Some singular curves and surfaces arising from invariants of complex reflection groups. (English) Zbl 1474.14010
Exp. Math. 30, No. 3, 429-440 (2021); correction ibid. 29, No. 3, 360 (2020).
Barth and Sarti used pencils of surfaces constructed from invariants of some finite Coxeter groups to obtain surfaces of degree 6, 10, 12 with a record number of nodes [A. Sarti, J. Algebra 246, No. 1, 429–452 (2001; Zbl 1064.14038)].
The author reports on his systematic exploration of pencils of curves and surfaces constructed from invariants of finite complex reflection subgroups of \(\mathrm{GL}_3(\mathbb C)\) and \(\mathrm{GL}_4(\mathbb C)\). Among the results are a curve of degree 14 with 42 cusps from the complex reflection group \(G_{24}\), the known upper bound being 55. Furthermore surfaces of degree 8, 12 and 24 with 48, 160 and 1440 singularities of type \(D_4\), respectively.
In the computations with MAGMA, reflection groups \(W\) are represented as subgroups of \(\mathrm{GL}_n(K)\) with \(K\) a number field depending on \(W\). The choice of model has a considerable impact on computation time, and on the form of the defining polynomials obtained. The author finds an equation for the Sarti surface of degree 12 with 600 nodes, defined over \(\mathbb Q\).
The strategy for finding highly singular elements of pencils is explained, but most of the MAGMA code is contained in two separate texts, not intended to be published, but available for readers interested in checking the computations by themselves, [“A surface of degree 24 with 1440 singularities of type \(D_4\)”, Preprint, arXiv:1804.08388; “Magma codes for ‘Some singular curves and surfaces arising from invariants of complex reflection groups”’, Preprint, hal.archives-ouvertes.fr/ hal-01897587].
The author reports on his systematic exploration of pencils of curves and surfaces constructed from invariants of finite complex reflection subgroups of \(\mathrm{GL}_3(\mathbb C)\) and \(\mathrm{GL}_4(\mathbb C)\). Among the results are a curve of degree 14 with 42 cusps from the complex reflection group \(G_{24}\), the known upper bound being 55. Furthermore surfaces of degree 8, 12 and 24 with 48, 160 and 1440 singularities of type \(D_4\), respectively.
In the computations with MAGMA, reflection groups \(W\) are represented as subgroups of \(\mathrm{GL}_n(K)\) with \(K\) a number field depending on \(W\). The choice of model has a considerable impact on computation time, and on the form of the defining polynomials obtained. The author finds an equation for the Sarti surface of degree 12 with 600 nodes, defined over \(\mathbb Q\).
The strategy for finding highly singular elements of pencils is explained, but most of the MAGMA code is contained in two separate texts, not intended to be published, but available for readers interested in checking the computations by themselves, [“A surface of degree 24 with 1440 singularities of type \(D_4\)”, Preprint, arXiv:1804.08388; “Magma codes for ‘Some singular curves and surfaces arising from invariants of complex reflection groups”’, Preprint, hal.archives-ouvertes.fr/ hal-01897587].
Reviewer: Jan Stevens (Göteborg)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 20-08 | Computational methods for problems pertaining to group theory |
| 14H20 | Singularities of curves, local rings |
Citations:
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