3·1. The boundary stratification of Hurwitz stacks

Recall that the space $\overline{\mathcal{H}}_{g,d,n,r}$ parametrises tuples

\begin{equation*}\left(C \xrightarrow{\pi} i\;\;\; D, p_1, \ldots, p_n \in C \right)\,,\end{equation*}

where $\pi$ is a degree d admissible cover from the genus g curve C to the genus 0 curve D and $p_1, \ldots, p_n$ are smooth points of C such that $p_{n-r+1}, \ldots, p_n$ map to the same point under $\pi$ . Let $\Gamma, \Gamma'$ be the dual graphs of C, D, respectively. Then the combinatorics of the cover $\pi$ can be described by a graph cover $\Gamma \to \Gamma'$ . The data of such a cover is given by:

1. (i) maps $V(\Gamma) \to V(\Gamma')$ , $H(\Gamma) \to H(\Gamma')$ from the vertices/half-edges of $\Gamma$ to those of $\Gamma'$ , corresponding to the action of $\pi$ on the components and nodes of C;

2. (ii) local degrees at vertices and half-edges of $\Gamma$ , describing the degree of $\pi$ on the components, and the ramification order at the nodes of C, respectively.

Associated to the graph cover $\Gamma \to \Gamma'$ , there exists a gluing morphism

(10) \begin{equation} \overline{\mathcal{H}}_{(\Gamma, \Gamma')} \to \overline{\mathcal{H}}_{g,d,n,r}\end{equation}

parametrising the closure of the locus in $\overline{\mathcal{H}}_{g,d,n,r}$ with dual graph cover $\Gamma \to \Gamma'$ . Here, the space $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ is a fibre product of spaces of Hurwitz covers (one for each vertex of $\Gamma$ ) over a product of moduli spaces of stable curves (one for each vertex of $\Gamma'$ ). The reason for this fibre product structure is that for any two vertices $v_1, v_2$ in $\Gamma$ mapping to the same vertex v in $\Gamma'$ , the associated subcurves $C_{v_1}, C_{v_2}$ of C must cover the same subcurve $D_v$ of D. For more details see [Reference Schmitt and van Zelm25, section 3·4] in the setting of Galois covers and [Reference Lian18, section 2·3·4] in the setting of Hurwitz covers.

In the computation below, we will see that the pullback of a boundary stratum in $\overline{\mathcal{M}}_{g,n}$ under the map

\begin{equation*}\epsilon_g \;:\; \overline{\mathcal{H}}_{g,d,n,r} \to \overline{\mathcal{M}}_{g,n}\end{equation*}

has an effective description via a disjoint union of gluing morphisms (10), see [Reference Lian18, proposition 3·2]. For the combinatorial description of this disjoint union, we need to make precise the concept that a given stable graph $\Gamma$ is a specialisation of a graph $\Gamma_0$ . This is encoded in the notion of a $\Gamma_0$ -structure on $\Gamma$ , which is specified by maps

\begin{equation*}\varphi_V \;:\; V(\Gamma) \twoheadrightarrow V(\Gamma_0) \text{ and }\varphi_H \;:\; H(\Gamma_0) \hookrightarrow H(\Gamma)\,,\end{equation*}

such that the graph obtained from $\Gamma$ by contracting all edges formed by half-edges not in the image of $\varphi_H$ is isomorphic to $\Gamma_0$ via the map $\varphi_V$ . See [Reference Graber and Pandharipande14, appendix A] for a formal definition and further discussions of this notion.

Then, when pulling back the boundary stratum of $\overline{\mathcal{M}}_{g,n}$ associated to the stable graph $\Gamma_0$ under the map $\epsilon_g$ , we obtain the disjoint union of boundary strata $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ such that $\Gamma$ carries a $\Gamma_0$ -structure satisfying that the composition

\begin{equation*}H(\Gamma_0) \to{\varphi_H} H(\Gamma) \to H(\Gamma')\end{equation*}

is surjective. However, instead of introducing these result for pullbacks of boundary strata under $\epsilon_g$ in full generality here, we recall the relevant formulas as they come up in the proof.

3·2. Recursive formulas for Tevelev degrees

The basic property satisfied by the Tevelev degree which is used in the proofs of Theorems 1-4 is the following recursion.

Proposition 7. Let $g, r \geq 1$ . Let $\ell \in \mathbb{Z}$ satisfy

\begin{equation*}n[g,\ell] - r+1=g+3+2\ell-r+1 \ \geq \ 3\, .\end{equation*}

Then, we have the recursion

(11) \begin{equation} \mathsf{Tev}_{g,\ell,r} = \mathsf{Tev}_{g-1,\ell,\max(1,r-1)} + \mathsf{Tev}_{g-1,\ell+1,r+1}\,.\end{equation}

Proof. Up to a combinatorial factor, $\mathsf{Tev}_{g,\ell,r}$ is defined as the degree of the map

\begin{equation*}\tau_{g,\ell,r} \;:\; \overline{\mathcal{H}}_{g,d[g,\ell],n[g,\ell],r} \to\overline{\mathcal{M}}_{g,n[\textrm{g},\ell]}\times\overline{\mathcal{M}}_{0,n[g,\ell]-r+1}\, .\end{equation*}

To prove the recursion (11), we compute the degree by analysing the fiber over a particular point

\begin{equation*}(C,D)\in \overline{\mathcal{M}}_{g,n[\textrm{g},\ell]}\times\overline{\mathcal{M}}_{0,n[g,\ell]-r+1}\, .\end{equation*}

The degree of $\tau_{g,\ell,r}$ is simply the degree of the zero cycle

\begin{equation*}\tau_{g,\ell,r}^* [(C,D)]\in \mathsf{CH}_0\left(\overline{\mathcal{H}}_{g,d[g,\ell],n[g,\ell],r} \right)\, .\end{equation*}

The actual fiber $\tau_{g,\ell,r}^{-1} [(C,D)]$ will have excess dimension, so some care must be taken in the analysis.

1. (i) The pointed curve C in $\overline{\mathcal{M}}_{g,n[\textrm{g},\ell]}$ is chosen to have the following form:

   (12)

   
   The curve C consists of a general curve $C' \in \overline{\mathcal{M}}_{g-1,n[\textrm{g},\ell]-1+2}$ attached at two points to a rational curve R. All the markings lie on C ^′ except for the last marking $p_{n[g,\ell]}$ which lies on R. The marking $p_{n[g,\ell]}$ is the last of the distinguished subset of r markings.

2. (ii) The pointed curve D in $\overline{\mathcal{M}}_{0,n[g,\ell]-r+1}$ is any general point (in particular, the curve D is nonsingular).

