Gonality and genus of canonical components of character varieties

We begin with some algebraic geometry. We follow Shafarevich [22] and Hartshorne [9]. The ‘natural models’ of the character varieties of interest are complex affine algebraic sets, which typically have singularities at infinity. We will rely on an analysis of rational maps between possibly singular affine sets. Such a map $f:X\dashrightarrow Y$ behaves (on most of $X$ and $f(Y)$) as a combination of a very nice map and a map with controlled bad parts. We now present some algebraic geometry to make this precise.

Let $\mathbb{C}[T]$ be the polynomial ring with coefficients in $\mathbb{C}$ in the variables $T_{1},\dots,T_{n}$. A (Zariski) closed subset of $\mathbb{C}^{n}$ is a subset $X\subset\mathbb{C}^{n}$ consisting of the common vanishing set of a finite number of polynomials in $\mathbb{C}[T]$. We let $\mathfrak{U}_{X}$ denote the ideal of $X$, the ideal in $\mathbb{C}[T]$ consisting of all polynomials which vanish on $X$. If $I$ is an ideal in $\mathbb{C}[T]$ we let $V(I)$ denote the vanishing set of $I$ in $\mathbb{C}^{n}$, and if $f\in\mathbb{C}[T]$ let $V(f)$ be the vanishing set of $f$ in $\mathbb{C}^{n}$. The coordinate ring of $X$ is $\mathbb{C}[X]=\mathbb{C}[T]/{\mathfrak{U}}_{X}$. This ring is the ring of all polynomials on $X$, up to the equivalence of being equal on $X$; adding any polynomial in ${\mathfrak{U}}_{X}$ to a polynomial $f$ does not change the value of $f$ on $X$. If a closed set $X$ is irreducible then the field of fractions of the coordinate ring $\mathbb{C}[X]$ is the function field or field of rational functions of $X$; it is denoted $\mathbb{C}(X)$.

A function $f$ defined on $X$ with values in $\mathbb{C}$ is regular if there exists a polynomial $F(T)$ with coefficients in $\mathbb{C}$ such that $f(x)=F(x)$ for all $x\in X$. A map $f:X\rightarrow Y$ is regular if there exist $n$ regular functions $f_{1},\dots,f_{n}$ on $X$ such that $f(x)=(f_{1}(x),\dots,f_{n}(x))$ for all $x\in X$. Any regular map $f:X\rightarrow Y$ defines a pullback homomorphism of $\mathbb{C}$-algebras $f^{*}:\mathbb{C}[Y]\rightarrow\mathbb{C}[X]$ as follows. If $u:X\rightarrow Z$ we define $v:Y\rightarrow Z$ by $v(x)=u(f(x))$. We define $f^{*}$ by $f^{*}(u)=v$. A rational map $f:X\dashrightarrow Y\subset\mathbb{C}^{m}$ is an $m$-tuple of rational functions $f_{1},\dots,f_{m}\in\mathbb{C}(X)$ such that, for all points $x\in X$ at which all the $f_{i}$ are defined, $f(x)=(f_{1}(x),\dots,f_{m}(x))\in Y.$ We define the pullback $f^{*}$ as above. A rational map $f:X\dashrightarrow Y$ is called dominant if $f(X)$ is (Zariski) dense in $Y$. A rational map $f:X\dashrightarrow Y$ is called birational if $f$ has a rational inverse, and in this case we say that $X$ and $Y$ are birational. A regular map $f:X\rightarrow Y$ is finite if $\mathbb{C}[X]$ is integral over $\mathbb{C}[Y]$. These definitions naturally extend to the projective setting. A morphism $\varphi:X\rightarrow Y$ is a continuous map such that for every open $V\subset Y$ and every regular function $f:Y\rightarrow\mathbb{C}$ the function $f\circ\varphi:\varphi^{-1}(Y)\rightarrow\mathbb{C}$ is regular.

