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Journal of the London Mathematical Society 2005 72(1):73-90; doi:10.1112/S0024610705006459
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© The London Mathematical Society

The Defining Ideal of a Set of Points in Multi-Projective Space

Adam van Tuyl

Department of Mathematical Sciences, Lakehead University Thunder Bay, ON P7B 5E1, Canada avantuyl{at}sleet.lakeheadu.ca

Received 27 October 2003. Revision received 9 September 2004.

The defining ideal IX of a set of points X in Pn1 x ... x Pnk is investigated with a special emphasis on the case when X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, the degrees of the generators of the associated ideal IX are determined. {nu}(IX) denotes the minimal number of generators of IX, and this description of the degrees is used to construct a function {upsilon}(s; n1,...,nk) with the property that {nu}(IX)≥ {upsilon}(s; n1,..., nk) always holds for s points in generic position in Pn1 x ... x Pnk. When k = 1, {upsilon}(s;n1) equals the expected value for {nu}(IX) as predicted by the ideal generation conjecture. If k ≥ 2, it is shown that there are cases with {nu}(IX) > {upsilon}(s; n1, ..., nk). However, computational evidence suggests that in many cases {nu}(IX) = {upsilon}(s; n1, ..., nk).


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