Elliptic Gromov-Witten Invariants of Del-Pezzo Surfaces

We obtain a formula for the number of genus one curves with a variable complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done using Getzler's relationship among cohomology classes of certain codimension 2 cycles in $\overline{M}_{1,4}$ and recursively computing the genus-one Gromov-Witten invariants of del Pezzo surfaces. Using completely different methods, this problem has been solved earlier by Bertram and Abramovich, Ravi Vakil, Dubrovin and Zhang and more recently using Tropical geometric methods by M. Shoval and E. Shustin. We also subject our formula to several low degree checks and compare them to the numbers obtained by the earlier authors.