Tropical methods for building real space sextics with totally real tritangent planes

TL;DR

This paper introduces tropical geometry techniques to construct real space sextic curves with all tritangent planes totally real, providing new examples and insights into their algebraic and arithmetic properties.

Contribution

It develops a method using tropical geometry to explicitly build real space sextics with all tritangents totally real, linking tropical and classical algebraic geometry.

Findings

Tropical tritangents form 15 equivalence classes with classical counterparts.

Liftings of tropical tritangents are defined over quadratic extensions.

Constructed examples of real sextics with 64 and 120 totally real tritangents.

Abstract

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.