ar X iv : m at h / 05 10 55 2 v 1 [ m at h . A C ] 2 6 O ct 2 00 5 BETTI NUMBERS AND DEGREE BOUNDS FOR SOME LINKED ZERO - SCHEMES

Abstract. In [8], Herzog and Srinivasan study the relationship between the graded Betti numbers of a homogeneous ideal I in a polynomial ring R and the degree of I. For certain classes of ideals, they prove a bound on the degree in terms of the largest and smallest Betti numbers, generalizing results of Huneke and Miller in [9]. The bound is conjectured to hold in general; we study this using linkage. If R/I is Cohen-Macaulay, we may reduce to the case where I defines a zero-dimensional subscheme Y. If Y is residual to a zero-scheme Z of a certain type (low degree or points in special position), then we show that the conjecture is true for IY .