  Let (X,L) be a polarised manifold. We show that K-stability and asymptotic
Chow stability of the blowup of X along a 0-dimensional cycle are closely
related to Chow stability of the cycle itself, for polarizations making the
exceptional divisors small. This can be used to give (almost) a converse to a
result of Arezzo and Pacard, and to give new examples of K\"ahler classes with
no constant scalar curvature representatives.
