Universal curve counting under conifold transitions

Abstract

For projective conifold transitions of Calabi–Yau threefolds, we lift the numerical degeneration formula to Pardon's rational universal curve-counting algebra. We assume a smoothing with smooth algebraic total space and torsion-free second homology at both endpoints. After removing the exceptional local-curve factors, effective lifts to the common blow-up give a support bound that makes specialization finite in every bounded target degree. We carry out this specialization completely in the primitive line class of a plane in the smoothing of a compact four-node transition. The source calculation uses an Euler-characteristic bound for pure curves on two intersecting rational surfaces, allowing arbitrary multiple exceptional components. Together with a finite genus-zero mirror calculation, this bound determines all 37 candidate universal coefficients: eight are nonzero, and their sum agrees with an independent stable-pair calculation on a plane in the smoothing. We also determine the smoothing coefficient in twice the line class. If the resolution has integral product exponents, then so does the smoothing.