A support theorem for nested Hilbert schemes of planar curves

TL;DR

This paper proves the smoothness of relative nested Hilbert schemes for certain families of planar curves and demonstrates that their pushforward decomposes into a direct sum of semisimple perverse sheaves without positive codimension support.

Contribution

It establishes conditions under which the relative nested Hilbert scheme is smooth and describes the decomposition of its pushforward sheaf into semisimple perverse sheaves.

Findings

Relative nested Hilbert schemes are smooth under certain assumptions.

The pushforward sheaf decomposes into a direct sum of semisimple perverse sheaves.

No summand is supported in positive codimension.

Abstract

Consider a family of integral complex locally planar curves. We show that under some assumptions on the basis, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension.