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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 ...
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 .
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.
Fulton and MacPherson extended the Chow ring to singular varieties by defining the "operational Chow ring" and more generally a bivariant theory associated to any morphism of schemes. [13] A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively.
Let : be a local-complete-intersection morphism that admits a global factorization: it is a composition where is a regular embedding and a smooth morphism. Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as: [ 5 ]
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them".
Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory, which Serre sought to address. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra.