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

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10.1016/0021-8693(80)90215-X

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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",

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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.

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UR - https://www.scopus.com/pages/publications/49149143742#tab=citedBy

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

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

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JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -

Principal ideals and smooth curves

Abstract

All Science Journal Classification (ASJC) codes

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