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Good evening,

I'm stuck in the following exercise in Huybrechts, Complex Geometry, chapter 2, page 103.

Let $X$ be a K3 surface, i.e. X is a compact complex surface with $K_X \cong \mathcal{O}_X$ and $h^1(X,\mathcal{O}_X)=0.$ Show that X is not the blow-up of any other smooth surface. Here $K_X$ is the canonical line bundle of $X.$

Does anyone have some ideas to solve this exercise?

Thanks in advance,

Duc Anh

Đức Anh
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    Do you know the formula for the canonical bundle of a blowup? –  Apr 01 '13 at 19:10
  • Yes, the proposition 2.5.5 in Huybrechts gives a formula, but I don't see relations :( – Đức Anh Apr 01 '13 at 19:15
  • Well, could the formula there give $O_X$ as a result? –  Apr 01 '13 at 19:26
  • Thank you. The formula is $K_{\hat{X}}\cong \sigma^{\ast}K_X\otimes\mathcal{O}{\hat{X}}(E)$ (here $\dim X =2$). So may $\sigma^{\ast}K_X\otimes\mathcal{O}{\hat{X}}(E)$ have a non-trival global section? – Đức Anh Apr 01 '13 at 19:36
  • Related: http://math.stackexchange.com/questions/284649/why-the-k3-surfaces-are-minimal-surfaces – Matt Apr 01 '13 at 23:15
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    Thank you all very much. I think I could solve the exercise, by showing that $\sigma^{\ast}K_X \otimes \mathcal{O}_{\hat{X}}(E)$ has a non-trival global section, while $\mathcal{O}_X$ does not. – Đức Anh Apr 02 '13 at 13:43

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By the adjunction formula we have $$2g(C) - 2 = C^2$$ for the self-intersection of any curve C on X. Hence X does not contain any -1 curve, which were necessary for a blow-up, q.e.d.

Jo Wehler
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