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

TL;DR

This paper introduces a geometric interpretation of line contributions on quintic threefolds, defines the dynamic Euler number, and computes the total types of lines, providing new tools for enumerative geometry over fields with characteristic not 2.

Contribution

It defines the type of a line on a quintic threefold and introduces the dynamic Euler number, enabling computation of the A^1-Euler number for non-isolated zeros.

Findings

Count of 2875 lines on the Fermat quintic threefold matches the dynamic Euler number.

Sum of line types on a general quintic is 1445<1> + 1430<-1> in GW(k).

Provides a quadratic count 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.