Suspending the Picard group: Trivial conormal bundles and efficient generation on affine threefolds

Let k be any field and let F ∈ k[z, w] define a nonempty smooth curve C = V(F) ⊂ 𝔸²ₖ. For the smooth affine threefold X_F = {xy = F(z, w)}, we give an explicit isomorphism Pic(C) ≅ CH²(X_F), sending the class of a closed point P to the class of Γ_P = P × 𝔸¹ ⊂ {x = 0}. The proof is a short application of the blow-up formula and localization. The ideal I_P of Γ_P has a free conormal module of rank two; it is generated by two elements exactly when [P] = 0 in Pic(C), and Γ_P is a set-theoretic complete intersection exactly when [P] is torsion. Two elliptic curves then give, over every field, explicit height-two prime ideals that require three generators; their cycle classes have exact order two or three, depending on the characteristic. In particular, these ideals have free conormal modules but are not efficiently generated. These examples answer negatively a hypersurface analogue of Murthy's complete-intersection conjecture; we also relate them to its formulations via surfaces and projective modules, and briefly to a possible functorial refinement.