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**Picard numbers of Delsarte surfaces.**
*(English)*
Zbl 1344.14025

The Picard number is one of the most fundamental invariants of an algebraic variety, yet is is notoriously difficult to compute in general. It thus came as a big surprise when T. Shioda [Am. J. Math. 108, 415–432 (1986; Zbl 0602.14033)] worked out an algorithm to compute the Picard number of what he called a Delsarte surface, i.e., a hypersurface \(X\) in \(\mathbb P^3\) given by an irreducible polynomial which comprises four monomials only. Indeed, Delsarte surfaces are Galois quotients of Fermat surfaces, so cycles on \(X\) (in particular transcendental ones) can be computed in terms of invariant cycles on the cover which in turn amounts to a purely combinatorial calculation (over algebraically closed fields of any characteristic, in fact).

Recently, the Delsarte approach was used by the reviewer to engineer complex quintic surfaces in \(\mathbb P^3\) (defined over \(\mathbb Q\)) with any given Picard number in the range \(1,\dots,45=h^{1,1}(X)\) (see [M. Schütt, J. Math. Soc. Japan 63, No. 4, 1187–1201 (2011; Zbl 1232.14022); Proc. Lond. Math. Soc. (3) 110, No. 2, 428–476 (2015; Zbl 1316.14073)]).

The paper under review set outs in a different direction; namely the author gives closed formulas for the Picard numbers of Delsarte surfaces of any degree \(n\geq 5\). More precisely, he singles out all those Delsarte surfaces whose singularities, if any, are no worse than rational double points (also known as ADE singularities which preserve the deformation class and thus basic invariants such as Betti and Hodge numbers). Then he computes the Picard numbers of the minimal desingularizations, based on a methods going back to Shioda’s work [loc. cit.] and extended by the author in [Elliptic Delsarte surfaces. Groningen: Rijksuniversiteit Groningen (PhD Thesis) (2011)].

Recently, the Delsarte approach was used by the reviewer to engineer complex quintic surfaces in \(\mathbb P^3\) (defined over \(\mathbb Q\)) with any given Picard number in the range \(1,\dots,45=h^{1,1}(X)\) (see [M. Schütt, J. Math. Soc. Japan 63, No. 4, 1187–1201 (2011; Zbl 1232.14022); Proc. Lond. Math. Soc. (3) 110, No. 2, 428–476 (2015; Zbl 1316.14073)]).

The paper under review set outs in a different direction; namely the author gives closed formulas for the Picard numbers of Delsarte surfaces of any degree \(n\geq 5\). More precisely, he singles out all those Delsarte surfaces whose singularities, if any, are no worse than rational double points (also known as ADE singularities which preserve the deformation class and thus basic invariants such as Betti and Hodge numbers). Then he computes the Picard numbers of the minimal desingularizations, based on a methods going back to Shioda’s work [loc. cit.] and extended by the author in [Elliptic Delsarte surfaces. Groningen: Rijksuniversiteit Groningen (PhD Thesis) (2011)].

Reviewer: Matthias Schütt (Hannover)

### MSC:

14J29 | Surfaces of general type |

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\textit{B. Heijne}, J. Math. Soc. Japan 68, No. 1, 101--118 (2016; Zbl 1344.14025)

### References:

[1] | N. Aoki, On some arithmetic problems related to the Hodge cycles on the Fermat varieties, Math. Ann., 266 (1983), 23-54. · Zbl 0506.14030 |

[2] | N. Aoki, Erratum: “On some arithmetic problems related to the Hodge cycles on the Fermat varieties”, Math. Ann., 267 (1984), 572. · Zbl 0534.14021 |

[3] | N. Aoki and T. Shioda, Generators of the Néron-Severi group of a Fermat surface, In Arithmetic and geometry, Vol.,I, Progr. Math., 35 , Birkhäuser Boston, Boston, MA, 1983, 1-12. · Zbl 0586.14028 |

[4] | A. Degtyarev, On the picard group of a delsarte surface, 2013, · Zbl 1289.35338 |

[5] | A. H. Durfee, Fifteen characterizations of rational double points and simple critical points, Enseign. Math. (2), 25 (1979), 131-163. · Zbl 0418.14020 |

[6] | B. Heijne, Elliptic Delsarte Surfaces, PhD thesis, Rijksuniversiteit Groningen, 2011. · Zbl 1267.11071 |

[7] | J. Kogure, The Picard number of non-singular complex Delsarte surfaces (in Japanese), (based on the author’s master thesis at Tokyo Univ.). In: Proceedings of Arithmetic Algebraic Geometry Symposium, 1987, Tokyo Univ. (ed. T. Shioda), 1987, 75-99. |

[8] | M. Reid, The Du Val singularities \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\), http://homepages.warwick.ac.uk/ masda/surf/more/duval.pdf. |

[9] | M. Schütt, Quintic surfaces with maximum and other Picard numbers, J. Math. Soc. Japan, 63 (2011), 1187-1201. · Zbl 1232.14022 |

[10] | T. Shioda, On the Picard number of a Fermat surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1982), 725-734. · Zbl 0567.14021 |

[11] | T. Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math., 108 (1986), 415-432. · Zbl 0602.14033 |

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