Smoothing Calabi–Yau threefolds with nodes and del Pezzo cone points

Abstract

We give a necessary and sufficient homological criterion for smoothing projective Calabi–Yau threefolds with nodes and anticanonical cones over smooth del Pezzo surfaces of degrees five, six and seven, or over a smooth quadric surface. For some choice of local smoothing components, the criterion requires a relation on a crepant resolution among link classes whose associated local parameters avoid every discriminant. The complexified relation space describes first-order deformations modulo locally trivial ones. Necessity follows from local period and vanishing-cycle arguments; sufficiency applies Friedman's unobstructedness theorem to a crepant partial resolution. Whenever the criterion holds, the deformation space on the chosen components is smooth. A nonsmoothable example, whose deformation space is reduced with four smooth components, shows why nonvanishing at every singular point is insufficient. We also construct a smoothable Batyrev hypersurface admitting no smoothing induced by proper ambient families obtained from Minkowski decompositions at a single lattice degree.