To compute the cycle $\tau_{g,\ell,r}^* [(C,D)]$ , let

be the stable graph of the curve (12), and let

\begin{equation*}\xi_{\Gamma_0} \;:\; \overline{\mathcal{M}}_{\Gamma_0} = \overline{\mathcal{M}}_{g-1,n[\textrm{g},\ell]-1+2} \times \overline{\mathcal{M}}_{0,3} \to \overline{\mathcal{M}}_{g,n[\textrm{g},\ell]}\end{equation*}

be the associated gluing map, sending the point $(C',R) \in \overline{\mathcal{M}}_{\Gamma_0}$ to the curve C. We consider the following commutative diagram:

(13)

The upper right vertical arrow is the map remembering the domain of the admissible cover together with the $n[g,\ell]$ markings and the

\begin{equation*}b[g,\ell] = 2 g - 2 + 2 d[g,\ell] = 4 g + 2 \ell\end{equation*}

simple ramification points. The bottom right vertical arrow is the map forgetting these extra markings, so the composition on the right is the familiar map $\epsilon_g$ . The disjoint union on the left-hand side of the diagram is over all stable graphs $\widehat{\Gamma}_0$ that can be obtained by distributing $b[g,\ell]$ legs to the two vertices of $\Gamma_0$ . The lower square in the diagram is then not quite Cartesian, but the disjoint union of the $\overline{\mathcal{M}}_{\widehat \Gamma_0}$ maps properly and birationally to the corresponding fibre product. This property will be sufficient for the intersection theoretic computations below (see [Reference Bae and Schmitt2, lemma C·7] for a formal argument).

The upper square is a fibre diagram on the level of sets, as proven in [Reference Lian18, proposition 3·2]. Here, the disjoint union is over boundary strata $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ of $\overline{\mathcal{H}}_{g,d[g,\ell],n[g,\ell],r}$ which, as we saw in Section 3·1, parametrise maps from a curve with stable graph $\Gamma$ mapping to a curve of stable graph $\Gamma'$ . The index set of the disjoint union consists of isomorphism classes of $\widehat{\Gamma}_0$ -structures on $\Gamma$ such that the induced map

\begin{equation*}E(\widehat{\Gamma}_0) \to E(\Gamma) \to E(\Gamma')\end{equation*}

is surjective. The lift of the upper fibre diagram to the level of schemes is described in [Reference Lian18, proposition 3·3].

Our strategy is now as follows. To compute

\begin{equation*}\tau_{g,\ell,r}^* [(C,D)] = \epsilon_g^* [C] \cdot \epsilon_0^* [D]\, ,\end{equation*}

we consider $[C] = (\xi_{\Gamma_0})_*[(C',R)]$ and use the diagram (13) to decompose $\epsilon_g^* [C]$ into terms supported on the spaces $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ . Only three types of contributions survive after taking the product with $\epsilon_0^* [D]$ . We begin by describing these three surviving cases and computing their contributions. Multiplicities and excess intersections appear.

Contribution 1. The Hurwitz cover degenerates as indicated in diagram (14) with the degrees of the map written in green.Footnote ¹³ The last r markings are in blue, and the last marking is starred.

(14)

Each of the two rational components mapping with degree 2 must carry two of the ramification points. The total number of loci of the form (14) is therefore given by

(15) \begin{equation} \frac{1}{2} \binom{b[d,\ell]}{2,2,b[d,\ell]-4} = \frac{1}{8} \frac{b[d,\ell]!}{(b[d,\ell]-4)!} \,,\end{equation}

with the factor $1/2$ corresponding to the fact that the two pairs of branch points going to the outer components can be exchanged without changing the locus.

For $r>1$ , the locus is parametrised by the space

(16) \begin{equation} \overline{\mathcal{H}}_{(\Gamma, \Gamma')} = \overline{\mathcal{H}}_{g-1,d[g,\ell]-1,n[g,\ell]-1+2,r-1} \times\underbrace{ \overline{\mathcal{H}}_{0,2,2,2} \times \overline{\mathcal{H}}_{0,2,2,2}}_{=\{\textrm{pt}\}}\,.\end{equation}

The unique point of $\overline{\mathcal{H}}_{0,2,2,2}$ is given by the map

\begin{equation*}(\mathbb{P}^1, 1, -1) \xrightarrow{z \mapsto z^2} \mathbb{P}^1\, ,\end{equation*}

and the cover has no remaining automorphisms.

On the other hand, for $r=1$ , the position of marking $p_{n[g,\ell]}$ is no longer determined by the map and markings on the genus $g-1$ component. If $r=1$ , let

\begin{equation*}\widehat{\epsilon}_0 \;:\; \overline{\mathcal{H}}_{g-1,d[g,\ell]-1,n[g,\ell]-1+2,r-1} \to\; \overline{\mathcal{M}}_{0,4g+2\ell-4+n[g,\ell]-1}\end{equation*}

be the map remembering both the $4g+2\ell-4$ branch points and the images of the $n[g,\ell]-1$ markings on the target of the cover. Then the locus fits into a diagram

(17)

where the right vertical arrow is the map forgetting the last marking.

Since, independent of r, the map (14) is unramified at all the nodes, we see from [Reference Lian18, proposition 3·3] that the locus $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ appears with multiplicity 1 in the upper fibre diagram (13). On the other hand, there are 8 different $\widehat{\Gamma}_0$ -structures on the graph $\Gamma$ of the domain curve in (14) that satisfy the assumptions of [Reference Lian18, proposition 3·2]. Indeed, the image of the first edge of $\widehat{\Gamma}_0$ can be chosen freely among the 4 nonseparating edges of $\Gamma$ , and then the second edge must go to one of the 2 edges on the other side. So overall, in the disjoint union in the top left of the diagram (13), there are

\begin{equation*}\frac{b[d,\ell]!}{(b[d,\ell]-4)!}\end{equation*}

many components that can contribute.

By the strategy explained above, we see that the contribution to the degree of $\tau_{g,\ell,r}^* [(C,D)]$ coming from each of the components is given by the degree of the map

(18) \begin{equation} \overline{\mathcal{H}}_{(\Gamma, \Gamma')} \to \underbrace{\overline{\mathcal{M}}_{g-1,n[g,\ell]+1}}_{=\overline{\mathcal{M}}_{\Gamma_0}} \times \overline{\mathcal{M}}_{0,n[g,\ell]-r+1}.\end{equation}

1. (i) For $r>1$ , the locus $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ is given by (16) and the above map fits as the top horizontal arrow into a commutative diagram

   
   mapping birationally onto the corresponding fibre product. So the degree of the top map is the same as the degree of the bottom map. Using
   \begin{equation*}d[g,\ell]-1 = d[g-1,\ell]\, , \ \ \ n[g,\ell]-1 = n[g-1,\ell]\, ,\end{equation*}
   the bottom map is precisely $\tau_{g-1,\ell,r-1}$ .

