A counterexample to integral product exponents in universal curve counting

Abstract

We construct a smooth projective Calabi–Yau threefold with torsion-free second homology whose product expansion in Pardon's scalar equivariant local-curve generators has a nonintegral exponent. The genus-ten exponent at three times a primitive class has fractional part $\frac{2}{3}$, disproving the integer-exponent assertion for these generators. The obstruction comes from an exact degree-three calculation in the local Gromov–Witten Frobenius algebra. Comparison with the integer elementary expansion of Ionel–Parker gives a necessary divisibility condition on the primitive genus-three count. We realize a count violating this condition on a first Hirzebruch surface in a compact toric hypersurface. A filtration by powers of the surface ideal controls transverse thickenings and disconnected products; the stable-pair moduli space at minimal Euler characteristic is a projective space of dimension thirteen, with virtual count $-14$. We also prove integrality in classes of divisibility at most two. The counterexample concerns the scalar local factors; the integer elementary expansion is an input to its proof.