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Resolutions of singularities.

Yoshifumi Tsuchimoto

\fbox{01. Nagata Hironaka.}

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$

\includegraphics[scale=0.4]{implicit.eps} \includegraphics[scale=0.4]{implicit2.eps}

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$ .