2019/06/13 by Nesterov, Denis, Oberdieck, Georg
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1906.05817
We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic fourfold is a non-singular curve of genus 631. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely 3780 elliptic curves of minimal degree with fixed (general) j-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kähler varieties with fixed j-invariant in terms of Gromov--Witten invariants. In K3[2]-type this leads to explicit formulas of these counts in terms of modular forms.