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

Sabrina Pauli

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.

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.

Paper Structure

This paper contains 24 sections, 24 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The local index
  3. Relative orientability
  4. Local coordinates
  5. The local index at an isolated, simple zero of $\sigma_f$
  6. Definition of the type
  7. The type is equal to the local index
  8. An algebraic interpretation of the product of resultants
  9. Proof of Theorem \ref{['thm: main thm']}
  10. The generic case
  11. $C$ has a tacnode
  12. $C$ is not birational onto its image
  13. The dynamic Euler number and excess intersection
  14. Fulton's intersection product
  15. The dynamic Euler number
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $X=\{f=0\}\subset\mathbb{P}^4$ be a general quintic threefold and let $l\subset X$ be a $k$-line on $X$. The type of $l$ is equal to the local index at the corresponding zero of the section $\sigma_f:\operatorname{Gr}(2,5)\rightarrow \mathcal{E}$.

Figures (1)

  • Figure 1: involution $i_M$

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 55 more