Principal ideals and smooth curves

A. V. Geramita; C. A. Weibel

doi:10.1016/0021-8693(80)90215-X

Principal ideals and smooth curves

A. V. Geramita, C. A. Weibel

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let A be the coordinate ring of an affine piece of a smooth curve, V, defined over either R or an algebraically closed field k. We ask which maximal ideals of A are principal. We give a complete determination if V has genus 0 or 1, and give partial results if V has genus >/ 2. We conjecture that if k is algebraically closed of characteristic 0, genus (V) >/ 2, then A has only finitely many principal maximal ideals. This conjecture is equivalent to the Mordell Conjecture of Diophantine Geometry.

Original language	English (US)
Pages (from-to)	235-248
Number of pages	14
Journal	Journal of Algebra
Volume	62
Issue number	1
DOIs	https://doi.org/10.1016/0021-8693(80)90215-X
State	Published - Jan 1980
Externally published	Yes

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Access to Document

10.1016/0021-8693(80)90215-X

Cite this

@article{bb1e3bfd7fc7446c845ae825b03f559b,

title = "Principal ideals and smooth curves",

abstract = "Let A be the coordinate ring of an affine piece of a smooth curve, V, defined over either R or an algebraically closed field k. We ask which maximal ideals of A are principal. We give a complete determination if V has genus 0 or 1, and give partial results if V has genus >/ 2. We conjecture that if k is algebraically closed of characteristic 0, genus (V) >/ 2, then A has only finitely many principal maximal ideals. This conjecture is equivalent to the Mordell Conjecture of Diophantine Geometry.",

author = "Geramita, {A. V.} and Weibel, {C. A.}",

note = "Funding Information: in part by the National in part by NSF Grant",

year = "1980",

month = jan,

doi = "10.1016/0021-8693(80)90215-X",

language = "English (US)",

volume = "62",

pages = "235--248",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - Principal ideals and smooth curves

AU - Geramita, A. V.

AU - Weibel, C. A.

N1 - Funding Information: in part by the National in part by NSF Grant

PY - 1980/1

Y1 - 1980/1

N2 - Let A be the coordinate ring of an affine piece of a smooth curve, V, defined over either R or an algebraically closed field k. We ask which maximal ideals of A are principal. We give a complete determination if V has genus 0 or 1, and give partial results if V has genus >/ 2. We conjecture that if k is algebraically closed of characteristic 0, genus (V) >/ 2, then A has only finitely many principal maximal ideals. This conjecture is equivalent to the Mordell Conjecture of Diophantine Geometry.

AB - Let A be the coordinate ring of an affine piece of a smooth curve, V, defined over either R or an algebraically closed field k. We ask which maximal ideals of A are principal. We give a complete determination if V has genus 0 or 1, and give partial results if V has genus >/ 2. We conjecture that if k is algebraically closed of characteristic 0, genus (V) >/ 2, then A has only finitely many principal maximal ideals. This conjecture is equivalent to the Mordell Conjecture of Diophantine Geometry.

UR - http://www.scopus.com/inward/record.url?scp=49149143742&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49149143742&partnerID=8YFLogxK

U2 - 10.1016/0021-8693(80)90215-X

DO - 10.1016/0021-8693(80)90215-X

M3 - Article

AN - SCOPUS:49149143742

SN - 0021-8693

VL - 62

SP - 235

EP - 248

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -

Principal ideals and smooth curves

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this