# Amoebas of Curves

The basic idea of tropical geometry is to study a complex plane curve by looking at its image in ${\displaystyle \mathbb {R} ^{2}}$ under the map:

{\displaystyle {\begin{aligned}\varphi :\ &(\mathbb {C} ^{*})^{2}\rightarrow \mathbb {R} ^{2}\\&(x,y)\mapsto (\log {|x|},\log {|y|})\end{aligned}}}
If ${\displaystyle C\subseteq (\mathbb {C} ^{*})^{2}}$ is a complex curve then its image ${\displaystyle \varphi (C)}$ is called the amoeba of ${\displaystyle C}$. Since ${\displaystyle C}$ has two real dimensions, we might expect the same to be true for its amoeba. Usually this will be the case, so that the amoeba of a complex curve forms a (real) surface in ${\displaystyle \mathbb {R} ^{2}}$.

Remark 14.1

This fact is not completely obvious and actually does not hold in all situations. For instance, it is clear that ${\displaystyle \varphi }$ contracts any radial circle ${\displaystyle S_{x}^{1}\subseteq \mathbb {C} _{x}^{*}\subseteq (\mathbb {C} _{xy}^{*})^{2}}$ to a point, and similarly for any circle ${\displaystyle S_{y}^{1}\subseteq (\mathbb {C} _{xy}^{*})^{2}}$. Thus there are entire tori ${\displaystyle S_{x}^{1}\times S_{y}^{1}}$ which are mapped to a single point under ${\displaystyle \varphi }$, so that ${\displaystyle \varphi }$ does not always preserve dimensions. However, if the curve we choose is sufficiently generic then it will be transverse to these tori, and then the image will have two real dimensions as desired.

We will now look at a couple of examples. In what follows we will use uppercase ${\displaystyle X}$ and ${\displaystyle Y}$ to denote the (real) co-ordinates on the codomain.

Example 14.1

Our first example is extremely simple: letting ${\displaystyle C=\{y=c\}}$ in ${\displaystyle (\mathbb {C} ^{*})^{2}}$ for some constant ${\displaystyle c}$, we have:

${\displaystyle \varphi (C)=\{Y=\log {|c|}\}}$
Thus the amoeba associated to the line ${\displaystyle C}$ is simply another line (though this time in ${\displaystyle \mathbb {R} ^{2}}$). This has real dimension ${\displaystyle 1}$, whereas ${\displaystyle C}$ has real dimension ${\displaystyle 2}$; the reason for this discrepancy is that ${\displaystyle C}$ contains radial circles ${\displaystyle S_{x}^{1}}$ which are contracted down to a point by ${\displaystyle \varphi }$ (see Remark on Contracting Circles above). In fact, ${\displaystyle \varphi :C\rightarrow \varphi (C)}$ is a fibration with circle fibres of the form ${\displaystyle S_{x}^{1}}$.

Example 14.2

Moving away from this degenerate example, let us consider a generic line ${\displaystyle C=\{ax+by=c\}\subseteq (\mathbb {C} ^{*})^{2}}$ (we assume from now on that ${\displaystyle a,b,c}$ are positive real numbers; the reason for this will soon become clear). As a variety this is isomorphic to ${\displaystyle \mathbb {P} ^{1}}$ minus three points (a line in ${\displaystyle \mathbb {C} ^{2}}$ is ${\displaystyle \mathbb {P} ^{1}}$ minus a single point, and we lose two more points by excluding the cases ${\displaystyle x=0}$ and ${\displaystyle y=0}$).

In order to study ${\displaystyle \varphi (C)}$ we examine what happens when ${\displaystyle |x|}$ or ${\displaystyle |y|}$ tends to ${\displaystyle 0}$ or ${\displaystyle \infty }$; note that this is equivalent to ${\displaystyle \log |x|}$ or ${\displaystyle \log |y|}$ tending to ${\displaystyle -\infty }$ or ${\displaystyle \infty }$ respectively, so that we are really examining the asymptotics of the amoeba. There are essentially three cases to consider.

First, consider the case ${\displaystyle |x|\gg 0}$ (equivalently ${\displaystyle |y|\gg 0}$), so that ${\displaystyle |a||x|\simeq |b||y|}$. Projecting along ${\displaystyle \varphi }$, this is equivalent to ${\displaystyle A+X\simeq B+Y}$ (where of course ${\displaystyle A=\log {a},B=\log {b}}$). Thus as ${\displaystyle x}$ and ${\displaystyle y}$ approach ${\displaystyle \infty }$, the amoeba of ${\displaystyle C}$ approaches the line ${\displaystyle \{X+A=Y+B\}}$ in ${\displaystyle \mathbb {R} ^{2}}$.

The second case to consider is when ${\displaystyle |x|\ll 1}$, so that ${\displaystyle |b||y|\simeq |c|}$. Again, projecting along ${\displaystyle \varphi }$ we see that this is equivalent to ${\displaystyle B+Y\simeq C}$, so that as ${\displaystyle x}$ approaches ${\displaystyle 0}$ (that is, as ${\displaystyle X}$ approaches ${\displaystyle -\infty }$) the amoeba of ${\displaystyle C}$ approaches the line ${\displaystyle \{Y+B=C\}}$:

The final case to consider is when ${\displaystyle |y|\ll 1}$. The same arguments as in the second case apply, and so we see that as ${\displaystyle y}$ approaches ${\displaystyle 0}$ (that is, as ${\displaystyle Y}$ approaches ${\displaystyle -\infty }$) the amoeba of ${\displaystyle C}$ approaches the line ${\displaystyle \{X+A=C\}}$.

Putting all of these together, we see that our amoeba looks something like this:

Notice the key role of the lines ${\displaystyle \{X+A=Y+B\}}$,${\displaystyle \{Y+B=C\}}$ and ${\displaystyle \{X+A=C\}}$, and more precisely the subsegments of these lines drawn in the above figure (which indicate the direction in which they govern the amoeba's asymptotics). Ideally we would like to forget about the amoeba itself and just concentrate on these line segments. This is made precise in the notion of the graph of an amoeba, which we now turn to.