Tropical tangents for complete intersection curves

TL;DR

This paper develops a method to compute the tropicalizations of tangent lines, dual, and tangential varieties of complete intersection curves in projective space, enabling degree and Newton polytope calculations without elimination theory.

Contribution

It introduces a procedure to compute tropicalizations of tangent-related varieties directly from hypersurface tropicalizations, avoiding elimination theory.

Findings

Computed degrees of dual and tangential varieties.

Described Newton polytope of the tangential variety.

Provided a method for tropicalization of Gauss map images.

Abstract

We consider the tropicalization of tangent lines to a complete intersection curve $X$ in $\mathbb{P}^n$. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of $X$ in terms of the tropicalizations of the hypersurfaces cutting out $X$. We apply this to obtain descriptions of the tropicalization of the dual variety $X^*$ and tangential variety $\tau(X)$ of $X$. In particular, we are able to compute the degrees of $X^*$ and $\tau(X)$ and the Newton polytope of $\tau(X)$ without using any elimination theory.