Moduli of cubic surfaces and their anticanonical divisors

Gallardo, Patricio; Martinez-Garcia, Jesus

doi:10.1007/s13163-019-00298-y

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Moduli of cubic surfaces and their anticanonical divisors

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Martinez-Garcia, Jesus
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s13163-019-00298-y
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arXiv:1607.03697.pdf
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Gallardo-Martinez-Garcia_Moduli of cubic surfaces and their anticanonical divisors_2019.pdf
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Citation

Gallardo, P., & Martinez-Garcia, J. (2019). Moduli of cubic surfaces and their anticanonical divisors. Revista Matemática Complutense, 32(3), 853-873. doi:10.1007/s13163-019-00298-y.

Cite as: https://hdl.handle.net/21.11116/0000-0004-D94D-2

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.