Hirzebruch's curves \(F_1, F_2, F_4, F_{14}, F_{28}\) for \(\mathbb{Q}(\sqrt 7)\) (Q2702556)

In this work, the author explores some modular curves that are related to Klein's quartic defined by the equation \(x^3y+ y^3z+ z^3x=0\) in the complex projective plane. The starting point is F. Hirzebruch's remarkable discovery that the complex plane \(\mathbb{P}^2\) is a minimal model of the symmetric Hilbert modular surface of level \(\sqrt{7}\) for the extended Hilbert modular group of the number field \(\mathbb{Q}(\sqrt{7})\), and that the identification is equivariant for natural actions of the modular group \(G= \text{PSL}_2 (\mathbb{F}_7)\). Hirzebruch's result enabled him to identify the Hilbert modular surface associated to this group explicitly, and to construct other interesting modular curves \(F_N\) for some numbers \(N\in \mathbb{N}\), which are now called ``Hirzebruch curves'' (or Hirzebruch surfaces, respectively). The first one among them, \(F_1\), is just Klein's quartic, which is well-understood, whereas the projective embeddings of the higher Hirzebruch curves have never been described that explicitly in the literature. NEWLINENEWLINENEWLINEThe purpose of the article under review is twofold. First, the author shows how to identify the images of the Hirzebruch curves \(F_N\) in \(\mathbb{P}^2\) for \(N=1,2,4,14\) and 28. Second, since the ultimate details of Hirzebruch's theory were never completely published, the author provides those details here, so to speak as a public service and source for references. As to the latter undertaking, he relied on some unpublished notes and private correspondences with F. Hirzebruch. Therefore this paper may be seen as both a further elaboration and enhancement of Hirzebruch's work on modular curves. NEWLINENEWLINENEWLINEThe article consists of sixteen sections. The first part is exclusively devoted to detailed proofs of Hirzebruch's published results (1971-1977), while the second part deals with the determination of the Hirzebruch curves \(F_1\), \(F_2\), \(F_4\), \(F_{14}\) and \(F_{28}\) as plane projective curves. Here the author has masterly worked in some classical related results by L. Berzolari (1903-1915), R. Fricke (1893), and E. Hecke (1935), as well as some computer-aided explicit calculations. There are also results regarding the nonsingular models of the curves \(F_2\) and \(F_4\), which turn out to be the Hessian of Klein's quartic. Finally, the author also studies the group of invariant line bundles and the Jacobian of this Hessian curve. NEWLINENEWLINENEWLINEAltogether, using his explicit identification of the surfaces \(F_N\) (for \(N=1,2,4,14,28\)), the author is able to complete Hirzebruch's description of the non-symmetric Hilbert modular surface. Indeed, this fine work is a very rewarding contribution to the theory of Klein's quartic and its related Hirzebruch curves. NEWLINENEWLINENEWLINEFor the entire collection see Zbl 0941.00006 and Zbl 0991.00005 for a review of the paperback edition.