Common Real Secants to Pairs of Real Twisted Cubic Curves

TL;DR

This paper explores the structure and classification of common secant lines to pairs of real twisted cubic curves, revealing the full symmetric monodromy group over complex numbers and demonstrating diverse real secant configurations.

Contribution

It introduces a detailed classification of real secant lines to real twisted cubics, including totally real, partially real, and minimally real lines, and constructs examples for each case.

Findings

The monodromy group of ten common secant lines over complex numbers is the full symmetric group.

For each number of totally real secants from 0 to 10, there exist pairs of real twisted cubics realizing that count.

Examples show the wide range of configurations of real secant lines to real twisted cubics.

Abstract

It is well established that a general pair of twisted cubic curves in complex projective space has ten common secant lines. As an initial investigation, we show that the monodromy group of the ten common secant lines over the complex numbers is the full symmetric group demonstrating that the common secant lines have no special structure over the complex numbers. We then investigate a novel question in real algebraic geometry: describe the possible collections of ten common secant lines to a pair of real projective twisted cubic curves. In addition to distinguishing between real and nonreal secant lines, we introduce a refinement of this classification which takes intersection points into account yielding totally real, partially real, and minimally real secant lines. Using computational algebraic geometry as well as combinatorics, we show that for each $k$ between 0 and 10, there exist pairs of real twisted cubic curves with exactly $k$ common totally real secant lines. We also obtain examples of real twisted cubics whose sets of common real secants cover a wide range of possibilities within our admissible classification of common real secant lines.