Common Real Secants of a Pair of Real Twisted Cubic Curves

This repository accompanies the paper https://arxiv.org/abs/2603.25003.

It provides:

1. Explicit parameter examples of different configurations of common real secant lines to a pair of real twisted cubic curves. The solutions were certified using alphaCertified.
2. Data verifying that the monodromy group of the ten secant lines is the full symmetric group, certified using CertifiedHomotopyTracking.jl.

Classification

Let $C_1$ and $C_2$ be real twisted cubic curves in $\mathbb{P}^{3}$, and let $\ell$ a common real secant line to both curves. The intersection of $\ell$ with each curve determines its type:

1. Totally real: both $C_1\cap \ell$ and $C_2\cap \ell$ consist of two real points,

2. Partially real: one intersection is real and the other is a pair of nonreal complex conjugate points,

3. Minimally real: both intersections consist of nonreal complex conjugate pairs.

Counting Secant Lines

For a generic pair of real twisted cubics, there are exactly 10 common secant lines. We represent their distribution using a $3$-tuple $(n_t,n_p,n_m)$ where $n_t$, $n_p$, and $n_m$ are the number of totally real, partially real, and minimally real common secant lines, respectively.

The total number of common real secant lines is $n_{\mathbb{R}}$ $: = n_t+n_p+n_m$, and the number of common nonreal secant lines is $10-n_{\mathbb{R}}$. Since nonreal solutions occur in complex conjugate pairs, $n_{\mathbb{R}}$ must be even, with $n_t,n_p,n_m\in$ {0,1,...,10}.

A 3-tuple $(n_{t},n_{p},n_{m})$ is admissible if $n_{t},n_{p},n_{m} \in$ {0,1,...,10} and $n_{\mathbb{R}}$ is even and at most 10. There are a total of $161$ distinct admissible $3$-tuples $(n_t,n_p,n_m)$. An admissible $3$-tuple $(n_{t},n_{p},n_{m})$ is realizable if there exists real twisted cubic curves $C_{1},C_{2}$ whose $10$ common secant lines yield that distribution.

Repository Structure

100000_run contains the 100000 parameters studied in the paper https://arxiv.org/abs/2603.25003.

example_parameters contains the explicit parameters realizing specific 3-tuples

Monodromy_computation.txt contains the certification data for the monodromy group

local_sampling_guide.txt is the guide to randomly sample around a given point