Real Space Sextics and their Tritangents

TL;DR

This paper develops algorithms to compute real tritangent planes of space sextic curves arising from del Pezzo surfaces, revealing the wide variation in the number of real tritangents and solving a problem posed in 1928.

Contribution

It introduces novel algorithms for identifying real tritangents of space sextics and analyzes their discriminants, addressing a longstanding mathematical question.

Findings

Number of real tritangents varies from 0 to 120.

Algorithms successfully compute real tritangents for specific cases.

The study confirms the wide variability in real tritangent counts.

Abstract

The intersection of a quadric and a cubic surface in 3-space is a canonical curve of genus 4. It has 120 complex tritangent planes. We present algorithms for computing real tritangents, and we study the associated discriminants. We focus on space sextics that arise from del Pezzo surfaces of degree one. Their numbers of planes that are tangent at three real points vary widely; both 0 and 120 are attained. This solves a problem suggested by Arnold Emch in 1928.