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A web article with many references explaining how the groupoid concept has led to notions of higher-dimensional groupoids, not available in group theory, with applications in homotopy theory and in group cohomology. Brown, R.; Higgins, P.J. (1981). "On the algebra of cubes". Journal of Pure and Applied Algebra. 21 (3): 233– 260.
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any smooth projective surface.
An arithmetic cycle of codimension p is a pair (Z, g) where Z ∈ Z p (X) is a p-cycle on X and g is a Green current for Z, a higher-dimensional generalization of a Green function. The arithmetic Chow group C H ^ p ( X ) {\displaystyle {\widehat {\mathrm {CH} }}_{p}(X)} of codimension p is the quotient of this group by the subgroup generated by ...
In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hypersurface (of dimension D) embedded in a higher dimensional space (of dimension D + 1) is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D=1 is the case of conic sections ...
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the Siegel modular variety. This is the problem underlying Siegel modular form theory.
An alternative approach, [4] motivated by the goal of constructing geometry over spaces of paths and higher-dimensional objects, has been developed by Saikat Chatterjee, Amitabha Lahiri, and Ambar N. Sengupta. The mathematical framework for traditional gauge theory places the gauge potential as a 1-form on a principal bundle over spacetime.
The Hodge conjecture generalises this statement to higher dimensions. In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops.
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