The inverse cross ratio is invariant under projective transformations and thus makes sense for points in the projective line.However, the formula above only makes sense for points in the affine line.In the slightly more general set-up below, the cross ratio makes sense for any four collinear points in projective space One just identifies the line containing the points with the projective line by a suitable projective transformation and then uses the formula above.…

More generally, it can be shown that affine proper morphisms are necessarily finite.For example, it is not hard to see that the affine line A 1 is not complete.In fact the map taking A 1 to a point x is not universally closed.…

When the field underlying all the constructions is F, the affine line is just a copy of F. The affine line is a subset of the projective line.Any finite list of points in the projective line can be moved into the affine line by a suitable projective transformation.…

This smooth completion can also be obtained as follows.Project the affine curve to the affine line using the x-coordinate.Embed the affine line into the projective line, then take the normalization of the projective line in the function field of the affine curve.…

But it is in some way inadequate to the needs of geometry to use only the affine line as target for functions, since we shall end up only with constants.Regular maps are, by definition, morphisms in the category of algebraic varieties.…

As an affine set X is not closed since any polynomial zero on the complement must be zero on the affine line.For another example, the complement of any conic in projective space of dimension 2 is affine.…

Project the affine curve to the affine line using the x-coordinate.Embed the affine line into the projective line, then take the normalization of the projective line in the function field of the affine curve.A smooth connected curve over an algebraically closed field is called hyperbolic if 2g-2+r>0 where g is the genus of the smooth completion and r is the number of added points.…

This definition generalizes the multiplicity of a root of a polynomial in the following way.The roots of a polynomial f are points on the affine line, which are the components of the algebraic set defined by the polynomial.…

Since the functions ce z are defined on the whole affine line A 1, the monodromy of this flat connection is trivial.

By contrast, the affine line A 1 is normal: it cannot be simplified any further by finite birational morphisms.A normal complex variety X has the property, when viewed as a stratified space using the classical topology, that every link is connected.…

What is nice is that the various objects fit together exactly, and together add up to a proof that this torus motion really exists.When the field underlying all the constructions is F, the affine line is just a copy of F. The affine line is a subset of the projective line.Any finite list of points in the projective line can be moved into the affine line by a suitable projective transformation.…

The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval, which is not an algebraic variety, with the affine line, which is.The theory requires a substantial amount of technique to set up, but has spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.…

Geometrically, the polynomial ring in two variables represents the two-dimensional affine space A 2, so the existence of this one-parameter family says that affine space admits non-commutative deformations to the space determined by the Weyl algebra.In fact, this deformation is related to the symbol of a differential operator and the fact that A 2 is the cotangent bundle of the affine line.Studying the Weyl algebra can lead to information about affine space: The Dixmier conjecture about the Weyl algebra is equivalent to the Jacobian conjecture about affine space.…

The tame fundamental group of some scheme U is a quotient of the usual fundamental group of U which takes into account only covers that are tamely ramified along D, where X is some compactification and D is the complement of U in X. For example, the tame fundamental group of the affine line is zero.From a categoric point of view, the fundamental group is a functor The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions).…

In fact the map taking A 1 to a point x is not universally closed.For example, the morphism is not closed since the image of the hyperbola uv = 1, which is closed in A 1 × A 1, is the affine line minus the origin and thus not closed.…

If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.For example, the normalization of the subscheme X of the affine plane A 2 defined by xy = 0 is the disjoint union of two copies of the affine line, mapping to the lines x = 0 and y = 0 in X.…