2. (ii) For $r=1$ , we must distinguish two more cases:

3. (a) For $n[g,\ell] \geq 4$ , the map (18) appears as the top horizontal arrow in the commutative diagram

   
   mapping birationally to the associated fibre product, and so the degree of (18) is given by the degree of the bottom map $\tau_{g-1,\ell,1}$ .

4. (b) For $n[g,\ell]=3$ , the map (18) factors through the left vertical map in (17). Since the vertical map drops dimension by 1, we know that the morphism (18) can no longer be dominant and thus yields a zero contribution. The vanishing here is compatible with our original definition, since $\mathsf{Tev}_{g-1,\ell,1}$ is defined to be zero since $n[g-1,\ell]=2$ .

Overall, we see that the total part of the degree arising from Contribution 1 is given by

\begin{equation*}\frac{b[d,\ell]!}{(b[d,\ell]-4)!} \deg \tau_{g-1,\ell,\max(1,r-1)} = b[d,\ell]! \cdot \mathsf{Tev}_{g-1,\ell,\max(1,r-1)}\,.\end{equation*}

Contribution 2. The second type of locus that can contribute parametrises covers of the form illustrated in diagram (19).

(19)

The rational component mapping with degree 2 must carry two of the ramification points. The total number of loci of the form (19) is given by

(20) \begin{equation} \binom{b[d,\ell]}{2} = \frac{1}{2} \frac{b[d,\ell]!}{(b[d,\ell]-2)!} \,.\end{equation}

The locus $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ is parametrised by the space

(21) \begin{equation} \overline{\mathcal{H}}_{g-1,d[g,\ell],n[g,\ell]+1,r+1} \times \overline{\mathcal{H}}_{0,2,3,2}\,.\end{equation}

The $r+1$ index in (21) is explained by the following basic important observation: all $r+1$ nodes of the genus $g-1$ domain component connecting to rational curves containing blue markings map to the same point (which is the node of the range curve).

The space $\overline{\mathcal{H}}_{0,2,3,2}$ , parametrises double covers

(22) \begin{equation} f \;:\; (\mathbb{P}^1, p, z, z', q_1, q_2) \to \mathbb{P}^1\end{equation}

such that $f(z)=f(z')$ and such that $q_1, q_2$ are ramification points of the cover. The natural map

\begin{equation*}\delta \;:\; \overline{\mathcal{H}}_{0,2,3,2} \to \overline{\mathcal{M}}_{0,4}\end{equation*}

sending the point (22) to

\begin{equation*}(\mathbb{P}^1,\; f(p),\; f(z),\; f(q_1),\; f(q_2)) \in \overline{\mathcal{M}}_{0,4}\end{equation*}

has degree 2, corresponding to the two choices z, z ^′ of preimage of f(z).

By the same argument as before, the loci (19) appear with multiplicity 1 in the upper fibre diagram (13). On the other hand, there are 2 $\widehat{\Gamma}_0$ -structures on the graph $\Gamma$ of the domain curve in (19), corresponding to the two choices of sending the edges of $\widehat{\Gamma}_0$ to the two nonseparating edges e, e ^′ of $\Gamma$ . However, these two $\widehat{\Gamma}_0$ -structures are isomorphic (since the cover $\Gamma \to \Gamma'$ has an automorphism exchanging the two edges e, e ^′). So overall, in the disjoint union in the top left of the diagram (13) there are

\begin{equation*}\frac{1}{2} \cdot \frac{b[d,\ell]!}{(b[d,\ell]-2)!}\end{equation*}

many components that can contribute.

However, a new phenomenon (not seen in Contribution 1) appears here: a dimension count shows that the loci (21) only have codimension 1 in their ambient space, whereas the gluing map $\xi_{\Gamma_0}$ had codimension 2. The excess class is given by

\begin{equation*}- \psi_{w} \otimes 1 - 1 \otimes \psi_{z},\end{equation*}

where w, z are the preimages of one of the nonseparating nodes in (19), see [Reference Schmitt and van Zelm25, section 4·2] and [Reference Lian18, proposition 3·3]. The class $\psi_{z}$ on $\overline{\mathcal{H}}_{0,2,3,2}$ is the pullback of $\psi_2$ on $\overline{\mathcal{M}}_{0,4}$ under the map $\delta$ above. We calculate

\begin{equation*}\int_{\overline{\mathcal{H}}_{0,2,3,2}} \psi_{z} = \deg(\delta) \cdot \int_{\overline{\mathcal{M}}_{0,4}} \psi_2 = 2 \cdot 1 = 2\,.\end{equation*}

The contribution to the degree of $\tau_{g,\ell,r}^* [(C,D)]$ coming from each locus above is given by the degree of the excess class capped with the preimage of a point under the map

\begin{equation*}\overline{\mathcal{H}}_{g-1,d[g,\ell],n[g,\ell]+1,r+1} \times \overline{\mathcal{H}}_{0,2,3,2} \rightarrow \overline{\mathcal{M}}_{g-1,n[g,\ell]+1} \times \overline{\mathcal{M}}_{0,n[g,\ell]-r+1}\, .\end{equation*}

Since the above map does not depend on the factor $\overline{\mathcal{H}}_{0,2,3,2}$ in the domain, the only nonzero contribution can come from the term $-1 \otimes \psi_{z}$ . After eliminating the factor $\overline{\mathcal{H}}_{0,2,3,2}$ by integrating $-1 \otimes \psi_{z}$ (and obtaining a factor $-2$ from the degree of $-\psi_z$ ), we are left with the degree of $\tau_{g-1,\ell+1,r+1}$ .

The total part of the degree arising from Contribution 2 is given by

\begin{equation*} \frac{1}{2} \cdot \frac{b[d,\ell]!}{(b[d,\ell]-2)!} \cdot (-2) \cdot \deg \tau_{g-1,\ell+1,r+1} = - b[d,\ell]! \cdot \mathsf{Tev}_{g-1,\ell+1,r+1}\,.\end{equation*}

Contribution 3. The third type of locus that can contribute parametrises covers of the form illustrated in diagram (23).

