We study the linkage classes of homogeneous ideals in polynomial rings An ideal is said to be homogeneously licci if it can be linked to a complete intersection using only homogeneous regular sequences at each step. We ask a natural question: if / is homogeneously licci, then can it be linked to a complete intersection by linking using regular sequences of forms of smallest possible degree at each step (we call such ideals minimally homogeneously licci)? In this paper we answer this question in the negative In particular, for every n > 28 we construct a set of n points in P3 which are homogeneously licci, but not minimally homogeneously licci. Moreover, we prove that one cannot distinguish between the classes of homogeneously licci and non-licci ideals based only on their Hilbert functions, nor distinguish between homogeneously licci and minimally homogeneously licci ideals based solely on the graded Betti numbers Finally, by taking hypersurface sections, we show that the natural question has a negative answer whenever the height of the ideal is at least three.
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