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. 



Non Smooth K3 surface?
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.


