Quadratic types and the dynamic Euler number of lines on a quintic threefold

Abstract

We provide a geometric interpretation of the local contribution of a line to the count of lines on a quintic threefold over a field k of characteristic not equal to 2, that is, we define the type of a line on a quintic threefold and show that it coincides with the local index at the corresponding zero of the section of Sym^5 S^* -> Gr(2, 5) defined by the threefold. Furthermore, we define the dynamic Euler number which allows us to compute the A^1-Euler number as the sum of local contributions of zeros of a section with non-isolated zeros which deform with a general deformation. As an example we provide a quadratic count of 2875 distinguished lines on the Fermat quintic threefold which computes the dynamic Euler number of Sym^5 S^* -> Gr(2, 5). Combining those two results we get that the sum of the types of lines on a general quintic threefold is 1445<1> + 1430<-1> in GW(k) when k is a field of characteristic not equal to 2 or 5.