Surface singularities, unexpected fillings, and line arrangements

A link of an isolated complex surface singularity (X, 0) is a 3-manifold Y which is the boundary of the intersection of X with a small ball centered at 0. Smoothings of the singularity give non-singular 4-manifolds, the Milnor fibers, with the same boundary Y. The Milnor fibers carry symplectic (even Stein) structures, and thus provide fillings of the canonical contact structure on Y; another Stein filling comes from the minimal resolution of (X, 0). An important question is whether all Stein fillings of the link come from this algebraic construction: this is true in some simple cases such as lens spaces. However, even in the “next simplest” case, for many rational singularities, we are able to construct “unexpected” Stein fillings that do not arise from Milnor fibers. To this end, we encode Stein fillings via curve arrangements, motivated by  T.de Jong-D.van Straten’s description of smoothings  of certain rational surface singularities in terms of deformations of associated singular plane curves. We then use classical projective geometry  to construct unexpected line arrangements and unexpected fillings. This is a topological story, with minimal input from algebraic geometry.  Joint work with L. Starkston.