RESOLUTIONS OF SINGULARITIES.

YOSHIFUMI TSUCHIMOTO

01. Nagata Hironaka. We refer to [1] as a general reference. Theorem 1.1 (Nagata). Every scheme X of finite type over a Noe-

therian integral scheme S is a open subscheme of a proper scheme X′ over S. of a complete

Theorem 1.2 (Hironaka). Every singular variety over a field 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 1.1. 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.

References

[1] Herwig Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand)., Bull. Am. Math. Soc., New Ser. 40 (2003), no. 3, 323–403 (English).