Slope stability and exceptional divisors of high genus

We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is slope stable but not K-stable. D. Panov was supported by EPSRC grant number EP/E044859/1 and J. Ross was partially supported by the National Science Foundation, Grant No. DMS-0700419.