Discriminants and motivic integration
Oscar Kivinen, Alexei Oblomkov, Dimitri Wyss
TL;DR
The paper develops a motivic framework for plane curve singularities by studying motivic integrations on symmetric powers of a formal deformation, identifying discriminant and orbifold measures with significant invariants: discriminant integrals recover principal Hilbert schemes and orbifold integrals yield the plethystic exponential of the motivic Igusa zeta function. It proves that the gel-form Poincaré series $P_{gel}(T)$ matches the motivic Poincaré series of Campillo–Delgado–Gusein-Zade and, after Euler specialization, recovers the Milnor monodromy zeta function, linking to knot Floer invariants via Gorsky–Nemethi. The work also introduces the orbifold framework, relates the orbifold form to the Gelfand form through the discriminant, and establishes connections to Fixed Point and Knot Floer theories, with explicit calculations in smooth and nodal examples that illustrate the framework. These results provide a new bridge between motivic invariants and low-dimensional topology, suggesting deep interplay between arc spaces, Hilbert schemes, and Floer-theoretic invariants.
Abstract
We study invariants of a plane cuve singularity $(f,0)$ coming from motivic integration on symmetric powers of a formal deformation of $f$. We show that a natural discriminant integral recovers the motivic classes of the principal Hilbert schemes of points on $f$, while the orbifold integral gives the plethystic exponential of the motivic Igusa zeta function of $f$. The latter result also holds in higher dimemsions. Combined with results of Gorsky and Némethi we obtain an interpretation of the discriminant integrals in terms of knot Floer homology, which is reminiscent of the relation between the cohomology of contact loci and fixed point Floer homology proven by de la Bodega and Poza.