# zbMATH — the first resource for mathematics

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].
##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 17C40 Exceptional Jordan structures 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 14M17 Homogeneous spaces and generalizations
##### Keywords:
Severi variety; space of orbits; moduli space