Slope Stability and Exceptional Divisors of High Genus

Understand

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.