Li-Zinger’s hyperplane property for reduced genus one GW-invariants of quintics states that the genus one GW-invariants of the quintic threefold is the sum of its reduced genus one GW-invariants and $1/12$ times its genus zero GW-invariants. We apply the theory of GW-invariants of stable maps with fields to give an algebro-geometric proof of this hyperplane property.
"An algebraic proof of the hyperplane property of the genus one GW-invariants of quintics." J. Differential Geom. 100 (2) 251 - 299, June 2015. https://doi.org/10.4310/jdg/1430744122