1997/03/21 by W. M. Oxbury, William Oxbury, Oxbury, William +2 · 16 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics #FOS: Mathematics #Genus #Heisenberg group #Homotopy and Cohomology in Algebraic Topology #Hypersurface #Invariant (physics) #Mathematical physics #Mathematics #Moduli space #Pure mathematics #Quartic function #Quartic surface #Rank (graph theory) #Vector bundle #alg-geom #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9703026
published in HAL (Le Centre pour la Communication Scientifique Directe) (Centre National de la Recherche Scientifique) · LaTeX, 36 pages, 2 figures
arxiv created 1997/03/21 · openalex publication_date 1997/03/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
If C is a curve of genus 4 without vanishing theta-nulls then there exists a unique (irreducible) Heisenberg-invariant quartic QC in |2Θ| = P15 such that Sing QC contains the image of SUC(2), the moduli space of rank 2 vector bundles with trivial determinant. Moreover, in each eigen-P7 of the Heisenberg action on |2Θ|, QC restricts to the classical Coble quartic of the corresponding Prym-Kummer variety. We compare QC with the hypersurface G3 in |2Θ| of divisors containing a translate of C in J(C), and show that in the eigen-P7s G3 recovers Beauville--Debarre's quadrisecant planes of the Prym-Kummers (this works for any genus). Using the Recillas construction this enables us to deduce, contrary to the analogous result for genus 3, that QC and G3 are distinct.