyle X=[Y/G]} , the Chow group of X is the same as the G - equivariant Chow group of Y .
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. [1] The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a ...
The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed ...
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory.Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves, and the moduli stack of elliptic curves.
That is, a scheme-theoretic multiplicity of an intersection may differ from an intersection-theoretic multiplicity, the latter given by Serre's Tor formula. Solving this disparity is one of the starting points for derived algebraic geometry, which aims to introduce the notion of derived intersection.
For example, if X and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. [ 1 ] If Spec B {\displaystyle \operatorname {Spec} B} is regularly embedded into a regular scheme , then B is a complete intersection ring .
Example: Let f be a regular function on a smooth variety (or stack). Then the set of critical points of f carries a symmetric obstruction theory in a canonical way. Example: Let M be a complex symplectic manifold. Then the (scheme-theoretic) intersection of Lagrangian submanifolds of M carries a canonical symmetric obstruction theory.
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