algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a Coble hypersurface is one of the hypersurfaces associated to the

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...

of a curve of

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

2 or 3 by

Arthur Coble Arthur Byron Coble (November 3, 1878 – December 8, 1966) was an American mathematician. He did research on finite geometries and the group theory related to them, Cremona transformations associated with the Galois theory of equations, and t ...

. There are two similar but different types of Coble hypersurfaces. *The

Kummer variety In mathematics, the Kummer variety of an abelian variety is its quotient by the map taking any element to its inverse. The Kummer variety of a 2-dimensional abelian variety is called a Kummer surface In algebraic geometry, a Kummer quartic su ...

of the Jacobian of a genus 3 curve can be embedded in 7-dimensional

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

under the 2-theta map, and is then the singular locus of a 6-dimensional quartic hypersurface , 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 , also called a Coble hypersurface.

References

* * *{{Citation , last1=Coble , first1=Arthur B. , title=Algebraic geometry and theta functions , orig-year=1929 , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, location=Providence, R.I. , series=American Mathematical Society Colloquium Publications , isbn=978-0-8218-1010-1 , mr=733252 , year=1982 , volume=10 Abelian varieties

See also

References