(23)

The rational component mapping with degree 2 must carry two of the ramification points. The total number of loci of the form (19) is given by

(24) \begin{equation} \binom{b[d,\ell]}{2} = \frac{1}{2} \frac{b[d,\ell]!}{(b[d,\ell]-2)!} \,.\end{equation}

The locus $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ is parametrised by the space

(25) \begin{equation} \overline{\mathcal{H}}_{g-1,d[g,\ell],n[g,\ell]+1,r+1} \times \underbrace{\overline{\mathcal{H}}_{0,2,2,2}}_{=\{\textrm{pt}\}}\,.\end{equation}

Again the multiplicity is 1, but there are 4 non-isomorphic $\widehat{\Gamma}_0$ -structures on the graph $\Gamma$ : one of the two edges of $\widehat{\Gamma}_0$ must go to the edge of $\Gamma$ connecting the genus $g-1$ vertex to the vertex containing the last marking. The other can go to either one of the two edges incident to the vertex with the degree 2 map. So overall, in the disjoint union in the top left of the diagram (13) there are

\begin{equation*} 2 \frac{b[d,\ell]!}{(b[d,\ell]-2)!}\end{equation*}

components that can contribute.

By the strategy explained above, the contribution to the degree of $\tau_{g,\ell,r}^* [(C,D)]$ coming from each locus above is given by the degree of the map

\begin{equation*}\overline{\mathcal{H}}_{g-1,d[g,\ell],n[g,\ell]+1,r+1} \to \overline{\mathcal{M}}_{g-1,n[g,\ell]+1} \times \overline{\mathcal{M}}_{0,n[g,\ell]-r+1}.\end{equation*}

The total part of the degree arising from Contribution 3 is given by

\begin{equation*} \frac{1}{2} \cdot 2 \cdot \frac{b[d,\ell]!}{(b[d,\ell]-2)!} \cdot 2 \cdot \deg \tau_{g-1,\ell+1,r+1} = 2 b[d,\ell]! \cdot \mathsf{Tev}_{g-1,\ell+1,r+1}\,.\end{equation*}

Exclusion of other components.

To conclude, we must prove that Contributions 1, 2 and 3 are the only loci in the diagram (13) that can contribute.

Let $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ be a locus in (13) dominating the moduli space

\begin{equation*}\overline{\mathcal{M}}_{g-1,n[g,\ell]+1} \times \overline{\mathcal{M}}_{0,n[g,\ell]-r+1}\,.\end{equation*}

There must then be a vertex $v_{g-1} \in V(\Gamma)$ such that the corresponding factor of $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ parametrises covers from a genus $g-1$ curve. Since the $n[g,\ell]-1$ first markings will go to the vertex $v_{g-1}$ after forgetting the ramification points and stabilising, their images in $\Gamma'$ will similarly stabilise to a unique vertex $v_0$ . Therefore, in the cover $\Gamma \to \Gamma'$ , the vertex $v_{g-1}$ maps to $v_0$ . Next, we observe that the map

(26) \begin{equation} \overline{\mathcal{H}}_{(\Gamma, \Gamma')} \to \overline{\mathcal{M}}_{g-1,n[g,\ell]+1} \times \overline{\mathcal{M}}_{0,n[g,\ell]-r+1}\end{equation}

only depends upon the factors of $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ lying over the vertex $v_0$ . Indeed, the map to $\overline{\mathcal{M}}_{g-1,n[g,\ell]+1}$ only depends on the factor for $v_{g-1}$ . On the other hand, since the stabilisation of $\Gamma'$ , after forgetting the branch points, consists of the single vertex $v_0$ , the factors of $\overline{\mathcal{H}}_{(\Gamma, \Gamma')}$ over $v_0$ determine the component of the map (26) to $\overline{\mathcal{M}}_{0,n[g,\ell]-r+1}$ .

Since the factors over $v_0$ form a finite cover of the moduli space $\overline{\mathcal{M}}_{0,n(v_0)}$ associated to $v_0$ , the valence $n(v_0)$ must satisfy

\begin{equation*}n(v_0) - 3 \geq \dim\ \overline{\mathcal{M}}_{g-1,n[g,\ell]+1} \times \overline{\mathcal{M}}_{0,n[g,\ell]-r+1} = b[g,\ell] + n[g,\ell] -r -4\end{equation*}

in order for the map (26) to be dominant.

A short computation shows that the valence condition is only possible in one of the following cases:

1. (i) $\Gamma'$ has two other vertices $w_1, w_2$ , both of valence 3;

2. (ii) $\Gamma'$ has one other vertex w, of valence 4;

3. (iii) $\Gamma'$ has one other vertex w, of valence 3.

We further observe that for a simple branch point at the vertex $v_0$ whose ramification point is not on the component associated to $v_{g-1}$ , the position of the branch point does not affect the image under the map (26). Therefore, in cases (i) and (ii) all the simple ramification points over $v_0$ are located at the vertex $v_{g-1}$ , whereas in case (iii) at most one of them can be on a separate component.

Finally, we observe that for every leaf of the graph $\Gamma'$ which is not equal to $v_0$ (so one of the vertices $w,w_1,w_2$ above), the leaf must contain at least two of the simple branch points. Indeed, by the condition that there is a $\widehat{\Gamma}_0$ structure on $\Gamma \to \Gamma'$ , there must be a circular path from $v_{g-1}$ to itself in $\Gamma$ whose edges surject to the edges of $\Gamma'$ . Thus over a leaf there must be a vertex with at least two edges adjacent, which needs at least two ramification points on that vertex.

Case (i). There are two possibilities $\Gamma'_{\!\!1}$ and $\Gamma'_{\!\!2}$ for the graph $\Gamma'$ .

By the general comments above, the two markings at $w_2$ must be branch points, and we see that there are at least $b[g,\ell]-3$ simple ramification points at $v_{g-1}$ . The configuration is only possible if the degree at the vertex $v_{g-1}$ remains at $d[g,\ell]$ , otherwise the number of ramification points would drop by at least 4. The marking at $w_1$ is then forced to be the image of the marking $p_{n[g,\ell]}$ , since otherwise this marking would be contained in the genus $g-1$ component of the curve, contradicting the shape (12) we have fixed before. We conclude that only the pair $(\Gamma, \Gamma'_{\!\!1})$ of Contribution 3 has the specified shape.

The markings at $w_1, w_2$ belonging to branch points, so we have precisely $b[g,\ell]-4$ branch points at $v_0$ . Then the component $v_{g-1}$ must have degree $d[g,\ell]-1$ , so that there is an additional vertex of genus 0 over $v_0$ mapping with degree 1. We are then forced to be in the case of Contribution 1.

Case (ii). The shape of the graph $\Gamma'$ is as follows:

Since there are at least $b[g,\ell]-3$ simple branch points at $v_0$ , the component $v_{g-1}$ must map with full degree $d[g,\ell]$ . Therefore, one of the three markings at w is forced to be the image of marking $p_{n[g,\ell]}$ . Moreover, the point $p_{n[g,\ell]}$ must lie on the unique loop in $\Gamma$ which goes through the vertex over w containing the two ramification points. We are in the case of Contribution 2.

