## affine line

16 examples (0.02 sec)
• However, the formula above only makes sense for points in the affine line.
• For example, it is not hard to see that the affine line A 1 is not complete.
• Any finite list of points in the projective line can be moved into the affine line by a suitable projective transformation.
• Project the affine curve to the affine line using the x-coordinate.
• 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.
• As an affine set X is not closed since any polynomial zero on the complement must be zero on the affine line.
• Embed the affine line into the projective line, then take the normalization of the projective line in the function field of the affine curve.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.