Multiplicity of the Lyashko-Looijenga Map for Laurent Polynomials

It is well-known that the multiplicities of the Lyashko-Looijenga map on the strata of compactified Hurwitz spaces (spaces of meromorphic functions on complex curves) is related to the Hurwitz numbers corresponding to the functions with prescribed branching types in these spaces. The multiplicity of the Lyashko-Looijenga map can be computed if we know the cohomology classes that are Poincaré dual to the homology classes of the strata in the compactified Hurwitz space, and this computation, in turn, is significantly simplified if the primitive strata in this space intersect transversally. We restrict our attention to the Hurwitz spaces $H_{1,n}$ ——- spaces of functions with $2$ poles (i.e. Laurent polynomials), one of which is simple, while the other has order $n$. We explore a suitable compactification of $H_{1,n}$ that arises from its connection to certain moduli spaces. For the case of $n = 3$, we use this compactification to compute the cohomology classes of the strata along with showing that the intersection of the primitive strata is transversal. We also use this computation to find the multiplicities of the Lyashko-Looijenga map on the generic stratum, the caustic and the Maxwell stratum.