  In their paper on multiplicity bounds (1998), 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 (1985). 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 I_Y.
