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I want to prove Castelnuovo's contraction theorem by Mori's contraction theorem.

Question. How can one show that a $(-1)$ curve on a smooth surface is an extremal ray?

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In general, if $C$ is an irreducible curve on a smooth surface $X$ such that $C^2 \leq 0$ then $[C]$ is in the boundary of $\overline{NE}(X)$, and if $C^2 <0$ then $[C]$ is extremal in $\overline{NE}(X)$.

See Lemma 1.22, p. 21 in

J. Kollár, S. Mori: Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press. viii, 254 p. (1998). ZBL0926.14003.

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    $\begingroup$ Thanks a lot!!! $\endgroup$
    – George
    Commented May 15 at 15:07
  • $\begingroup$ @George, if you like and find useful Francesco's answer, accepting it is perhaps a nice thing to do. $\endgroup$
    – Daniele Tampieri
    Commented May 16 at 10:18

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