Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a special case of a conjecture of Oblomkov and Shende identifies the Euler numbers of the Hilbert schemes with the "U(infinity)" invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.Comment: 16 page

Liaison Linkages

The complete classification of hexapods - also known as Stewart Gough platforms - of mobility one is still open. To tackle this problem, we can associate to each hexapod of mobility one an algebraic curve, called the configuration curve. In this paper we establish an upper bound for the degree of this curve, assuming the hexapod is general enough. Moreover, we provide a construction of hexapods with curves of maximal degree, which is based on liaison, a technique used in the theory of algebraic curves.Comment: 40 pages, 6 figure