Tritangents and Their Space Sextics

TL;DR

This paper develops algorithms for constructing and analyzing space sextic curves, including computing tritangents and Steiner systems, with applications to reconstructing original geometric configurations, all implemented in magma.

Contribution

It introduces new algorithms for constructing space sextic curves, computing their tritangents and Steiner systems, and reconstructing original configurations from these data.

Findings

Algorithms successfully compute 120 tritangents and Steiner systems.

Reconstruction algorithms can recover original points or certify impossibility.

All methods are implemented in magma for practical use.

Abstract

Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta characteristics. In this paper we revisit both results from the computational perspective. Specifically, we give an algorithm to construct space sextic curves that arise from blowing up projective plane at eight points and provide algorithms to compute the 120 tritangents and their Steiner system of any space sextic. Furthermore, we develop efficient inverses to the aforementioned methods. We present an algorithm to either reconstruct the original eight points in the projective plane from a space sextic or certify that this is not possible. Moreover, we extend a construction of Lehavi which recovers a space sextic from its tritangents and Steiner system. All algorithms in this paper have been implemented in magma.