This paper takes a new look at ideals generated by 2×2 minors of 2×3 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 compl...
This paper takes a new look at ideals generated by 2×2 minors of 2×3 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.
Citation
Planas, F.; O'Carroll, L. Ideals of Herzog-Northcott type. "Proceedings of the Edinburgh Mathematical Society", 19 Gener 2011, vol. 54, núm. 01, p. 161-186.