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We study the moduli space of pairs consisting of a smooth cubic surface and a smooth hyperplane section, via a Hodge theoretic period map due to Laza, Pearlstein, and the second‐named author. The construction associates to such a pair a so‐called Eckardt cubic threefold, admitting an involution, and the period map sends the pair to the anti‐invariant part of the intermediate Jacobian of this cubic threefold, with respect to this involution. Our main result is that the global Torelli theorem holds for this period map; that is, the period map is injective. To prove the result, we describe the anti‐invariant part of the intermediate Jacobian as a Prym variety of a branched cover. Our proof uses results of Naranjo–Ortega, Bardelli–Ciliberto–Verra, and Nagaraj–Ramanan, on related Prym maps. In fact, we are able to recover the degree of one of these Prym maps by describing positive‐dimensional fibers, in the same spirit as a result of Donagi–Smith on the degree of the Prym map for connected étale double covers of genus 6 curves.

The moduli space of cubic surface pairs via the intermediate Jacobians of Eckardt cubic threefolds