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Pré-Publication, Document De Travail Année : 2023

The Coble Quadric

La Quadrique de Coble

Résumé

Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in P^8 as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU_C(2, L), the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of G(2, 8). In fact, each point p ∈ C defines a natural embedding of SU_C(2, O(p)) in G(2, 8). We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of SU_C(2, O(p)), and thus deserves to be coined the Coble quadric of the pointed curve (C, p).
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Dates et versions

hal-04158249 , version 1 (11-07-2023)

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  • HAL Id : hal-04158249 , version 1

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Vladimiro Benedetti, Daniele Faenzi, Michele Bolognesi, L Manivel. The Coble Quadric. 2023. ⟨hal-04158249⟩
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