Case (iii). The graph $\Gamma'$ has the following shape:

The markings at w are branch points, so the remaining $b[g,\ell]-2$ branch points are at $v_0$ . As we have seen before, in case (iii), at most one of the associated ramification points can be on a component not equal to $v_{g-1}$ , which still forces $b[g,\ell]-3$ ramification points on $v_{g-1}$ . The component $v_{g-1}$ must map with full degree $d[g,\ell]$ . Hence we arrive at a contradiction, since the marking $p_{n[g,\ell]}$ lies over $v_0$ and thus in $v_{g-1}$ . The curve can not have the shape (12).

Contributions 1,2 and 3 are therefore the only nonvanishing terms in the intersection. After summing the three contributions to the degree of $\tau_{g,\ell,r}$ and dividing by $b[d,\ell]!$ , we obtain the recursion stated in the Proposition.

4·1. The genus 0 case

The recursion of Proposition 7 reduces the calculation of the degree of $\tau_{g,\ell,r}$ to the genus $g=0$ case. Indeed, for $n[g,\ell]-r+1 \geq 3$ , we can apply the recursion and lower the genus and, for $n[g,\ell]-r+1 < 3$ , $\mathsf{Tev}_{g,\ell,r}$ is defined to be 0. In genus 0, since

\begin{equation*}1 \leq r \leq d[0,\ell] = \ell+1\, \end{equation*}

by (3), we must have $\ell\geq 0$ . The Tevelev degrees in genus 0 for $\ell\geq 0$ are determined by the following result.

Proposition 8. For $\ell \geq 0$ and $1\leq r \leq \ell +1$ we have

\begin{equation*} \mathsf{Tev}_{0,\ell,r}=1.\end{equation*}

Proof. Let $(\mathbb{P}^1, p_1,\ldots,p_n)$ be a genus 0 curve with n distinct markings, for

\begin{equation*}n=n[0,\ell]=3+2\ell \geq 3\, .\end{equation*}

Let $\mathcal{M}_{d,r}$ be the moduli space of maps

\begin{equation*}\pi \;:\; (\mathbb{P}^1,p_1,\ldots,p_n) \to \mathbb{P}^1\end{equation*}

of degree $d[0,\ell]=1+\ell$ satisfying

\begin{equation*}\pi(p_{n-r+1}) = \cdots = \pi(p_n)= [1,0]\in \mathbb{P}^1\, .\end{equation*}

The moduli space $\mathcal{M}_{d,r}$ is the nonsingular $n-r$ dimensional quasi-projective subvariety

\begin{equation*}\mathcal{M}_{d,r} \subset\mathbb{P}(H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(d)) \oplus H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(d)))\ \end{equation*}

defined by pairs of sections $(s_0,s_1)$ of $H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(d))$ satisfying:

1. (i) $s_0$ and $s_1$ have no common zeros;

2. (ii) $s_1$ vanishes at the r points $p_{n-r+1}$ ,…, $p_n$ .

Conditions (ii) are linear, so $\mathcal{M}_{d,r}\subset \mathbb{P}^{2d+1-r}= \mathbb{P}^{n-r}$ .

The tangent space of $\mathcal{M}_{d,r}$ at $[\pi]\in \mathcal{M}_{d,r}$ is

\begin{equation*} \mathsf{Tan}_\pi =H^0(\mathbb{P}^1, \pi^*(\mathsf{Tan}_{\mathbb{P}^1})(-p_{n-r+1}-\cdots-p_n))\, .\end{equation*}

The degree of the line bundle $\pi^*(\mathsf{Tan}_{\mathbb{P}^1})(-p_{n-r+1}-\ldots-p_n)$ is

\begin{equation*}2(1+\ell) - r =n-r-1\, .\end{equation*}

Since the points $p_1,\ldots,p_{n-r}$ impose independent conditions on sections of $\pi^*(\mathsf{Tan}_{\mathbb{P}^1})(-p_{n-r+1}-\cdots-p_n)$ , the differential of the evaluation map

\begin{equation*}\text{ev}\;:\;\mathcal{M}_{d,r} \to (\mathbb{P}^1)^{n-r}\, , \ \ \ \ [\pi]\longmapsto (\pi(p_1),\cdots,\pi(p_{n-r}))\end{equation*}

is surjective.

Let $x_1\,\ldots,x_{n-r}\in \mathbb{P}^1$ be general points. For $1\leq i \leq n-r$ , let

\begin{equation*}\mathsf{H}_i\subset \mathcal{M}_{d,r}\end{equation*}

be the locus of maps $\pi$ satisfying

\begin{equation*}\pi(p_i)=x_i\, .\end{equation*}

The closure $\overline{\mathsf{H}}_i\subset\mathbb{P}^{2d+n-r}$ is a linear space of codimension 1. Since

(27) \begin{equation}\mathsf{ev}^{-1}(x_1,\ldots,x_{n-r})\subset \mathcal{M}_{d,r}\end{equation}

is both transverse and linear, $\mathsf{ev}^{-1}(x_1,\ldots,x_{n-r})$ is either empty or a reduced point. Since the image of ev is dense, $\mathsf{ev}^{-1}(x_1,\ldots,x_{n-r})$ must be a reduced point.Footnote ¹⁴

Finally, a dimension count shows the single reduced point (27) corresponds to a map with only simple ramification points. Therefore, $\mathsf{Tev}_{0,\ell,r}=1$ .

4·2. Induction argument

We are ready to completely calculate the Tevelev degrees in case $\ell \geq 0$ . Because of the form of the recursion (11), we consider all the cases with $\ell \geq 0$ at once instead of proving Theorems 1, 2 and 3 separately.

Theorem. For $g \geq 0$ , $ \ell \geq 0$ and $1 \leq r \leq \ell +1 +g$ , we have

\begin{equation*}\mathsf{Tev}_{g,l,r}=\begin{cases}2^g- \sum_{j=0}^{r-2} \left(\begin{matrix} {g} \\ {j} \end{matrix}\right) &\text{for } \ell =0,\\ \mathsf{Tev}_{g,0,\max(r-\ell,1)} & \text{for } \ell \leq r,\\ 2^g & \text{for } \ell\geq r.\end{cases}\end{equation*}

Proof. We proceed by induction on g. The $g=0$ case is covered by Proposition 8. Suppose $g > 0$ .

