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The question of whether the varieties ′ and appearing above are non-singular is an important one. It seems natural to hope that if we start with smooth , then we can always find a minimal model or Fano fibre space inside the category of smooth varieties. However, this is not true, and so it becomes necessary to consider singular varieties also.
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 ...
A polarisation of an abelian variety is an isogeny from an abelian variety to its dual that is symmetric with respect to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the associated graph morphism is ample (so it is analogous to a positive-definite quadratic form). Polarised abelian varieties have ...
since the first two -terms are zero, which follows from being of genus , and the second calculation follows from the Riemann–Roch theorem, we have convexity of . Then, any nodal map can be reduced to this case by considering one of the components C i {\displaystyle C_{i}} of C {\displaystyle C} .
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.
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]
In contrast to positively curved varieties such as del Pezzo surfaces, a complex algebraic K3 surface X is not uniruled; that is, it is not covered by a continuous family of rational curves. On the other hand, in contrast to negatively curved varieties such as surfaces of general type, X contains a large discrete set of rational curves ...
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 ...