Coble hypersurface

In algebraic geometry, a Coble hypersurface is one of the hypersurfaces associated to the Jacobian variety of a curve of genus 2 or 3 by Arthur Coble.

There are two similar but different types of Coble hypersurfaces.

• The Kummer variety of the Jacobian of a genus 3 curve can be embedded in 7-dimensional projective space under the 2-theta map, and is then the singular locus of a 6-dimensional quartic hypersurface (Coble 1982), called a Coble hypersurface.
• Similarly the Jacobian of a genus 2 curve can be embedded in 8-dimensional projective space under the 3-theta map, and is then the singular locus of a 7-dimensional cubic hypersurface (Coble 1917), also called a Coble hypersurface.

See also

• Coble curve (dimension 1)
• Coble surface (dimension 2)
• Coble variety (dimension 4)

References

• Beauville, Arnaud (2003), "The Coble hypersurfaces", Comptes Rendus Mathematique 337 (3): 189–194, doi:10.1016/S1631-073X(03)00302-9, ISSN 1631-073X, MR 2001133
• Coble, Arthur B. (1917), "Point Sets and Allied Cremona Groups (Part III)", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 18 (3): 331–372, doi:10.2307/1988959, ISSN 0002-9947, JSTOR 1988959, MR 1501073
• Coble, Arthur B. (1982) [1929], Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications 10, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1010-1, MR 733252