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RESEARCH PRODUCT
The Coble Quadric
Vladimiro BenedettiDaniele FaenziMichele BolognesiL Manivelsubject
Coble hypersurfacesMathematics - Algebraic Geometrydegeneracy loci[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]FOS: Mathematics14h60 22E46Moduli spaces of stable bundlessubvarieties of Grassmannians[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]Hecke linesself-dual hypersurfacesAlgebraic Geometry (math.AG)description
Given a smooth genus three curve $C$, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in $\mathbb{P}^8$ 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 $p \in C$ defines a natural embedding of SU$_C(2, \mathcal{O}(p))$ 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$_C(2, \mathcal{O}(p))$, and thus deserves to be coined the Coble quadric of the pointed curve $(C, p)$.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2023-01-01 |