One of the foundations of Gromov-Witten theory is the use (due to Kontsevich I think) of localization to calculate the number of degree $n$ curves on a general quintic 3-fold. When calculating the number of degree 2 curves, one evaluates an integral that gives the "expected" answer, 609250+2875/8. The extra term 2875/8 comes from the fact that we are integrating over the space of stable maps, so there is a contribution from double covers of lines. There are 2875 lines on a general quintic 3-fold, and the "Aspinwall-Morrison formula" says the contribution of each double cover is 1/8.
Now, one can apply exactly the same localization computation (with the same graphs) to find degree 2 curves on a generic degree $d$ hypersurface in $\mathbb{P}^{(2d+2)/3}$. For an octic 5-fold, we get the number 21553860841856. (A google search of this number turns up 1 hit, a physics paper on quantum cohomology.) On an octic 5-fold, we expect there to be a 1-dimensional family of lines. Since this is not finite, we may expect that double covers of these lines do not contribute to the integral, so perhaps 21553860841856 is in fact the correct number.
The next case is a generic degree 11 7-fold. This has a 2-dimensional family of lines, so we would again expect the number to be the correct count of degree 2 curves. However, we get 220133473099468969402439/32, not an integer! (This is all assuming my calculations are correct.)
What is going on here? Do we get a mysterious contribution from double covers of some of the lines? The next number (d=14, dim=9) gives an integer again, but then the next one (d=17, dim=11) gives 3777401804278045968840390365196911549093621/64, again not an integer.
