TY - JOUR

T1 - IDEALS OF HERZOG-NORTHCOTT TYPE

AU - O'Carroll, Liam

AU - Planas-Vilanova, Francesc

PY - 2011/2

Y1 - 2011/2

N2 - This paper takes a new look at ideals generated by 2 x 2 minors of 2 x 3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

AB - This paper takes a new look at ideals generated by 2 x 2 minors of 2 x 3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

UR - http://www.scopus.com/inward/record.url?scp=82455192658&partnerID=8YFLogxK

U2 - 10.1017/S0013091509001321

DO - 10.1017/S0013091509001321

M3 - Article

VL - 54

SP - 161

EP - 186

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

ER -