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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 ...
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 study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology. The study of quantum cohomology, Gromov–Witten invariants and mirror symmetry gave a significant progress in Clemens conjecture. Enumerative geometry is very closely tied to intersection theory. [1]
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".
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 Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne–Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for ...
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 ...
Justifying this calculus was the content of Hilbert's 15th problem, and was also the major topic of the 20 century algebraic geometry. [ 1 ] [ 2 ] In the course of securing the foundations of intersection theory, Van der Waerden and André Weil [ 3 ] [ 4 ] related the problem to the determination of the cohomology ring H*(G/P) of a flag ...