Criterion for the accumulation of v-walls

Abstract

In this thesis we consider walls in the slice of geometric stability conditions for Chern characters on a principally abelian polarised surface. It is known that there is a vertical line of stability conditions in our choice of parametrisation, determined entirely numerically, which must intersect every wall, apart from a single vertical one. Furthermore if this line has rational horizontal coordinate β_ , then there can only be finitely many walls. In principal, when this horizontal coordinate is not rational, then there could be infinitely many walls. This has also been shown for certain examples and classes of Chern characters, but not exhaustively for all Chern characters where the aforementioned vertical line of stability conditions has irrational horizontal coordinate.

The main conclusion of this thesis is that all other classes of Chern characters not considered also have infinitely many walls. That is, a Chern character v on a principally polarised abelian surface 𝕋 with NS(𝕋) = ℤℓ has finitely many walls if and only if β_(v) is rational. The other aspect of this thesis is developing methods to computing possibilities for walls more efficiently.