User:Bensculfor/Affine variety
A summary of eventual additions to the article affine variety.
To-do list[edit]
Examples[edit]
- 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[edit]
- 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[edit]
- 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)[edit]
- 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[edit]
Definition[edit]
An example: the tangent to an affine plane curve[edit]
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 −fx(x) and 1 respectively. Then the tangent space to C at the point p = (a,b) is the vanishing set defined by −fx(p) (x−a) + (y−b). This can be rewritten as the solution set of y = fx(p) x + (afx(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 Cy(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[edit]
- Add example to second paragraph
- Mention dimension of product (that dim V × W = dim V + dim W for V, W regular)