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Common Real Secants to Pairs of Real Twisted Cubic Curves

Saima Aslam, Matthew Faust, Jonathan D. Hauenstein, Jordy Lopez Garcia, Bryson Kagy, Margaret H. Regan, Charles W. Wampler, Albert Zhang

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.

Common Real Secants to Pairs of Real Twisted Cubic Curves

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 between 0 and 10, there exist pairs of real twisted cubic curves with exactly 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.

Paper Structure

This paper contains 12 sections, 7 theorems, 32 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Secants of twisted cubics
  3. Monodromy
  4. Heuristic searching for triangular loops
  5. Certification
  6. Realizable 3-tuples
  7. Uniform sampling
  8. Localizing sampling
  9. Totally real secant lines
  10. Discussion and Open Problems
  11. Acknowledgments
  12. Tables of 3-tuples

Key Result

Theorem 1.2

Two general twisted cubic curves in ${\mathbb P}^3({\mathbb C})$ have ten common secant lines.

Figures (5)

  • Figure 1: Illustration of the standard twisted cubic curve $C_0$ together with a secant line $\ell$.
  • Figure 2: Realized 3-tuples are plotted with $\bullet$. Admissible but not yet realized 3-tuples are plotted with $\times$. The figure on the left shows the 3-tuples with $n_{{\mathbb R}}\in \{0, 2,4,6,8\}$. The figure on the right shows the 3-tuples with $n_{{\mathbb R}} = 10$.
  • Figure 3: Ten totally real common real secant lines between $C_0$ and $C_{1}$.
  • Figure 4: Illustration of loops $\gamma_{1}$ and $\gamma_{2}$
  • Figure 5: Frequency distribution of the number of real common secant lines among 100,000 uniform random samples.

Theorems & Definitions (20)

  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 2.4: 10 totally real lines
  • Proposition 2.5
  • ...and 10 more