Critical values of coamoebas of codimension two affine planes

TL;DR

This paper investigates the topology of critical loci of coamoebas associated with generic affine planes in four-dimensional space, providing insights into their geometric and topological properties.

Contribution

It offers a detailed description of the critical loci of coamoebas for codimension two affine planes, advancing understanding of their topology.

Findings

Characterization of critical loci topology

Description of coamoeba structures in four-space

Insights into affine plane coamoeba topology

Abstract

We describe the topology of critical loci of coamoeba of generic affine planes in four-space.

Equations

$$(x,y)\mapsto (x,y,x+y-1,x+ky-a)$$

$$C_l=\{p\in \mathbb{R}{P}^2|\exists (l_0,l,l_1,l_a)\in {\mathcal{B}}_Z:\varphi ({\ell }_0,\ell ,{\ell }_1,{\ell }_a)=p\}$$

$$\tilde{\varphi }(l_0,l,l_1,l_a)=(\varphi (l_0,l,l_1,l_a),l).$$

Full text

Consider a generic affine plane $V$ in $\mathbb{C}^{4}$. Denote by $V^{\times}$ the intersection of $V$ with the algebraic torus $(\mathbb{C}^{\times})^{4}$ (recall that $\mathbb{C}^{\times}=\mathbb{C}\backslash\{0\}$). There are maps $Log:(\mathbb{C}^{\times})^{4}\rightarrow\mathbb{R}^{4},\ 2Arg:(\mathbb{C}^{\times})^{4}\rightarrow(\mathbb{R}P^{1})^{4}$ and $Arg:(\mathbb{C}^{\times})^{4}\rightarrow(S^{1})^{4}$. Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be the restrictions of these maps to $V^{\times},$ we call them amoeba, rolled coamoeba and (usual) coamoeba map respectively. Note that their loci of critical points are identical – denote by $Z$ this locus. Fix the notations $\mathcal{A}_{Z}$, $\mathcal{B}_{Z}$, $\mathcal{C}_{Z}$ for the critical values of $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}.$

Proof of Theorem 1

Without loss of generality we assume that $V$ is parametrized as

[TABLE]

for generic $a,k\in\mathbb{C}$. This allows to draw the associated linkage (see Figure 1).

From now on we denote by $(l_{0},l,l_{1},l_{a})$ a point in $(\mathbb{R}P^{1})^{4},$ where we think of every of the four lines as passing throug the origin in $\mathbb{C}.$ Consider $\mathbb{R}P^{2}$ as the compactification of $\mathbb{C}$ by the line $\mathbb{R}P^{1}$ at infinity.

Lemma 1**.**

If $(l_{0},l,l_{1},l_{a})\in\mathcal{B}_{Z}$ then there exist a unique point $p\in\mathbb{R}P^{2}$ such that the closures of $l_{0},l_{1}+1,l_{a}+a\subset\mathbb{C}$ intersect at $p.$

Therefore, we have a map $\phi:\mathcal{B}_{Z}\rightarrow\mathbb{R}P^{2}$ assigning the intersection point $p$ to a critical value. It is easy to see that $\phi$ is birational.

Remark 3**.**

For any $l\in\mathbb{R}P^{1}$ the locus

[TABLE]

is a conic curve. Moreover, the map $l\in\mathbb{R}P^{1}\mapsto C_{l}\in\mathbb{R}P^{5}$ is linear.

In other words, $\{C_{l}\}_{l\in{\mathbb{R}P^{1}}}$ is a pencil of conics.

Remark 4**.**

The base points of this pencil are $0,\ 1,\ a$ and $d,$ where the fourth point is constructed on Figure 2.

Denote by $\tilde{C}=\{(p,l)\in\mathbb{R}P^{2}\times\mathbb{R}P^{1}|\,p\in C_{l}\}$ the universal curve of this pencil.

Remark 5**.**

$\tilde{C}$* is a blow-up of $\mathbb{R}P^{2}=\mathbb{C}\cup\mathbb{R}P^{1}$ at four points $0,\ 1,\ a,\ d\in\mathbb{C}$.*

Finally, the map $\phi$ extends to a homeomorphism $\tilde{\phi}:\mathcal{B}_{Z}\to\tilde{C}$ given by

[TABLE]

Proof of Lemma 1

Lemma 2**.**

The map $\mathcal{B}$ is rational.

Proof.

Note that $2arg(z)=[{Re(z)}:{Im(z)}]\in\mathbb{R}P^{1}$, for a non-zero complex number $z$, and $\mathcal{B}$ is a composition of the coordinatewise $2arg$ and the affine embedding of $V$ to $\mathbb{C}^{4}$.

∎

Lemma 3**.**

The map $\mathcal{B}$ is birational.

Proof.

For a generic point $(l_{0},l,l_{1},l_{a})\in(\mathbb{R}P^{1})^{4}$ we construct a unique preimage under $\mathcal{B}$. Recall that in this text $\mathbb{R}P^{1}$ is seen as a real projectivisation of $\mathbb{C}$.

Let $\mathbb{C}/l$ be the set of all lines $L\subset\mathbb{C}$ parallel to $l.$ Denote by $x(L)$ the intersection of $L\in\mathbb{C}/l$ with $l_{0}$ and let $x(L)+y(L)$ be the intersection of $L$ with $l_{1}+1$. Define $\lambda(l_{0},l,l_{1})\subset\mathbb{C}$ as the locus of points of the form $z(L)=x(L)+ky(L).$ If $l_{0},\ l_{1}$ and $l$ are generic then $\lambda(l_{0},l,l_{1})\subset\mathbb{C}$ is a line.

