The Nikulin-Yau-Zaslow formula

Location: 11th Floor Lecture Hall Speaker Francois Greer, Michigan State University Session Chair Kristin DeVleming, University of California San Diego Abstract Let (S,L) be a general primitively polarized K3 surface of degree 2n. The number of rational curves in the linear system |L| is equal to the nth Fourier coefficient of a weakly holomorphic modular form of weight -12, namely the inverse of the discriminant. This fact is known as the Yau-Zaslow formula. If we assume further that (S,L) admits a symplectic involution i preserving L, then the linear system |L| will contain a finite number of hyperelliptic curves C of genus g, whose hyperelliptic involution is equal to the restriction of i to C. We show that these hyperelliptic curve counts are encoded in the Fourier-Jacobi coefficients of a weakly holomorphic Jacobi form of weight -8 in two variables. The proof is close in spirit to the original proof of Yau-Zaslow due to Beauville, and no virtual classes are involved. This is based on joint work with Stephen Pietromonaco.