Counterexamples to the Polishchuk--Van den Bergh conjecture on curves

Abstract

We construct effective finite group actions on smooth projective complex curves for which Conjecture A of Polishchuk and Van den Bergh fails. For every $h\geq1$, an explicit $S_3$ action on the smooth projective model of $y^2=t^{12h-5}+t$ has quotient of genus $h$ and exactly one ramification orbit, with inertia of order three. The fixed quotient for the three-cycle class consists of two points, but its derived category admits no $\C$-linear exact fully faithful embedding into the equivariant derived category. More generally, when the coarse quotient has positive genus, we prove that the maximum number of pairwise completely orthogonal exceptional objects is $\sum_i\lfloor e_i/2\rfloor$, where $e_i$ are the inertia orders. We also characterize conjugacy-class decompositions with embeddings linear over the coarse quotient: such a decomposition exists if and only if $|[g]\cap H_i|\leq\lfloor e_i/2\rfloor$ for every nonidentity conjugacy class and every branch inertia group $H_i$. The sufficiency proof uses coloured paths of type $A$ and derived reflection functors, and produces kernels on the natural fibre products. The known ordered curve decomposition still yields a decomposition indexed by the connected components of the fixed quotients.