User:Bensculfor/Affine variety

A summary of eventual additions to the article affine variety.

Examples

 * More elementary examples (plane curves)
 * work through an example (affine subvarieties of the complex plane)
 * give a non-example (e.g. $V(x^{2}-1)$ )

Structure sheaf

 * Rewrite to be more elementary
 * Define sheaf (roughly)
 * Start with showing local ring at a point
 * Show restriction/gluing
 * Keep some of the category theory at the end

Tangent space

 * Define in terms of derivations, then show that this space is spanned by the partial derivatives.
 * Add general plane curve paragraph (give example of a singularity).

Dimension section (new, after Tangent space)

 * Krull dimension
 * Proper chain of nonempty subvarieties (i.e. topological dimension)
 * Smooth vs. singular points (Krull dim \leq \dim T_xV)
 * Mention codimension

An example: the tangent to an affine plane curve
If we have an equation $y = f (x)$ (where $f$ is a polynomial in one variable), this corresponds to the hypersurface $C$ in the affine complex plane $C^{2}$ defined by $y − f (x).$ The partial derivatives with respect to $x$ and $y$ are $−f_{x}(x)$ and 1 respectively. Then the tangent space to $C$ at the point $p = (a,b)$ is the vanishing set defined by $−f_{x}(p) (x−a) + (y−b).$ This can be rewritten as the solution set of $y = f_{x}(p) x + (af_{x}(p)+b).$ If we consider only the real points (i.e. the R-rational points) of the tangent line and the curve, this agrees with the definition of the tangent line to a function $f : R → R$ given by differential calculus. As $C_{y}(p) = 1$ at every point $p$ on $C$, the tangent space never vanishes, so the curve is non-singular everywhere.

A general affine plane curve $F(X,Y)$ cannot be expressed in this form.

Product of affine varieties

 * Add example to second paragraph
 * Mention dimension of product (that $dim V × W = dim V + dim W$ for $V,$ $W$ regular)