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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 ...
Later Andrew Kresch (1999) extended the theory to a stack admitting a stratification by quotient stacks. For higher Chow groups (precursor of motivic homologies ) of algebraic stacks, see Roy Joshua's Intersection Theory on Stacks:I and II.
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.
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.
"K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems". arXiv: math/9908097. Lowrey, Parker; Schürg, Timo (2012-08-30). "Grothendieck-Riemann-Roch for derived schemes". arXiv: 1208.6325 . Vakil, Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry
schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks. Edidin (2003) and Fantechi (2001) give a brief introductory accounts of stacks, Gómez (2001) , Olsson (2007) and Vistoli (2005) give more detailed introductions, and Laumon & Moret-Bailly (2000) describes the more advanced theory.
The quotient stack of, say, an algebraic space X by an action of a group scheme G. / / The GIT quotient of a scheme X by an action of a group scheme G. L n An ambiguous notation. It usually means an n-th tensor power of L but can also mean the self-intersection number of L.
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 ...