User:Bensculfor/Affine variety

A summary of eventual additions to the article affine variety.

To-do list

Examples

• More elementary examples (plane curves)
• work through an example (affine subvarieties of the complex plane)
• give a non-example (e.g. V(x²-1) )

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

Tangent space

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 C² 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)