Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces. (English) Zbl 1539.14063
The moduli space of stable surfaces, is a modular compactification of the Gieseker moduli space of canonical models of surfaces of general type. Unlike in the case of stable curves, the geometry of the surfaces parametrised by the boundary and the component structure of the boundary has remained elusive so far.
The authors consider the most popular test case, surfaces with \(K_X^2 =1\) and \(\chi(\mathcal O_X)=3\) and construct eight new divisors in the closure of the Gieseker component.
The general element in each of these divisors is constructed as follows: the authors write down explict degenerations to surfaces with an unimodal, non-log-canonical singularity and then construct the stable replacement via a weighted blow up. They also give a geometric description of the surfaces in the divisor in most cases.
The remaining geometric descriptions and a proof that these are the only divisors arising in this way was given in [S. Rollenske and D. Torres, “I-surfaces from surfaces with one exceptional unimodal point”, Preprint, arXiv:2305.01316].
The authors consider the most popular test case, surfaces with \(K_X^2 =1\) and \(\chi(\mathcal O_X)=3\) and construct eight new divisors in the closure of the Gieseker component.
The general element in each of these divisors is constructed as follows: the authors write down explict degenerations to surfaces with an unimodal, non-log-canonical singularity and then construct the stable replacement via a weighted blow up. They also give a geometric description of the surfaces in the divisor in most cases.
The remaining geometric descriptions and a proof that these are the only divisors arising in this way was given in [S. Rollenske and D. Torres, “I-surfaces from surfaces with one exceptional unimodal point”, Preprint, arXiv:2305.01316].
Reviewer: Sönke Rollenske (Marburg)
MSC:
| 14J10 | Families, moduli, classification: algebraic theory |
| 14D06 | Fibrations, degenerations in algebraic geometry |
| 14E05 | Rational and birational maps |
| 14D07 | Variation of Hodge structures (algebro-geometric aspects) |
Keywords:
Hodge theory; Horikawa surface; moduli space; stable pair compactification; unimodal singularityReferences:
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