Coble hypersurface

In algebraic geometry, a Coble hypersurface is one of the hypersurfaces associated to the Jacobian variety of a curve of

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 Mathématique, 337 (3): 189–194,
  S2CID 18266649
• Coble, Arthur B. (1917), "Point Sets and Allied Cremona Groups (Part III)",
  MR 1501073
• Coble, Arthur B. (1982) [1929], Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications, vol. 10, Providence, R.I.:
  MR 0733252