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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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1 Answer 1

up vote 7 down vote accepted

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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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. –  Yoyontzin Oct 11 '12 at 14:05
    
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... –  Sándor Kovács Oct 12 '12 at 0:41
    
ok, Thanks. I think that your answer also is related with what I was thinking. Thanks a lot! –  Yoyontzin Oct 14 '12 at 2:03

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