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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.
A 3-dimensional foliated chart with n = 3 and q = 1. The plaques are 2-dimensional and the transversals are 1-dimensional. A rectangular neighborhood in R n is an open subset of the form B = J 1 × ⋅⋅⋅ × J n, where J i is a (possibly unbounded) relatively open interval in the ith coordinate axis. If J 1 is of the form (a,0], it is said ...
Gneiss, a foliated metamorphic rock. Quartzite, a non-foliated metamorphic rock. Foliation in geology refers to repetitive layering in metamorphic rocks. [1] Each layer can be as thin as a sheet of paper, or over a meter in thickness. [1] The word comes from the Latin folium, meaning "leaf", and refers to the sheet-like planar structure. [1]
A rationally connected variety V is a projective algebraic variety over an algebraically closed field such that through every two points there passes the image of a regular map from the projective line into V. Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety. [3]
For example, if two points in a variety approach each other in an algebraic family, the limiting subvariety is a single point, the limiting algebraic cycle is a point with multiplicity 2, and the limiting subscheme is a 'fat point' which contains the tangent direction along which the two points collided.
Sometimes one writes [] for the class of a subvariety in the Chow group, and if two subvarieties and have [] = [], then and are said to be rationally equivalent. For example, when X {\displaystyle X} is a variety of dimension n {\displaystyle n} , the Chow group C H n − 1 ( X ) {\displaystyle CH_{n-1}(X)} is the divisor class group of X ...
The map is said to be ramified at a point P in S′ if there exist analytic coordinates near P and π(P) such that π takes the form π(z) = z n, and n > 1. An equivalent way of thinking about this is that there exists a small neighborhood U of P such that π( P ) has exactly one preimage in U , but the image of any other point in U has exactly ...
The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties .