GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913758079614976 |
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| author | Bassi, Lucas Li |
| author_facet | Bassi, Lucas Li |
| contents | We study the behaviour on some nodal hyperplanes of the isomorphism, described in a paper of 2019 by Boissière, Camere and Sarti, between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of $K3^{[2]}$-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6; along those hyperplanes the automorphism degenerates by jumping to another family. We generalize their result to singular nodal cubic threefolds having one singularity of type $A_i$ for $i=2, 3, 4$ providing birational maps between the loci of cubic threefolds where a generic element has an isolated singularity of the types $A_i$ and some moduli spaces of hyperkähler fourfolds of $K3^{[2]}$-type with non-symplectic automorphism of order three belonging to different families. In order to treat the $A_2$ case, we introduce the notion of Kähler cone sections of $K$-type generalizing the definition of $K$-general polarized hyperkähler manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11149 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type Bassi, Lucas Li Algebraic Geometry 14J17, 14J30, 14J42, 14J50 We study the behaviour on some nodal hyperplanes of the isomorphism, described in a paper of 2019 by Boissière, Camere and Sarti, between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of $K3^{[2]}$-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6; along those hyperplanes the automorphism degenerates by jumping to another family. We generalize their result to singular nodal cubic threefolds having one singularity of type $A_i$ for $i=2, 3, 4$ providing birational maps between the loci of cubic threefolds where a generic element has an isolated singularity of the types $A_i$ and some moduli spaces of hyperkähler fourfolds of $K3^{[2]}$-type with non-symplectic automorphism of order three belonging to different families. In order to treat the $A_2$ case, we introduce the notion of Kähler cone sections of $K$-type generalizing the definition of $K$-general polarized hyperkähler manifolds. |
| title | GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type |
| topic | Algebraic Geometry 14J17, 14J30, 14J42, 14J50 |
| url | https://arxiv.org/abs/2301.11149 |