Character varieties are naturally affine sets. It is useful to consider an embedding of such an affine set into projective space. Given an affine variety $X$, a projective completion of $X$ is the Zariski closure of an embedding of $X$ in some projective space. A smooth projective model of $X$ is a smooth projective variety birational to a projective completion of $X$. Such a model is unique up to isomorphism if $X$ is a curve. The degree of any projective curve $X$, denoted $d(X)$ is the maximum number of points of intersection of $X$ with a hyperplane not containing any component of $X$. This depends on the specific embedding of $X$.

The gonality of a curve is, in the most basic sense, the degree of a map from that curve to $\mathbb{C}^{1}$. Intuitively the degree of a map is the number of preimages of ‘most points’ in the image. Let $X$ and $Y$ be irreducible varieties of the same dimension and $f:X\dashrightarrow Y$ a dominant rational map. The degree of the field extension $f^{*}(\mathbb{C}(Y))\subset\mathbb{C}(X)$ is finite and is called the degree of $f$. That is,

This is well defined for rational and regular maps.

We are now in a position to see that dominant rational maps between possibly singular curves can be decomposed in a pleasing manner.

The algebraic definition of gonality is as follows.

Here, $x\in\mathbb{C}(X)$ is generated by a single element. Since $\mathbb{C}(X)$ is the function field of $X$ and consists of the rational functions on $X$, we can reformulate the definition; the gonality of $X$ is the minimal degree of a dominant rational map from $X$ to $\mathbb{P}^{1}$. Therefore, gonality is a birational invariant; two varieties which are birational (but not necessarily isomorphic) have equal gonality. If $X$ is an affine curve, $X$ is birational to a smooth projective model and as such the gonality of $X$ and $X^{\prime}$ are equal, and this is the minimal degree of a dominant rational map from $X$ to $\mathbb{C}$.

By Noether’s Normalization theorem, if $X$ is complex projective variety of dimension $n$, then there is a linear subspace disjoint from $X$ such that the projection onto that subspace is finite to one and surjective. This is also true for affine varieties; if $Z$ is an affine variety of dimension $n$ there is a finite degree map from $Z$ onto a dense subset of $\mathbb{C}^{n}$. Therefore, gonality is always finite.

The following lemma will be essential for our gonality bounds.

We now introduce some notation. Commonly $\alpha(x)\in\Theta(\beta(x))$ is written to mean that there are positive constants $c_{1}$ and $c_{2}$ independent of $x$ such that

Notice that if $\alpha(x)\in\Theta(\beta(x))$ then also

so $\beta(x)\in\Theta(\alpha(x))$. This defines an equivalence relation.

We now compute the gonality of the curves given by $x^{p}-y^{q}$ and $x^{p}y^{q}-1$ in $\mathbb{C}^{2}(x,y)$. First we establish some notation for these sets.

We will now prove a few lemmas which will be essential in bounding the gonalities we are interested in. In $\S~\ref{section:gonalityindehnsurgeryspace}$ we will relate the situation below to that of the character varieties of interest. Recall the definition of $\mathcal{C}(r)$ and $\mathcal{H}(r)$ from Definition 3.7. By Lemma 3.8 the gonality of (each irreducible component of) $\mathcal{C}(r)\subset\mathbb{C}^{2}$ is equal to one.

By Definition 3.7, with $\mathcal{H}(r)\subset$ $\mathbb{C}^{4}(x_{1},y_{1},x_{2},y_{2})$ and $\mathcal{C}(r)\subset\mathbb{C}^{2}(x_{2},y_{2})$ then $\mathcal{C}(r)=\varpi_{2}(\mathcal{H}(r)).$

Assume that $M$ is a finite volume hyperbolic 3-manifold whose boundary consists of exactly $k$ torus cusps. Fix a framing for each cusp; let $T_{i}$ be the $i^{th}$ connected component of $\partial M$ and let $\mu_{i}$ and $\lambda_{i}$ be an oriented choice of meridian and longitude for $T_{i}$. For $r_{i}=p_{i}/q_{i}\in\mathbb{Q}\cup\{1/0\}$, $i=1,\dots,k$, written in lowest terms (with $q$ non-negative), let $M(r_{1},\dots,r_{k})$ denote the manifold obtained by performing $r_{i}$ Dehn filling on the $i^{th}$ cusp of $M$. We write $r_{i}=-$ to indicate that the $i^{th}$ cusp remains complete and is not filled. By Thurston’s Hyperbolic Dehn Surgery theorem, for all but finitely many values of $r$ in $\mathbb{Q}\cup\{1/0\}$ the manifold obtained by $r$ filling of one cusp of $M$ is hyperbolic.

