Web25 Jan 2024 · Let X be a smooth variety over a field k. The index of X over k is the gcd of the degrees [κ(x) : k] over all closed points x of X. The index is 1 if and only if X has a zero cycle of degree 1. If k is perfect, then the index of X is a birational invariant on smooth varieties over k: The reason is that given a nonempty open U of X and a closed point x in X you can … WebProposition 8.6. The set of smooth points of any variety is Zariski dense. Proof. Since the dimension of the Zariski tangent space is upper semi-continuous, and always at least the dimension of the variety, it su ces to prove that every irreducible variety contains at least one smooth point. By (8.5) we may assume that Xis a hypersurface ...
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Web23 Jun 2024 · A singular point, or singularity, of an algebraic variety is a point at which smoothness is violated. More accurately, let $ X $ be an algebraic variety or a scheme of finite type over a field $ k $. Then a point $ x \in X $ is said to be singular if the corresponding local ring $ {\mathcal O} _{X,x} $ is not regular (regularity of a local … Web1 Mar 2000 · A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working … pusheen and stormy
Smooth varieties with torus actions - ScienceDirect
A smooth scheme over a field is regular and hence normal. In particular, a smooth scheme over a field is reduced. Define a variety over a field k to be an integral separated scheme of finite type over k. Then any smooth separated scheme of finite type over k is a finite disjoint union of smooth varieties over k. For … See more In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. … See more A scheme X is said to be generically smooth of dimension n over k if X contains an open dense subset that is smooth of dimension n over k. … See more • Étale morphism • Dimension of an algebraic variety • Glossary of scheme theory See more First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space A over k for some natural number n. Then X is the closed subscheme defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial … See more • Affine space and projective space are smooth schemes over a field k. • An example of a smooth hypersurface in projective space P over k is the Fermat hypersurface x0 + ... See more Web3 Nov 2024 · Sometimes a smooth algebraic variety may also be called algebraic manifold. An abstract k k-prevariety in the sense of Serre is a locally ringed space which is locally isomorphic to affine k k-variety. The category of k k-prevarieties has a product which is obtained by locally gluing products in the category of affine k k-varieties. Web28 Oct 2024 · The second potential source of confusion is that a regular scheme over a perfect field is necessarily smooth over that field, but for an imperfect field this fails. See … security training las vegas