Projective duals to algebraic and tropical hypersurfaces

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 $\text{Trop}(X^{\ast })$ 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 $\text{Trop}(X^{\ast })$.

47 pages, 13 figures; v2 minor revisions; accepted to PLMS