Root Systems and Periods on Hirzebruch Surfaces

Abstract

§ I. Introduction Cayley classified all nonsingular cubic surfaces in three-dimensional complex projective space by using the configuration of the 27 lines on the surface [3]. The symmetry of these 27 lines can be described by the Weyl group and root system of type EG ([4]). Furthermore these objects, namely the 27 lines, the Weyl group of type EG, and root system of type EG, have natural realization in the Picard group of cubic surface ([8]). In [10] a fine moduli space M of marked cubic surfaces was constructed explicitly in such a manner that the relation between the geometrical structure of M and the structure of root system became clear. On the other hand the fine moduli spaces for certain classes of rational surfaces were constructed in terms of root system and periods, which are integrals of a meromorphic 2-form over 2-cycles on the surface corresponding to roots ([7]). The moduli space M was reconstructed in terms of the root system and the periods in the same way (Appendix in [10]). In this paper, we discuss the moduli problem for certain class of rational surfaces in terms of the root system of type A. Let X be the rational surface obtained from the w-th Hirzebruch surface or rational ruled surface with invariant n by blowing up n points. The relation between the Hirzebruch surfaces with n points blown up and the root system of type An-i is similar to the relation between the cubic surface and the root system of type EG. We prove a Torelli theorem for the pairs of X and a certain anticanonical divisor on X by using the structure of the root system of type An-l in the Picard group of X. We shall construct a family p: £-»S of the Hirzebruch surfaces with n points blown up and study a period mapping for the fibration p: £-»S, where the base space S is the quotient space of a maximal torus of the simple Lie group of type An-i by its Weyl group. The fiber & of p can be regarded as a compactificati on of the fiber of semi-universa l deformation of the simple sur