We fix a finite presentation for $\Gamma\cong\pi_{1}(M)$, with $M_{i}=[\mu_{i}]$, $L_{i}=[\lambda_{i}]$ amongst the generators for $i=1,\dots,k$. If $\Gamma$ has presentation

then by van Kampen’s theorem, a presentation for

is given by

where $w_{i}=M_{i}^{p}L_{i}^{q}$ unless $r=p/q=-$ in which case it is empty.

We state the next lemma for convenience. The proof is clear.

We will now define various varieties associated to the fundamental group of a 3-manifold $M$ (see [25].) For a finitely generated group $\Gamma$, and a homomorphism $\rho:\Gamma\rightarrow\text{SL}_{2}(\mathbb{C}),$ the set

naturally carries the structure of an affine algebraic set defined over $\mathbb{C}$. This is called the representation variety, $R(\Gamma)$. Given an element $\gamma\in\Gamma$ we define the trace function $I_{\gamma}:R(\Gamma)\rightarrow\mathbb{C}$ as $I_{\gamma}(\rho)=\text{tr}(\rho(\gamma))$. Similarly, given a representation $\rho\in R(\Gamma)$ we define the character of $\rho$ to be the function $\chi_{\rho}:\Gamma\rightarrow\mathbb{C}$ given by $\chi_{\rho}(\gamma)=\text{tr}(\rho(\gamma))$. The $\text{SL}_{2}(\mathbb{C})$-character variety of $\Gamma$ is the set of all characters,

Such a set is called a natural model for the character variety, as opposed to a set isomorphic to one obtained in this manner. The functions $t_{\gamma}:R(\Gamma)\rightarrow X(\Gamma)$ defined by $t_{\gamma}(\rho)=\chi_{\rho}(\gamma)$ are regular maps. The coordinate ring of $X(\Gamma)$ is $T\otimes\mathbb{C}$ where $T$ is the subring of all regular functions from $R(\Gamma)$ to $\mathbb{C}$ generated by 1 and the $t_{\gamma}$ functions.

If $\Gamma$ is isomorphic to $\Gamma^{\prime}$, then the sets $R(\Gamma)$ and $R(\Gamma^{\prime})$ are isomorphic, as are $X(\Gamma)$ and $X(\Gamma^{\prime})$. We write $R(M)$ or $X(M)$ to denote a specific $R(\pi_{1}(M))$ or $X(\pi_{1}(M))$, respectively, which is well-defined up to isomorphism. The inclusion homomorphism $i:\pi_{1}(\partial M)\rightarrow\pi_{1}(M)$ induces a map $i^{*}:X(M)\rightarrow X(\pi_{1}(\partial M))$ by $\chi_{\rho}\mapsto\chi_{\rho\mid_{\pi_{1}(\partial M)}}.$ We write $A(M)$ to denote the closure of the image of this map.

The representation variety $R(M)$ naturally sits in $\mathbb{C}^{4n}(g_{11},g_{12},g_{13},g_{14},\dots,g_{n4})$ where $g_{i1},g_{i2}$, $g_{i3}$ and $g_{i4}$ are the four entries of the $2\times 2$ matrix $\rho(\gamma_{i})$. Fix a generating set for $\pi_{1}(\partial M)$, $\{M_{1},L_{1},\dots,M_{k},L_{k}\}$ where each pair $\{M_{i},L_{i}\}$ generates $\pi_{1}(T_{i})$, the fundamental group a connected component of $\partial M$. Let $m_{i}^{\pm 1}$ be the eigenvalues of $\rho(M_{i})$ and $l_{i}^{\pm 1}$ be the eigenvalues of $\rho(L_{i})$.

