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 $1\leq n\leq g-1$, both $\mathrm{Perf}(\mathrm{Hilb}^n(C))$ and $\mathrm{D^b_{coh}}(\mathrm{Hilb}^n(C))$ 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 $\mathrm{D^b_{coh}}$ case. Our results hold in arbitrary characteristic.