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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.
Stacks are the underlying structure of algebraic stacks (also called Artin stacks) and Deligne–Mumford stacks, which generalize schemes and algebraic spaces and which are particularly useful in studying moduli spaces. There are inclusions: schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks.
The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry. Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach.
A data structure known as a hash table.. In computer science, a data structure is a data organization and storage format that is usually chosen for efficient access to data. [1] [2] [3] More precisely, a data structure is a collection of data values, the relationships among them, and the functions or operations that can be applied to the data, [4] i.e., it is an algebraic structure about data.
In algebraic geometry, a Deligne–Mumford stack is a stack F such that the diagonal morphism F → F × F {\displaystyle F\to F\times F} is representable , quasi-compact and separated. There is a scheme U and étale surjective map U → F {\displaystyle U\to F} (called an atlas ).