Define the following functions on $\mathbb{C}(g_{11},\dots,g_{n4},m_{1}^{\pm 1},l_{1}^{\pm 1},\dots,m_{k}^{\pm 1},l_{k}^{\pm 1})$

The extended representation variety, $R_{E}(M)$ is the algebraic set in $\mathbb{C}^{4n}\times(\mathbb{C}^{*})^{2k}$ cut out by $\mathfrak{U}_{R(M)}$ and the equations above for $i=1,\dots,k$. The extended character variety $X_{E}(M)$ is defined similarly. The eigenvalue variety, $E(M)$, is the closure of the image of $R_{E}(M)$ under the projection to the coordinates $m_{1},l_{1},\dots,m_{k},l_{k}$. It naturally sits in $(\mathbb{C}^{*})^{2k}(m_{1}^{\pm 1},l_{1}^{\pm 1},\dots,m_{k}^{\pm 1},l_{k}^{\pm 1})$. We have the following commuting diagram where the maps $p_{1}$, $p_{2}$, and $p_{3}$ are all finite to one of the same degree.

An irreducible component of $R(M)$ containing a discrete and faithful representation is denoted $R_{0}(M)$, and the image in $X(M)$ is denoted $X_{0}(M)$ and called a canonical component. For a given $M$ there can be more than one canonical component, as there are multiple lifts of a discrete faithful representation to $\text{SL}_{2}(\mathbb{C})$. We call an irreducible component of $A(M)$ a canonical component and write $A_{0}(M)$ if it is the image of a canonical component in $X(M)$. This definition also extends to the extended varieties.

We can define the $\text{PSL}_{2}(\mathbb{C})$ representation and character varieties in a similar way. (See [2].) We denote the $\text{PSL}_{2}(\mathbb{C})$ character variety of $M$ by $Y(M)$ and a canonical component by $Y_{0}(M)$. We can form the restriction variety $B(M)$ analogous to $A(M)$ as above (and also form $B_{0}(M)$ when appropriate). The group $\text{PSL}_{2}(\mathbb{C})$ is isomorphic to $\rm{Isom}^{+}(\mathbb{H}^{3})$, the group of orientation preserving isometries of hyperbolic half space. Since the index $[\rm{Isom}(\mathbb{H}^{3}):\rm{Isom}^{+}(\mathbb{H}^{3})]=2$, for $M$ a finite volume hyperbolic 3-manifold with at least one cusp there are two $\text{PSL}_{2}(\mathbb{C})$ characters corresponding to discrete faithful representations of $\pi_{1}(M)$. These correspond to different orientations of $M$. The authors know of no examples where these two characters lie on different irreducible components of $Y(M)$. The coordinate ring of the quotient is $T_{e}\otimes\mathbb{C}$ where $T_{e}$ is the subring of $T$ consisting of all elements invariant under the action of $\mu_{2}$. Culler [6] (Corollary 2.3) showed that a faithful representation of any discrete torsion-free subgroup to $\text{PSL}_{2}(\mathbb{C})$ lifts to $\text{SL}_{2}(\mathbb{C})$, and moreover (Theorem 4.1) that a representation $\rho:\Gamma\rightarrow\text{PSL}_{2}(\mathbb{C})$ lifts to a representation to $\text{SL}_{2}(\mathbb{C})$ if and only if every representation in the path component containing $\rho$ lifts.

By the proof of Thurston’s Dehn Surgery Theorem ([24] Theorem 5.8.2), any neighborhood of the character of a discrete faithful representation in $Y(M)$ contains all but finitely many Dehn filling characters. Since a discrete faithful character is always a simple point of $Y(M)$ [21] it follows $Y_{0}(M(-,r))$ is contained in some $Y_{0}(M)$ for all but finitely many $r$. Therefore, for all but finitely many $r$, $X_{0}(M(-,r))$ is contained in some canonical component of $X(M)$. (Due to lifting restrictions it is not necessarily true that all but finitely many $X_{0}(M(-,r))$ are contained in a given $X_{0}(M)$.)