Step 1. We consider the case $ \ell \geq r \geq 1$ . If we start with $ \ell \geq r$ , the same inequality persists for the recursion, and we have

\begin{equation*} \mathsf{Tev}_{g,\ell,r}= \mathsf{Tev}_{g-1,\ell,\max(1,r-1)}+ \mathsf{Tev}_{g-1,\ell+1,r+1}= 2^{g-1}+ 2^{g-1}=2^g\,. \end{equation*}

Step 2. We consider case $0\leq \ell \leq r$ . We have already checked the case $\ell =r$ . Suppose $ \ell < r$ . If we start with $ \ell < r$ , then the non-strict inequality $ \ell \leq r$ persists in the first step of the recursion. We have

\begin{align*}\mathsf{Tev}_{g,\ell,r}& =\mathsf{Tev}_{g-1,\ell ,\max(r - 1,1)} + \mathsf{Tev}_{g-1,\ell+1,r+1}\\& = 2^{g-1}- \sum_{j=0}^{ \max(r-3-\ell,-1 - \ell) } \left(\begin{matrix} { g-1} \\ {j} \end{matrix}\right) + 2^{g-1} - \sum_{j=0}^{ r-2-\ell } \left(\begin{matrix}{ g-1} \\ {j} \end{matrix}\right) \\& = 2^g - \sum_{j=0}^{ r-2-\ell } \Bigg[ \left(\begin{matrix} { g-1} \\ {j-1} \end{matrix}\right) + \left(\begin{matrix} {g-1} \\ {j} \end{matrix} \right) \Bigg]\\ &= 2^g - \sum_{j=0}^{ r-2-\ell } \left( \begin{matrix} { g} \\ {j} \end{matrix}\right) \ , \end{align*}

where in the first equality we are using the inductive step applied to g.

The proof is complete.

4·3. Lines in projective space

By the formulas of Theorems 2 and 3, we have

(28) \begin{equation}\mathsf{Tev}_{g,\ell,g+1+\ell}=1\, .\end{equation}

The Tevelev degree is defined via the map

\begin{equation*}\tau_{g,\ell,g+1+\ell} \;:\; \overline{\mathcal{H}}_{g,g+1+\ell,g+3+2\ell,g+1+\ell} \to\overline{\mathcal{M}}_{g,g+3+2\ell}\times\overline{\mathcal{M}}_{0,3+\ell}\, .\end{equation*}

There is a simple geometric reason for the Tevelev degree to be 1 here.

Let $(C,p_1,\ldots,p_{g+3+2\ell})$ be a general nonsingular pointed curve. The last $g+1+\ell$ markings (associated to r) determine a line bundle L on C. Consider the complete linear series $\mathbb{P}(H^0(C,L))$ of dimension $1+\ell$ by Riemann–Roch.

1. (i) There is a distinguished element

   \begin{equation*}\zeta =[p_{3+\ell}+ \cdots +p_{g+3+2\ell}]\in \mathbb{P}(H^0(C,L))\, .\end{equation*}

2. (ii) There are $2+\ell$ distinguished hyperplanes

   \begin{equation*}\mathsf{H}_1,\ldots, \mathsf{H}_{2+\ell} \subset \mathbb{P}(H^0(C,L))\end{equation*}
   determined by the points $p_1,\ldots,p_{2+\ell} \in C$ .

3. (iii) There is a projective space $\mathbb{P}(\mathsf{Tan}_\zeta)$ of dimension $\ell$ parametrising lines

   \begin{equation*}\mathsf{P}\subset \mathbb{P}(H^0(C,L))\end{equation*}
   passing through $\zeta$ .

For a general such line $\mathsf{P}$ , we obtain a $3+\ell$ pointed curve of genus 0,

\begin{equation*}(\mathsf{P}, \zeta, \mathsf{P}\cap \mathsf{H}_1,\ldots, \mathsf{P}\cap\mathsf{H}_{2+\ell})\, .\end{equation*}

The associated rational map

(29) \begin{equation}\mathbb{P}(\mathsf{Tan}_\zeta)\dashrightarrow\mathcal{M}_{0,3+\ell}\end{equation}

is easily seen to be birational (of degree 1). The degree is $\mathsf{Tev}_{g,\ell,g+1+\ell}$ .

The map (29) has the flavour of a Gale transformation. In the $\ell=1$ case, a direct connection to the Gale transformation can be made. Whether the birationality of (29) can be explained for all $\ell$ in terms of the Gale transformation is a question left to the reader.

5·1. Notation

We consider here the case $\ell \leq 0$ and prove Theorem 4. Before presenting the proof, we briefly recall the notation of Section 1·4.

For $\ell \leq 0$ and $r \geq 1 $ , we have

\begin{equation*}n[g, \ell]=3+g+2 \ell\, .\end{equation*}

After imposing $n[g, \ell]-(r-1) \geq 3$ , we obtain the condition

\begin{equation*}g \geq r-2 \ell-1\, .\end{equation*}

Following the notation of Section 1·4,

\begin{equation*}g[ \ell, r]= r - 2 \ell -1\, . \end{equation*}

We also have

\begin{equation*}d[g[\ell,r],\ell]=g[\ell,r]+1+\ell =r-\ell>0\,.\end{equation*}

Define the infinite vectors $\mathsf{T}_{\ell, r}$ and $\mathsf{E}_r$ by

\begin{equation*}\mathsf{T}_{\ell,r}[j]= \mathsf{Tev}_{g[\ell,r]+j, \ell,r}\ \ \ \text{and} \ \ \ \mathsf{E}_{r}[j] = 2^{r+j-1} - \sum_{i=0}^{r-2}\binom{r+j-1}{i}\,\end{equation*}

for $j \geq 0$ . We set

\begin{equation*}\mathsf{T}_{\ell,r}[j]= 0\ \ \ \text{and} \ \ \ \mathsf{E}_{r}[j]=0\end{equation*}

for $j < 0$ . For all $j \in \mathbb{Z}$ and $r \geq 1$ , we have

(30) \begin{equation} \mathsf{E}_{r+1}[j+1] = \mathsf{E}_{r+1}[j] +\mathsf{E}_{r}[j+1]\, .\end{equation}

5·2. Proof of Theorem 4 for $\ell\leq 0$

By Theorems 2 and 3 for $\ell \geq 0$ , we already know

\begin{equation*}\mathsf{T}_{0,r}= \mathsf{E}_{r} =\sum_{\gamma\in \mathsf{P}(0,r)}\mathsf{E}_{\mathsf{Ind}(\gamma)} \end{equation*}

holds.

We prove the formula of Theorem 4 for the coefficient

\begin{equation*}\mathsf{T}_{\ell,r}[j]\ \ \text{for}\ j\geq 0\end{equation*}

by induction on $g=g[\ell,r]+j$ . When $g[\ell,r]+j=0$ , we must have $(\ell,r)=(0,1)$ and $j=0$ . So the base of the induction is established.

