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arXiv · 1607.03697

Moduli of cubic surfaces and their anticanonical divisors

Patricio Gallardo·Jesus Martinez-Garcia
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Abstract

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.

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Patricio Gallardo, Jesus Martinez-Garcia. 2019-02-12. Moduli of cubic surfaces and their anticanonical divisors. https://doi.org/10.1007/s13163-019-00298-y

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