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Enriques-Kodaira Classification

• Work over \({\mathbf{C}}\) for simplicity, take all dimensions over \({\mathbf{C}}\).

• Minimal model program : classifying complex projective varieties.

• Stratify the moduli space of varieties by \({\mathbf{k}}{\hbox{-}}\)dimension.

• Dimension 1:

  • Smooth Algebraic curves = compact Riemann surfaces, classifed by genus
  • Roughly known by Riemann: moduli space of smooth projective curves \({\mathcal{M}}_g\) is a connected open subset of a projective variety of dimension \(3g-3\).

• Dimension 2:

  • Smooth Algebraic Surfaces : Hard. See Enriques classification.

  • Setting of classical theorem: always 27 lines on a cubic surface!

  • Example Clebsch surface, satisfies the system \begin{align*} \begin{array}{l} x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0 \\ \\ x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0 \end{array} \end{align*}

Kodaira dimension