Let (X,L) be a polarised manifold. We show that K-stability and as- ymptotic Chow stability of the blowup of X along a 0-dimensional cycle are closely related to Chow stability of the cycle itself, for polarisations making the exceptional divisors small. This can be used to give (almost) a converse to the results of Arezzo and Pacard (2004 and 2007) and to give new examples of K ̈ahler classes with no constant scalar curvature representatives.
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