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Moduli of cubic surfaces and their anticanonical divisors. (English) Zbl 1441.14153

Summary: We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using geometric invariant theory and the anticanonical polarization. The construction depends on a weight on the divisor. For smaller weights the stable pairs consist of mildly singular surfaces and very singular divisors. Conversely, a larger weight allows more singular surfaces, but it restricts the singularities on the divisor. The one-dimensional space of stability conditions decomposes in a wall-chamber structure. We describe all the walls and relate their value to the worst singularities appearing in the compactification locus. Furthermore, we give a complete characterization of stable and polystable pairs in terms of their singularities for each of the compactifications considered.

MSC:

14L24 Geometric invariant theory
14J10 Families, moduli, classification: algebraic theory
14J26 Rational and ruled surfaces
14J45 Fano varieties
14Q10 Computational aspects of algebraic surfaces

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