Abstract :
The defining ideal IX of a set of points X in Pn1 ×. . .×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. ν(IX)
denotes the minimal number of generators of IX, and this description of the degrees is used to
construct a function v(s; n1, . . . , nk ) with the property that ν(IX) v(s; n1, . . . , nk) always holds
for s points in generic position in Pn1 × . . . × Pnk. When k=1, v(s; n1) equals the expected value
for ν(IX) as predicted by the ideal generation conjecture. If k 2, it is shown that there are
cases with ν(IX) > v(s; n1, . . . , nk ). However, computational evidence suggests that in many cases
ν(IX)=v(s; n1, . . . , nk ).
