A problem on an exponential domain.

*(English)*Zbl 0915.14006
Arslanov, M. M. (ed.) et al., Algebra and analysis. Proceedings of the international centennial Chebotarev conference, Kazan, Russia, June 5-11, 1994. Berlin: Walter de Gruyter. 23-29 (1996).

Let \({\mathcal T}_e= {\mathcal J}_e+ i{\mathcal P}_e\), where \({\mathcal J}_e\) is the 27-dimensional exceptional real Jordan algebra of \(3\times 3\) Hermitian matrices over the Cayley division algebra \({\mathcal C}\) and \({\mathcal P}_e\) its cone of positive elements. Let \(\Gamma_e\) be a particular arithmetic subgroup of \(\text{Hol}({\mathcal T}_e)\) previously considered by W. L. Baily jun. [Ann. Math., II. Ser. 92, 512-549 (1970; Zbl 0202.07901)]. In this paper the author raises the problem of interpreting the space of orbits of \(\Gamma_e\) on \({\mathcal T}_e\) as a moduli space for some family of polarized allgebraic varieties. As a step in answering this problem, the author considers Zak’s four Severi varieties [R. Lazarsfeld and A. Van de Ven, “Topics in the geometry of projective spaces. Recent work of F. L. Zak” (1984; Zbl 0564.14007)].

Denote these by \(S_1, \dots, S_4\). \(S_1\) is the Veronese embedding of \(\mathbb{P}^2\) in \(\mathbb{P}^5\); by taking intersections with quadric hypersurfaces \(Q\) he obtains genus 3 curves, whose moduli space is (essentially) given by the orbits of \(\text{Sp}(6,\mathbb{Z})\) on the Siegel space of degree 3. The bulk of the paper is spent examining the case of \(S_2\), the Segre embedding of \(\mathbb{P}^2\times\mathbb{P}^2\) in \(\mathbb{P}^8\). Here the author takes the intersection of the image with \(Q\cap H\), \(Q\) a quadric surface and \(H\) a hyperplane, to obtain a moduli problem for \(K3\) surfaces. He describes what is known, and not known, here. He raises the question of whether something similar exists for \(S_3\) and \(S_4\), being related to the domain \({\mathcal T}_e\), and concludes by remarking that “this is still totally unexplored territory”.

For the entire collection see [Zbl 0859.00022].

Denote these by \(S_1, \dots, S_4\). \(S_1\) is the Veronese embedding of \(\mathbb{P}^2\) in \(\mathbb{P}^5\); by taking intersections with quadric hypersurfaces \(Q\) he obtains genus 3 curves, whose moduli space is (essentially) given by the orbits of \(\text{Sp}(6,\mathbb{Z})\) on the Siegel space of degree 3. The bulk of the paper is spent examining the case of \(S_2\), the Segre embedding of \(\mathbb{P}^2\times\mathbb{P}^2\) in \(\mathbb{P}^8\). Here the author takes the intersection of the image with \(Q\cap H\), \(Q\) a quadric surface and \(H\) a hyperplane, to obtain a moduli problem for \(K3\) surfaces. He describes what is known, and not known, here. He raises the question of whether something similar exists for \(S_3\) and \(S_4\), being related to the domain \({\mathcal T}_e\), and concludes by remarking that “this is still totally unexplored territory”.

For the entire collection see [Zbl 0859.00022].

Reviewer: Steven H.Weintraub (MR 99c:32046)