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One says that “the affine plane does not have a good intersection theory”, and intersection theory on non-projective varieties is much more difficult. A line on a P 1 × P 1 (which can also be interpreted as the non-singular quadric Q in P 3) has self-intersection 0, since a line can be moved off itself. (It is a ruled surface.)
Angelo Vistoli () develops the basic theory (mostly over Q) for the Chow group of a (separated) Deligne–Mumford stack.There, the Chow group is defined exactly as in the classical case: it is the free abelian group generated by integral closed substacks modulo rational equivalence.
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.
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
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 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 general, they cannot be represented by schemes or even algebraic spaces, but in many cases, they have a natural structure of an algebraic stack. Algebraic stacks and their use to analyze moduli problems appeared in Deligne-Mumford (1969) as a tool to prove the irreducibility of the (coarse) moduli space of curves of a given genus.
Given an (affine or projective) algebraic variety (or more generally: any scheme or algebraic stack), the category of quasi-coherent sheaves on is a Grothendieck category. [ 4 ] Given a small site ( C , J ) (i.e. a small category C together with a Grothendieck topology J ), the category of all sheaves of abelian groups on the site is a ...