On the lower bounds for real double Hurwitz numbers

Abstract

As the real counterpart of double Hurwitz number, the real double Hurwitz number depends on the distribution of real branch points. We consider the problem of asymptotic growth of real and complex double Hurwitz numbers. We provide a lower bound for real double Hurwitz numbers based on the tropical computation of real double Hurwitz numbers. By using this lower bound and J. Rau’s result ( Math Ann 375: 895-915, 2019), we prove the logarithmic equivalence of real and complex Hurwitz numbers.

Appendix A. Real double Hurwitz numbers via factorization

In this appendix, we give an equivalent description of real double Hurwitz number via symmetric group.

Definition A.1

A real factorization of type $(g,\lambda ,\mu ;s)$ is a tuple $(\gamma ,\sigma _1,\tau _1,\ldots ,\tau _r,\sigma _2)$ of elements of the symmetric group $\mathcal {S}_d$ satisfying:

• $\sigma _2\cdot \tau _r\cdot \cdots \cdot \tau _1\cdot \sigma _1={\text {id}}$;

• $r=l(\lambda )+l(\mu )+2g-2$;

• $\mathcal {C}(\sigma _1)=\lambda$, $\mathcal {C}(\sigma _2)=\mu$, $\mathcal {C}(\tau _i)=(2,1,\ldots ,1)$, $i=1,\ldots ,r$;

• the subgroup generated by $\sigma _1$, $\sigma _2$, $\tau _1,\ldots ,\tau _r$ acts transitively on the set $\{1,\ldots ,d\}$.

• $\gamma$ is an involution (i.e. $\gamma ^2={\text {id}}$) satisfying: $\gamma \circ \sigma _1\circ \gamma =\sigma _1^{-1}$ and

  $$\begin{aligned}{} & {} \gamma \circ (\tau _i\circ \cdot \cdot \cdot \circ \tau _{1}\circ \sigma _1)\circ \gamma = (\tau _i\circ \cdot \cdot \cdot \circ \tau _{1}\circ \sigma _1)^{-1}, \text { for } i=1,\ldots ,s, \text { and}\\{} & {} \gamma \circ (\tau _j\circ \cdot \cdot \cdot \circ \tau _{s+1})\circ \gamma = (\tau _j\circ \cdot \cdot \cdot \circ \tau _{s+1})^{-1}, \text{ for } j=s+1,\ldots ,r. \end{aligned}$$

We denote by $\mathcal {F}^\mathbb {R}(g,\lambda ,\mu ;s)$ the set of all real factorizations of type $(g,\lambda ,\mu ;s)$.

Lemma A.2

Let g, d, $\lambda$ and $\mu$ be as above, then

$$\begin{aligned} H^\mathbb {R}_g(\lambda ,\mu ;s)=\frac{1}{d!}|\mathcal {F}^\mathbb {R}(g,\lambda ,\mu ;s)|. \end{aligned}$$

Proof

The proof of this lemma is essentially the same as the proof of [10, Lemma 2.3 and Construction 2.4]. So we only give a sketch here.

We fix r real points $p_1<\ldots<p_{r-s}<0<p_{r-s+1}<\ldots <p_r$ on $\mathbb {R}P^1\setminus \{\infty \}$. Let $p_0$ be a real point such that $p_{r-s}<p_0<0$. We choose $p_0$ as the base point. Let $l_0,l_1,\ldots ,l_r$ be $r+1$ loops depicted in Fig. 14.

Fig. 14

Generators of $\pi _1(\mathbb {C}P^1\setminus \{0,\infty ,p_1,\ldots ,p_r\},p_0)$

It is easy to see that $l_0,l_1,\dots ,l_r$ generate the fundamental group $\pi _1(\mathbb {C}P^1\setminus \{0,\infty ,p_1,\ldots ,p_r\},p_0)$. The action of complex conjugation on $\pi _1(\mathbb {C}P^1\setminus \{0,\infty ,p_1,\ldots ,p_r\},p_0)$ is determined by:

$$\begin{aligned}&{\text {conj}}(l_i\cdots l_0)=(l_i\cdots l_0)^{-1}, \qquad 0\le i\le s; \nonumber \\&{\text {conj}}(l_j\cdots l_{s+1})=(l_j\cdots l_{s+1})^{-1}, \qquad s+1\le j\le r. \end{aligned}$$

(1)

A real factorization $(\gamma ,\sigma _1,\tau _1,\ldots ,\tau _r,\sigma _2)$ of type $(g,\lambda ,\mu ;s)$ induces a real cover as follows: From the classical Hurwitz construction (see [11] or [5, Chapter 7]), we know that a tuple $(\sigma _1,\tau _1,\ldots ,\tau _r,\sigma _2)$ satisfying the first four conditions in Definition A.1 induces a cover $\pi :C\rightarrow \mathbb {C}P^1$ with ramification profiles $\lambda$ and $\mu$ over 0 and $\infty$, respectively, and simple ramification over $\underline{p}$. Moreover, $\pi ^{-1}(p_0)$ are labelled, i.e. $\pi ^{-1}(p_0)=\{q_1,\ldots ,q_d\}$, and the monodromy actions of the loops $l_0,\ldots ,l_r$ are represented by $\sigma _1,\tau _1,\ldots ,\tau _r$, respectively. Suppose that $p\in C$ is an unramified point. Choose a path $\alpha$ in $\mathbb {C}P^1\setminus \{0,\infty ,p_1,\ldots ,p_r\}$ from $p_0$ to $\pi (p)$. Lift $\alpha$ to a path $\tilde{\alpha }$ in C with endpoint p. Let $q_k$ be the starting point of $\tilde{\alpha }$. Let $\beta ={\text {conj}}(\alpha )$ be the conjugated path of $\alpha$. Then, lift $\beta$ to a path $\tilde{\beta }$ with starting point $q_{\gamma (k)}$. Let $\bar{p}$ be the endpoint of $\tilde{\beta }$. We define $\tau (p)=\bar{p}$. The fifth condition in Definition A.1 implies that $\tau (p)$ is well defined. Then, one can extend $\tau$ to C by standard arguments. From the construction, we know $\pi \circ \tau ={\text {conj}}\circ \pi$. Actually, this construction gives a map $\psi :\mathcal {F}^\mathbb {R}(g,\lambda ,\mu ;s)\rightarrow \mathcal {R}$, where $\mathcal {R}$ is the set of isomorphism classes of real Hurwitz covers of type $(g,\lambda ,\mu ,\underline{p})$. By a similar argument to the proof of [10, Lemma 2.3], we have $\psi :\mathcal {F}^\mathbb {R}(g,\lambda ,\mu ;s)/\mathcal {S}_d\rightarrow \mathcal {R}$ is bijective, and ${\text {Stab}}_{\mathcal {S}_d}(T)={\text {Aut}}(T)$, where $T\in \mathcal {F}^\mathbb {R}(g,\lambda ,\mu ;s)$ is a factorization. Then, we get Lemma A.2. $\square$