Resolutions of singularities.

Yoshifumi Tsuchimoto

We refer to [1] as a general reference.

THEOREM 01.1 (Nagata)   Every scheme $X$ of finite type over a Noetherian integral scheme $S$ is a open subscheme of a proper scheme $X'$ over $S$ . of a complete

THEOREM 01.2 (Hironaka)   Every singular variety over a field $\mathbbm{k}$ of characteristic 0 has a "resolution of singularities".

Examples of singular curves: $y^2=x^2+x^3$ , $y^2=x^3$

Blow up: Add an extra variable $u=y/x$ .

Sometimes we need to blow up several times to obtain regular curve.

EXERCISE 01.1   Resolve the singularity of $y^3=x^5$ .

An example of a singular surface: $z^2=y^2-x^2$

Blow up: Add extra variables $u=y/x, v=z/x$ .