The realization of hyperelliptic curves through endomorphisms of Kronecker modules
✍ Scribed by Frank Okoh; Frank Zorzitto
- Publisher
- Elsevier Science
- Year
- 2010
- Tongue
- English
- Weight
- 352 KB
- Volume
- 432
- Category
- Article
- ISSN
- 0024-3795
No coin nor oath required. For personal study only.
✦ Synopsis
Let K be an algebraically closed field and A the Kronecker algebra over K. A general problem is to study the endomorphism algebras of A-modules M that are extensions of finite-dimensional, torsion-free, rank-one A-modules P, by infinite-dimensional, torsion-free, rankone A-modules N. Such endomorphism algebras can be studied by means of a quadratic polynomial f (Y) in one variable Y over the rational function field K(X). We call this f (Y) the regulator of the extension. We prove that if the regulator has non-zero discriminant, then End M is a Noetherian, commutative K-algebra. We also prove that, subject to a regulator with non-zero discriminant, End M is affine over K if and only if End N is affine, in which case End M is the coordinate ring of a hyperelliptic curve.