Abstract |
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 ([3]),
Ravi Vakil ([23]), Dubrovin and Zhang ([8]) and more recently
using Tropical geometric methods by
M. Shoval and E. Shustin ([22]). We also subject our formula to several low degree
checks and compare them to the numbers obtained by the earlier authors.
Our numbers agree with the numbers obtained by Ravi Vakil, except for one number where we
get something different. We give geometric reasons to explain why our answer is likely to be correct
and hence conclude that the number written by Ravi Vakil is likely to be a minor typo (since our numbers are
consistent with the other numbers he has obtained).
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