The moduli space of cubic surface pairs via the intermediate Jacobians of Eckardt cubic threefolds
| Autor: | Sebastian Casalaina-Martin, Zheng Zhang |
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| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Cubic surface Intermediate Jacobian General Mathematics 010102 general mathematics Hyperplane section Prym variety 01 natural sciences Injective function Torelli theorem Moduli space Mathematics - Algebraic Geometry Mathematics::Algebraic Geometry Genus (mathematics) 0103 physical sciences FOS: Mathematics 4J30 14J10 14K10 14H40 010307 mathematical physics 0101 mathematics Algebraic Geometry (math.AG) Mathematics |
| DOI: | 10.48550/arxiv.2002.09861 |
| Popis: | 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; i.e., 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 \'etale double covers of genus 6 curves. Comment: 33 pages, AMS LaTeX, final version, minor update of the published version: fixed several typos, added reference to Sacc\`a for Proposition 3.10, clarified statement of Corollary 5.17 |
| Databáze: | OpenAIRE |
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