Chow Shadows of a Nodal Threefold

Let (Y) be a projective complex threefold with (n) ordinary double points and a projective small resolution (X → Y). The coherent recorder (ι_Y Perf(Y) D^bCoh(Y)) has degree-zero shadow [Sh_0(ι_Y) !(K_0(Y) → G_0(Y)).]We identify this shadow modulo its maximal divisible subgroup with the group generated by the exceptional classes ([OC_i(-1)]) in (K_0(X)). Its integral dual consequently has rank equal to the Chow rank (c(Y)) of the exceptional curves. We then construct a smooth projective family of Calabi–Yau threefolds with four marked exceptional ((-1,-1))-curves, constant homological rank, and varying Chow rank. The distinguishing cycle is the pullback of a degree-zero Jacobian class of a genus-four curve. These arguments give the two parts of Conjecture 8.32 of The Fibre of a World. We also prove its local Hypothesis 8.24 for the split node over fields of characteristic different from two. The final sections distinguish the proved statements from further questions and record corrections to the original manuscript.