Suppose now $g[\ell,r]+j>0$ . The proof of the induction step is split into two cases:

Case 1. Suppose $r=1$ and $ \ell \leq 0$ .

1. (i) If $j=0$ , we have

   \begin{align*}\mathsf{T}_{\ell,1}[0]&= \mathsf{Tev}_{g[\ell,1],\ell,1}\\ &= \mathsf{Tev}_{g[\ell,1]-1,\ell+1,2} \\&= \mathsf{T}_{\ell+1,2}[0] \\&=\sum_{\gamma\in \mathsf{P}(\ell +1 ,2)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[0] \\&=\sum_{\gamma\in \mathsf{P}(\ell ,1)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[0]\, .\end{align*}
   The second equality follows from (11) since
   \begin{equation*}\mathsf{Tev}_{g[\ell,1]-1,\ell,1}=\mathsf{T}_{\ell,1}[-1]=0\, .\end{equation*}
   The last equality follows from $\mathsf{E}_{\mathsf{Ind}(\gamma)}[0]=1$ for all $\gamma\in \mathsf{P}(\ell ,1) \cup \mathsf{P}(\ell +1 ,2)$ and the fact that $\mathsf{P}(\ell +1 ,2)$ and $\mathsf{P}(\ell ,1)$ are in bijection.

2. (ii) If $j > 0$ , we have

   \begin{align*} \mathsf{T}_{\ell,1}[j]&= \mathsf{Tev}_{g[\ell,1]+j,\ell,1}\\ & =\mathsf{Tev}_{g[\ell,1]+j-1,\ell,1}+\mathsf{Tev}_{g[\ell,1]+j-1,\ell+1,2}\\&=\mathsf{T}_{\ell,1}[j-1]+\mathsf{T}_{\ell+1,2}[j] \\& =\sum_{\gamma\in \mathsf{P}(\ell,1)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[j-1]+ \sum_{\gamma\in \mathsf{P}(\ell +1 ,2)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[j] \\& =\sum_{\gamma\in \mathsf{P}(\ell,1)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[j-1]+ \sum_{\gamma\in \mathsf{P}(\ell ,1)}\mathsf{E}_{\mathsf{Ind}(\gamma)-1}[j] \\& = \sum_{\gamma\in \mathsf{P}(\ell,1)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[j]\, .\end{align*}
   In the fifth equality, we use the natural bijection between $\mathsf{P}(\ell +1 ,2)$ and $\mathsf{P}(\ell ,1)$ given by completing a path that ends in $(\ell +1 ,2)$ to a path that ends $(\ell ,1)$ by stepping with $\mathsf{D}$ . In the last equality, we use (30).

Case 2. Suppose $r > 1$ and $\ell\leq 0$ .

For all $j \geq 0$ , we have a similar chain of equalities:

\begin{align*}\mathsf{T}_{\ell,r}[j]&= \mathsf{Tev}_{g[\ell,r]+j,\ell,r}\\ & =\mathsf{Tev}_{g[\ell,r]+j-1,\ell,r-1}+\mathsf{Tev}_{g[\ell,r]+j-1,\ell+1,r+1} \\&=\mathsf{T}_{\ell,r-1}[j]+\mathsf{T}_{\ell+1,r+1}[j]\\& =\sum_{\gamma\in \mathsf{P}(\ell,r-1)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[j]+ \sum_{\gamma\in \mathsf{P}(\ell +1 ,r+1)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[j]\\& = \sum_{\gamma\in \mathsf{P}(\ell,r)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[j]\, ,\end{align*}

which completes the induction step.

5·3. Refined Dyck path counting

For $\ell \leq 0$ and $r \geq 1$ , we can write

(31) \begin{equation}\mathsf{T}_{\ell,r}= \sum_{s=1}^{r- \ell +1} c_{\ell,r}^s \mathsf{E}_s\end{equation}

by Theorem 4 and Corollary 5. By definition,

\begin{equation*}\mathsf{E}_1[j]=2^j\, .\end{equation*}

For $s\geq 2$ , define the vectors $\mathsf{E}'_{\!\!s}$ by

\begin{equation*}\mathsf{E}'_{\!\!s} [j]= \mathsf{E}_s[j] - 2^{s-1} \mathsf{E}_1[j]\, .\end{equation*}

Then, $\mathsf{E}'_{\!\!s}[j]$ a polynomial in j of degree $s-2$ . The vectors $\mathsf{E}_s$ are therefore independent, and the coefficients $c^s_{\ell,r}$ in (31) are unique.

Table 1. Values of $\mathsf{T}_{\ell,r}$ for $\ell<0$

By Theorem 4, the coefficient $c^s_{\ell,r}$ is the number of paths $\gamma\in \mathsf{P}(\ell,r)$ of index s and therefore defines a refined Dyck path counting problem. The solution is given by the following result.

Before starting the proof, we require additional notation and an auxiliary result. For $(u,v),(u',v') \in \mathcal{A}$ and $k \in \mathbb{Z}$ , let

\begin{equation*}\mathcal{D}(k\;;\; u,v,u',v')\end{equation*}

be the set of paths $\gamma$ in $\mathcal{A}$ from (u,v) to (u ^′, v ^′) which take steps U and D and meet the $r=1$ axis

\begin{equation*}\{ \, (\ell,1)\, | \, \ell\leq 0\, \}\subset \mathcal{A}\end{equation*}

exactly k times. Let $d(k\;;\; u,v,u',v')= |\mathcal{D}(k\;;\; u,v,u',v')|$ .

Proof. Identity (i) is trivial. Identities (ii)-(iv) have been proven, for example, in [Reference Scott26] by B. Scott. We sketch here his proof adapted to our notation.

Identity (ii). Every Dyck path in $\mathcal{A}$ from (0,1) to (u ^′, v ^′) can be written as a sequence of $-u'$ letters D and $- u'+v'-1$ letters U. For any such sequence, changing every D with an U and every U with a D yields a Dyck path in $\mathcal{A}$ from (0,v ^′) and $(u'-v'+1,1)$ . Moreover, the number of intersections with the $r=1$ axis of the old and the new paths are the same. Therefore, we have bijection between $\mathcal{D}(k;\;0,1,u',v')$ and $\mathcal{D}(k;\;0,v',u'-v'+1,1)$ .

