Rational curves in a quadric threefold via an $\text{SL}(2,\mathbb{C})$-representation

TL;DR

This paper investigates rational curves in a quadric threefold using an $ ext{SL}(2, ext{C})$-representation, revealing geometric structures and cohomology of moduli spaces for low-degree curves.

Contribution

It introduces a novel approach to studying rational curves in a quadric threefold via $ ext{SL}(2, ext{C})$-representations, especially for degree 3 curves and their moduli spaces.

Findings

Confirmed fixed rational curves of low degree under torus action.

Identified the moduli space of twisted cubic curves as a projective bundle.

Derived the cohomology ring of the moduli space.

Abstract

In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group $\text{SL}(2,\mathbb{C})$. Most of them are confirmations of very well-known facts. If the degree of a rational curve is $3$, it is confirmed using the Lagrangian's geometric properties that the moduli space of twisted cubic curves in $Q_3$ has a specific projective bundle structure. From this, we can immediately obtain the cohomology ring of the moduli space.