Tropical secant graphs of monomial curves

TL;DR

This paper constructs an explicit tropical secant graph for monomial curves, enabling effective computation of multidegrees and Chow polytopes, and generalizes previous degree formulas using advanced tropical geometry techniques.

Contribution

It provides the first explicit construction of the tropical secant graph for monomial curves and develops algorithms for computing key invariants, extending prior theoretical results.

Findings

Explicit tropical secant graph construction for monomial curves

Algorithms for multidegree and Chow polytope computation

Generalization of Ranestad's degree formula

Abstract

The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is based on a refinement of the theory of geometric tropicalization due to Hacking, Keel and Tevelev.