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Hi,

My question is related to Algebraic Surfaces. I have seen that we always consider K3 surfaces which are smooth, but I wonder how can we define a non-smooth K3 surface.

The problem I see is on the canonical bundle. So if we suppose $X$ is an algebraic surface ofer a field $K$ and suppose we hace that the singular divisor is normal crossing. Then we have logarithmic differentials. So I wonder if the definition of non-smooth K3 surface is the one that I am imagine, that is: $X$ is a $K3$ surface if $H^1(X, \mathfrak O_X)=0$ and $\Omega^2_X(log)$ is trivial...

Thanks.

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An algebraic (possibly singular) $K3$ surface is a normal algebraic surface whose minimal resolution is a smooth $K3$ surface. You can obtain these for example by blowing down $(-2)$-curves on a smooth algebraic $K3$. As long as you only do that you can even restrict the kind of singularities you allow. For example blowing down a $(-2)$-curve (and certain configurations) will lead to Du Val (=rational Gorenstein=rational double points) singularities.

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  • $\begingroup$ I see, thanks. Do you have a reference for this? I would like also to know if people work with logarithmic geometry on singular K3 surfaces. Thanks a lot. $\endgroup$
    – Rogelio Yoyontzin
    Oct 11, 2012 at 14:05
  • $\begingroup$ I don't know a good reference. I learnt this definition from my advisor when I was a graduate student. As a result I put it in a paper that was based on my thesis and no one has complained yet... $\endgroup$
    – Sándor Kovács
    Oct 12, 2012 at 0:41
  • $\begingroup$ ok, Thanks. I think that your answer also is related with what I was thinking. Thanks a lot! $\endgroup$
    – Rogelio Yoyontzin
    Oct 14, 2012 at 2:03
  • $\begingroup$ This is quite late, but here's a reference for anywhere in the future: Definition 3.2 in Yuya Matsumoto. $\mu_p$- and $\alpha_p$-actions on K3 surfaces in characteristic $p$. J. Algebraic Geom., 32(2):271–322, 2023 $\endgroup$
    – PCeltide
    Dec 16 at 13:33
  • $\begingroup$ arXiv: arxiv.org/abs/1812.03466 $\endgroup$
    – PCeltide
    Dec 16 at 18:38

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