The lines $\lambda(l_{0},l,l_{1})$ intersects $l_{a}+a$ at $z(\hat{L})$ for a unique $\hat{L}\in\mathbb{C}/l.$ The preimage of $(l_{0},l,l_{1},l_{a})$ under $\mathcal{B}$ is $(x(\hat{L}),y(\hat{L}))\in V.$∎

Now we analyze how this invertion procedure can fail if the genericity assumption on $(l_{0},l,l_{1},l_{a})$ is droped. If $l_{0}=l=l_{1}\neq\mathbb{R}$ then $\lambda(l_{0},l,l_{1})$ is empty and $(l_{0},l,l_{1},l_{a})$ has no preimages under $\mathcal{B}$. The point $(\mathbb{R},\mathbb{R},\mathbb{R},l_{a})$ has infinetly many preimages for any $l_{a}\in\mathbb{R}P^{1}$ and, therefore, is a critical value of $\mathcal{B}$. Points of the form $(\mathbb{R},l,\mathbb{R},l_{a})$ for $l\neq\mathbb{R}$ have no preimages under the rolled coamoeba map.

If $l_{0}$ and $l_{1}+1$ intersect at a single point $p\in\mathbb{C}$ then the locus $\lambda(l_{0},l,l_{1})$ is either $\{p\}$ or a line. The former happens only if $p$ belongs to the circle circumscribing the triangle with vertices [math], $1$ and $k.$ If $\lambda(l_{0},l,l_{1})$ has no intersections with $l_{a}+a$ then $(l_{0},l,l_{1},l_{a})$ is not a value of $\mathcal{B}.$ If $\lambda(l_{0},l,l_{1})\subset l_{a}+a$ then $(l_{0},l,l_{1},l_{a})$ has infinetly many preimages.

Finally, for $l\neq l_{0}=l_{1}\neq\mathbb{R}$ the locus $\lambda(l_{0},l,l_{1})$ is a line parallel to $l_{0}.$ If this line intersects $l_{a}+a$ at a unique point then $(l_{0},l,l_{1},l_{a})$ is a regular value, if $\lambda(l_{0},l,l_{1})\cap(l_{a}+a)=\emptyset,$ it is not a value, and if $\lambda(l_{0},l,l_{1})=l_{a}+a,$ it is a critical value (with infinetly many preimages).

Corollary 2**.**

For $(l_{0},l,l_{1},l_{a})\in\mathcal{B}_{Z}$ the lines $l_{0},$ $l_{1}+1$ and $l_{a}$ intersect at a single point or parallel.

Remark 6**.**

In the complete analogy with what we just did with respect to a pair of points [math] and $1$ (by means of the locus $\lambda(l_{0},l,l_{a})$ and its study), the inversion construction and the analysis of its degeneration can be performed with respect to a pair of $1$ and $a$ or a pair of $a$ and $0.$

Proof of Theorems 2 and 3

Lets construct the compactification of $Z$ as the total space of a circle bundle over $\mathcal{B}_{Z}.$ Note that the fibers over $c\in\mathcal{B}_{Z}$ of the projection $\mathcal{B}|_{\mathcal{Z}}$ consist of linkage configurations with triangles spanned by vertices $x,x+y,x+ky\in\mathbb{R}P^{2},$ where all such triangles are related through homotety with center at $p=\phi(c)$ if $p\in\mathbb{C}$ or through the parallel transport along $p\in\mathbb{R}P^{1}$ otherwise. In particular, the fiber is a projective line of triangles besides at most 5 special triangles, that correspond to degenerate linkage configurations when either

• •

$x=0$,

• •

$y=0,$ i.e. the triangle degenerates to $p$,

• •

$x+y=1$,

• •

$x+ky=a$,

• •

$x$, $x+y,$ or $x+ky\in\mathbb{R}P^{1},$ i.e. the triangle has points at infinity.

We denote the corresponding sections by $s_{0},\ s_{p},\ s_{1},\ s_{a}$ and $s_{\infty}.$ While going along the circle-fiber, crossing one of the first four sections changes the argument of the corresponding coordinate to the opposite one. Passing through $s_{\infty}$ changes arguments of all four coordinates.

Proof of Theorem 4

On Figure 4 we plot the loci where the five sections $s_{0},\ s_{p},\ s_{1},\ s_{a}$ and $s_{\infty}$ pairwise coincide on $\mathcal{B}_{Z}$ (represented as $\mathbb{R}P^{2}$ blown-up at $0,\ 1,\ a,$ and $d$).

Remark 7**.**

Every pair of sections coincides along a topological circle and very triple has no common intersections.

Therefore, there exist a unique compactification $\hat{\mathcal{C}}_{Z}$ of $\mathcal{C}_{Z}$ to which the the projection $\mathcal{C}_{Z}\rightarrow\mathcal{B}$ extends as an unramified five-fold covering. The fibers are represented by five intervals (some of which can be degenerate if a pair of sections concides over a given point) between the five sections of $\hat{Z}\rightarrow\mathcal{B}_{Z}$.

Computing monodromies we deduce that $\hat{\mathcal{C}}_{Z}$ is connected (see Figure 5 for more details). Thus, $\hat{\mathcal{C}}_{Z}$ is homeomorphic to $\mathbb{R}P^{2}$ blown-up at sixteen points by classification of topological surfaces.