Matroids and semirings attached to toric singularity arrangements

Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the value semigroup, or its compactification, the value semiring. One natural problem is to explicitly determine the value semirings of distinguished infinite classes of singularities, with a view to understanding their asymptotic properties. In this paper, we establish a matroidal framework for resolving this problem for singularities determined by arrangements of toric branches; and we obtain precise quantitative results in the case of line arrangements. Our results have implications for the topology of Severi varieties of unisingular rational curves in projective space.