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Suppose that $X$ is an algebraic K3 surface (say polarized). If the singular divisor of $X$ is normal crossing... Do we have a moduli space parametrizing such $K3$ surfaces? If yes do we have a Toroidal compactification of it?

Any reference?

Thanks.

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yes! For a modern account, have a look at Martin Olsson: "Semistable degenerations and period spaces for polarized K3 surfaces", Duke Math. J. 125, 121-203 (2004). There, the problem is analyzed from the point of view of logarithmic geometry and you find a discussion of the toroidal point of view (plus references). –  Christian Liedtke Oct 14 '12 at 3:43
    
great! thanks a lot! I will take a look! –  Yoyontzin Oct 14 '12 at 4:36
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