Some Singular Curves and Surfaces Arising from Invariants of Complex Reflection Groups

Abstract

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups.(1) There is an important exception to this remark: all the singular points of the surface of degree 8 with 48 singularities of type D 4 constructed in Example 4.3 have rational coordinates.

Complex reflection groups and K3 surfaces I

“…Remark 2.5. The previous lemma might be used to explain the construction of several singular curves and surfaces constructed by the two authors [Sar01], [Bon19]. Let us explain how to proceed.…”

Complex reflection groups and K3 surfaces II. The groups $G_{29}$, $G_{30}$ and $G_{31}$

“…ANR-20-CE40-0026-01 (SMAGP). authors who constructed highly singular surfaces from invariants of complex reflection groups [20], [3]. Most (but not all) of the singularities constructed this way can be obtained from [5,Corollary 2.4].…”

“…requires computer calculations. We use here the software Magma [8] (as well as some specific functions described in [4]).…”

“…In §4. 3 we describe an elliptic fibration on this surface and thanks to that we compute the transcendental lattice. This result is new and summarized in the following theorem: Theorem 1.1.…”

“…These values of λ and the singular fibers of an elliptic fibration are resumed in Table 7. 3 Finally in the Appendix we collect several useful results that allow to find the equations of the K3 surfaces and the elliptic fibrations. We remark that in the case of G 31 we described a one parameter family of K3 surfaces, we believe that this family is not isotrivial, but we could not prove it, for G 30 this was shown in [1].…”