Abstract
Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs (X,H) consisting of a degree two K3 surface X and an ample divisor H . Specifically, we construct and describe explicitly a geometric compactification \overline{\mathcal P}_2 for the moduli of degree two K 3 pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space \mathcal F_2 of degree two K 3 surfaces. Using this map and the modular meaning of \overline{\mathcal P}_2 , we obtain a better understanding of the geometry of the standard compactifications of \mathcal F_2 .
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