Understanding divisors in tropical geometry

Question

I've been studying tropical geometry recently, and I found the divisor idea, but I don't really get it, and also, I didn't find good examples. If you can explain me this idea, and give me some intuitive examples, I'd be so grateful. Thanks in advance.

Answer 1

You want intuitive examples and it's simplest to have intuitive examples using (tropical) curves.

Definitions. A tropical divisor on a tropical curve $\Gamma$ is a "formal sum" of points $p\in\Gamma$ with integer coefficients, i.e., $D$ is a divisor on $\Gamma$ iff $$ D = \sum_{p\in\Gamma} a_p\,[p]\,, $$ where all coefficients $a_p\in\Bbb Z$ and only finitely many of the $a_p$ are non-zero for any given $D$. (In other words, divisors are elements of the free abelian group on the set of points $\{[p]\}$ of $\Gamma$, let's denote this by $\Bbb Z\Gamma$.)

A principal divisor on a tropical curve $\Gamma$ is the divisor associated with a tropical rational function. (A tropical rational function is just a piece-wise linear function $f: \Gamma \to \Bbb R$ with integer slopes.) The principal divisor $(f)$ of $f$ is formed by taking the "formal sum" of the points where $f$ has non-zero slope changes (critical points), weighted by the magnitude of these slope changes. (Formally, as before, the principal divisor $(f)$ of $f$ is given by an element of a subgroup of $\Bbb Z\Gamma$, i.e., it has the form: $$ (f) = \sum_{p \in \Gamma} \Delta_p(f)\,[p]\,, $$ where $\Delta_p(f)$ is the change in the slope of $f$ at point $p$.)

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Example. Consider the tropical line $\Gamma$ defined by $ \min(x, y, 0) = 0 $. One possible tropical rational function on $\Gamma$ is $ f(x, y) = \min(x, y) $. We find the principal divisor $(f)$ by identifying the points where the slope of $f$ changes and the magnitude of these changes. With some quick work, this is: $$ (f) = 1 \cdot [(0, 0)] - 1 \cdot [(-1, 0)] - 1 \cdot [(0, -1)] $$ This indicates that the principal divisor consists of:

• A point at the origin $(0, 0)$ with coefficient $1$,
• Points at $(-1, 0)$ and $(0, -1)$ each with coefficient $-1$.

The divisor indicates that the function has a critical point at the origin and decreases in slope by $1$ unit in both the $x$- and $y$-direction as you move away from the origin along the negative axes.

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It is to be noted that the above definitions also generalize for higher dimensions using classical algebraic geometric language. Since the OP is asking for a "reputable source", here I list a couple resources on tropical geometry (TG):

• A great introductory reference (with minimal prerequisites) is Introduction to Tropical Geometry by Maclagen & Strumfels.
• Tropical Geometry and Its Applications is a great exposition by G. Mikhalkin (at some point you must start reading his work if you want to keep going into TG).

There are other more formal/advanced treatments of TG as well. But I think the OP is looking for introductory/intuitive references and these are a great place to start. If OP wants more advanced treatments, one could start by looking at the references section of this nLab article. Hope this answers OP's questions.