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We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix.
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This content will become publicly available on January 1, 2026
The birational geometry of moduli of cubic surfaces and cubic surfaces with a line
Abstract We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the moduli space of unmarked cubic surfaces.
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- Award ID(s):
- 2101631
- PAR ID:
- 10589003
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Moduli
- Volume:
- 2
- ISSN:
- 2949-7647
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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