A support theorem for Hilbert schemes of planar curves, II

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

Counting Rational Curves on K3 Surfaces With Finite Group Actions

Göttsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge structure. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum _{n=0}^{\infty }\sum _{i=0}^{\infty }(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the Hodge structure of Hilbert schemes of points to the Hodge structure of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. Finally, we give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.

Fourier–Mukai and autoduality for compactified Jacobians. I

To every singular reduced projective curve X one can associate, following Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier–Mukai) transform with kernel the Poincaré sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence of line bundles coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagné, Kleiman, Rocha, and Sawon.The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi–Pantev.

Fourier–Mukai Transform on a Compactified Jacobian

We use Fourier–Mukai transform to compute the cohomology of the Picard bundles on the compactified Jacobian of an integral nodal curve $Y$. We prove that the transform gives an injective morphism from the moduli space of vector bundles of rank $r \ge 2$ and degree $d$ ($d$ sufficiently large) on $Y$ to the moduli space of vector bundles of a fixed rank and fixed Chern classes on the compactified Jacobian of $Y$. We show that this morphism induces a morphism from the moduli space of vector bundles of rank $r \ge 2$ and a fixed determinant of degree $d$ on $Y$ to the moduli space of vector bundles of a fixed rank with a fixed determinant and fixed Chern classes on the compactified Jacobian of $Y$.