Semiorthogonal indecomposability for Hilbert schemes of points on integral locally planar curves

Abstract

Let $C$ be an integral projective curve of arithmetic genus $g$ with locally planar singularities over an algebraically closed field. We prove that for every $1\le n\le g-1$, both $\mathrm{Perf}({\mathrm{Hilb}}^n(C))$ and ${\mathrm{D}}_{\mathrm{coh}}^{\mathrm{b}}({\mathrm{Hilb}}^n(C))$ are semiorthogonally indecomposable. We also establish the corresponding relative $S$-linear statement for a flat family of such curves over a connected base $S$, with admissibility required in the ${\mathrm{D}}_{\mathrm{coh}}^{\mathrm{b}}$ case. Our results hold in arbitrary characteristic.

8 pages. Comments welcome!