Projective duals to algebraic and tropical hypersurfaces

TL;DR

This paper explores the tropical analogue of projective duality for hypersurfaces, providing explicit descriptions for curves and surfaces under certain conditions, and extending classical duality results to tropical geometry.

Contribution

It introduces a tropical dual variety concept, explicitly describes it for curves and surfaces, and connects tropical duality with classical geometric formulas.

Findings

Explicit description of Trop(X*) for curves and surfaces

Transformation of Newton polygons under duality

Extension of classical duality formulas to tropical setting

Abstract

We study a tropical analogue of the projective dual variety of a hypersurface. When $X$ is a curve in $\mathbb{P}^2$ or a surface in $\mathbb{P}^3$, we provide an explicit description of $\text{Trop}(X^*)$ in terms of $\text{Trop}(X)$, as long as $\text{Trop}(X)$ is smooth and satisfies a mild genericity condition. As a consequence, when $X$ is a curve we describe the transformation of Newton polygons under projective duality, and recover classical formulas for the degree of a dual plane curve. For higher dimensional hypersurfaces $X$, we give a partial description of $\text{Trop}(X^*)$.