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.
|Number of pages||21|
|Journal||Revista Matemática Complutense|
|Early online date||25 Feb 2019|
|Publication status||Published - 1 Sept 2019|
- Classification of singularities
- Cubic surfaces
- Fano varieties
- Geometric invariant theory
- Moduli of pairs
ASJC Scopus subject areas
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Martinez Garcia, J. (Creator) & Gallardo, P. (Creator), University of Bath, 12 Dec 2017