Identity (iii). Let $\gamma \in \mathcal{D}(k;\;u,v,u',v')$ . For $1\leq i \leq t$ , let $(\ell_i,1)$ be the first t points at which $\gamma$ meets the $r=1$ axis. Here,

\begin{equation*}\ell_1 < \cdots < \ell_t\, .\end{equation*}

Each of the points $(\ell_i,1)$ is immediately followed by a step U in $\gamma$ . Removing these t steps U and translating $\gamma$ upwards t units yields a path in $\mathcal{D}(k-t;\;u,v+t,u',v')$ . The operation is a bijection.

Identity (iv). Using Identity (iii), we have

\begin{align*}d(k;\;0,1,u',1)&=d(1;\;0,k,u',1) \\&= \binom {-2 u'-k}{-u'-k+1} - \binom{-2 u' -k}{-u'} \\& = \binom{-2 u'-k}{-u'-1} \frac{k-1}{-u'}\, .\end{align*}

The second equality is by directly counting the paths using a standard reflection trick (see [Reference Comtet8, p.22]).

Proof of Proposition 9. For $\ell=0,-1$ the result is trivial, so we assume $\ell < -2$ . We write

\begin{align*}c_{\ell,r}^s& = | \{\gamma\in \mathsf{P}(\ell,r) \;:\; \mathsf{Ind}(\gamma)=s\}| \\& = \sum_{h=1}^{s-1} | \{\gamma\in \mathsf{P}(\ell,r) \;:\; | \gamma \cap \{ \ell=0\}|=h+1 \text{ and } | \gamma \cap \{ r=1\}|=s-h\}| \\&= \sum_{h=1}^{s-1} d(s-h-1;-1,h,\ell,r).\end{align*}

By Lemma 10, for $ h \in \{1,..., s-2 \}$ , the following chain of equalities holds:

\begin{align*}d(s-h-1;-1,h,\ell,r)&= d(s-2;\;-1,1,\ell,r) \\&=d(s-2;\;0,1,\ell+1,r) \\&=d(s-2;\;0,r,\ell+2-r,1) \\&=d(s+r-3;\;0,1,\ell+2-r,1) \\& =\binom{|2 \ell| +r-s-1}{|\ell|-s+2} \frac{s+r-4}{|\ell|-2+r}\, .\end{align*}

For $h=s-1$ , a simple direct count yields

\begin{align*}d(0;-1,s-1,\ell,r) &=d(0;\;0,s-1,\ell+1,r) \\&=\binom{|2 \ell| +r-s-1}{|\ell|-1} - \binom{|2 \ell| +r-s-1}{|\ell|-2+r}\end{align*}

which is the number of allowed paths from $(0,s-2)$ to $(\ell+1,r-1)$ .

The result follows by summing all the terms.

5·4. Fixed j

An interesting direction is to compute $\mathsf{T}_{\ell,r}[j]$ for fixed j in terms of $\ell$ and r. The simplest example is Castelnuovo’s formula for $\ell<0$ :

\begin{equation*}\mathsf{T}_{\ell,1}[0]= |\mathsf{P}(\ell,1)| = \frac{1}{|\ell|+1}\binom{|2\ell|}{|\ell|}\,.\end{equation*}

Proposition 11. For $ \ell <0$ , we have

\begin{align*}& \mathsf{T}_{\ell,1}[1]= \\ & \quad |\mathsf{P}(\ell,1)| + \frac{3}{ 2 | \ell | +1 } \left(\begin{matrix} {{2 |\ell| + 1}} \\ {{|\ell | -1}} \end{matrix}\right) + \frac{4 (2 |\ell| -1)(2 | \ell | + 1) \left(\left(\begin{matrix}{{2 |\ell| - 2}} \\ {{|\ell | -1}} \end{matrix}\right) - \left(\begin{matrix}{{2 |\ell| - 2}} \\ {{|\ell |}} \end{matrix} \right)\right)}{(|\ell| +1)(| \ell |+2)} \, .\end{align*}

Proof. The case $l =-1 $ can be checked by hand, so we assume $ l < -1$ . Using Theorem 4, we have

\begin{equation*}\mathsf{T}_{\ell,1}[1]= \sum_{\gamma\in \mathsf{P}(\ell,1)}\mathsf{E}_{\mathsf{Ind}(\gamma)}[1]= |\mathsf{P}(\ell,1)| + \sum_{\gamma\in \mathsf{P}(\ell,1)} \mathsf{Ind}(\gamma)\, .\end{equation*}

For $ \gamma \in \mathsf{P}(\ell,1) $ , define a return step of $\gamma$ to be a step $\mathsf{D}=(-1,-1)$ of $\gamma$ ending in the subset of $\mathcal{A}$ with $r=1$ . Let $ r(\gamma)$ to be the number of return steps in $\gamma$ . Then,

(32) \begin{equation} \mathsf{Ind}( \gamma) = r( \gamma) + | \gamma \cap \{ \ell =0 \} |\, .\end{equation}

By [Reference Deutsch10, section 6·6] the first term of (32) is

\begin{equation*}\sum_{\gamma\in \mathsf{P}(\ell,1)} r ( \gamma) = \frac{3}{ 2 | \ell | +1 } \left(\begin{matrix} {{2 |\ell| + 1}} \\ {{|\ell | -1}} \end{matrix}\right)\, .\end{equation*}

The second term of (32) is

\begin{align*}\sum_{\gamma\in \mathsf{P}(\ell,1)} & | \gamma \cap \{ \ell = 0 \}| \\& = \sum_{k=2}^{|\ell| + 1 } k \Bigg[ \left(\begin{matrix} {{2 |\ell| - (k-1)}} \\ {{|\ell |}} \end{matrix}\right) - \left(\begin{matrix} {{2 |\ell| -(k-1)}} \\ {{|\ell | +1}} \end{matrix}\right) - \Bigg( \left(\begin{matrix} {{2 |\ell| - k}} \\ {{|\ell | }} \end{matrix}\right) - \left(\begin{matrix} {{2 |\ell| -k}} \\ {{|\ell | +1}} \end{matrix}\right) \Bigg) \Bigg] \\& = \sum_{k=2}^{|\ell| + 1 } k \Bigg[ \left(\begin{matrix} {{2 |\ell| - k}} \\ {{|\ell |-1 }} \end{matrix}\right) - \left(\begin{matrix} {{2 |\ell| - k}} \\ {{|\ell | }} \end{matrix}\right) \Bigg] \\& = \frac{4 (2 |\ell| -1)(2 | \ell | + 1) \left(\left(\begin{matrix}{{2 |\ell| - 2}} \\ {{|\ell | -1}} \end{matrix}\right) - \left(\begin{matrix} {{2 |\ell| - 2}} \\ {{|\ell |}} \end{matrix} \right) \right)}{(|\ell| +1)(| \ell |+2)}\, .\end{align*}

After summing the terms, we obtain the result.