In the study of moduli spaces generally, it is common to find that sophisticated geometric properties of a space are encoded in simpler topological invariants of associated moduli spaces. Our purpose here is to investigate this phenomenon for the Hilbert schemes of points on an integral plane curve C. The simplest topological invariant of all is the Euler characteristic, and it has become increasingly clear that these should be collected into the generating function Z(C) = sumd qdchi(C[ d]).;A planar curve singularity can by definition by found on a curve locally embedded in a surface; restricting to the boundary of a small ball centered around the singularity gives a link in the three-sphere. The HOMFLY polynomial is an invariant of links specializing variously to the Alexander and Jones polynomials. We conjecture that Z(C) is, up to a normalization factor, the coefficient of the lowest degree power of a in the product of the HOMFLY polynomials of the links of the singularities of C. We introduce certain nested Hilbert schemes to account for the higher order terms in a. We prove this conjectural matching when the singularities are unibranch and of the form x k = yn. In the limit a → -1, the conjecture asserts the equality of the Alexander polynomial and the generating function of Euler numbers of the Cartier locus of the Hilbert scheme; this had been previously shown by Campillo, Delgado, and Gusein-Zade, albeit in other language. Further evidence comes from matching certain symmetry conditions between Z(C) and the HOMFLY polynomial. Finally, one can lift the conjectural matching to one between the homology of the Hilbert schemes with the Khovanov-Rozansky homology of the links. The evidence for the lifted version of the conjecture is much thinner and the calculations involved significantly more difficu we will only give the briefest of sketches here.;A suggestion which can be traced back at least to the physicists Gopakumar and Vafa, but which comes into mathematics by the work of Pandharipande and Thomas, is that if we expand Z(C) = sum nh(C)(1 -- q) 2h-2 qg-h, where g is the arithmetic genus of the curve C, then the integers nh( C) measure in some sense the number of smooth curves of genus h as which the singular curve C should be counted. We provide this notion with a deformation-theoretic interpretation: nh(C) is the multiplicity of the locus parameterizing curves of geometric genus ≤ h in the base of a versal deformation of C. The proof uses a technical result of independent interest: the total space of the relative Hilbert schemes of ≤ d points over a family of integral planar curves is smooth along the fibre over any point where the map from the base to the versal deformation of the singularities of the curve has general d dimensional image.;The smoothness result allows us to investigate how the cohomology of C[d] varies with C. Consider a family of integral plane curves C → B such that the relative Hilbert scheme pi [d] : Cd B → B has smooth total space. The general results on perverse sheaves by Beilinson, Bernstein, and Deligne ensure that Rpd *C decomposes as a direct sum of IC sheaves; we show here that all summands are supported over all of B. Roughly speaking, this means that all the information is already present in the locus of smooth curves -- unfortunately, it does not mean that this information is easy to extract. It follows that the cohomologies of the C[ d], for all d, are encoded by the perverse filtration on the cohomology of the compactified Jacobian of C.
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