2 Reduction to genus zero

In this section, we reduce the enumerative problem to genus zero via a standard limit linear series degeneration, see for example [Reference Lian12, Reference Osserman18]. We begin by recalling some notation, while assuming throughout some familiarity with basics of the theory of limit linear series [Reference Eisenbud and Harris7].

We recall the usual notation for Schubert cycles in the Grassmannian $\operatorname {\mathrm {Gr}}(r+1,d+1):=\operatorname {\mathrm {Gr}}(r+1,V)$ , where V is a $(d+1)$ -dimensional vector space, following [Reference Eisenbud and Harris9, §4]. For a nonincreasing sequence $\mu :=(\mu _0\geq \mu _1\geq \cdots \geq \mu _r)$ and a flag ${\textbf F}: V=V_{d+1}\supset V_d\supset \ldots \supset V_1\supset V_0=0$ , we introduce the Schubert cycle:

$$ \begin{align*} \Sigma_{\mu}=\Sigma_{\mu}({\textbf F}):=\Bigl\{\Lambda \in \operatorname{\mathrm{Gr}}(r+1,d+1): \mbox{dim } \bigl(\Lambda \cap V_{d-r+1+i-\mu_i}\bigr)\geq i+1, \ \mbox{ for } i=0,\ldots, r\Bigr\}. \end{align*} $$

Note that $\mbox {codim}\bigl (\sigma _{\mu }, \operatorname {\mathrm {Gr}}(r+1,d+1)\bigr )=|\mu |=\mu _0+\cdots +\mu _r$ . If $\mu =(1,\ldots , 1, 0)=:1^r$ , in projective terms, $\Sigma _{1^r}$ consists of r-dimensional subspaces $L=\mathbb P(\Lambda )\subset \mathbb P(V)\cong \mathbb P^d$ intersecting a fixed codimension $2$ subspace along an r-dimensional locus. We set $\sigma _{\mu }:=[\Sigma _{\mu }]\in \operatorname {{\mathrm {CH}}}^{|\mu |}\bigl (\operatorname {\mathrm {Gr}}(r+1,g+1)\bigr )$ .

For a smooth curve C and a linear series $\ell =(L,V)\in G^r_d(C)$ , we denote by:

$$ \begin{align*} \alpha^{\ell}(p):=\bigl(0\leq \alpha_0^{\ell}(p)\leq \alpha_1^{\ell}(p)\leq \cdots \leq \alpha_r^{\ell}(p)\leq d-r\bigr) \end{align*} $$

the ramification sequence at a point $p\in C$ . Keeping with the tradition of [Reference Eisenbud and Harris7] or [Reference Griffiths and Harris11], we write ramification indices of linear series nondecreasingly, whereas indices indexing Schubert cycles are written nonincreasingly. We formalise this practice as follows:

We introduce the proper stack of limit linear series of type $g^r_d$ :

$$ \begin{align*} \sigma \colon \widetilde{\mathcal{G}}^r_d\rightarrow \mathcal{M}_g^{\mathrm{ct}} \end{align*} $$

over the moduli space $\mathcal {M}_g^{\mathrm {ct}}$ of curves of compact type. For a curve C of compact type, we denote by $\overline {G}^r_d(C)$ the variety of limit linear series on C. For pairwise distinct smooth points $p_1, \ldots , p_n\in C_{\mathrm {reg}}$ and Schubert indices $\alpha ^i=\bigl (0\leq \alpha _0^i\leq \cdots \leq \alpha _r^i\leq d-r\bigr )$ , where $i=1, \ldots , n$ , we set:

$$ \begin{align*} \overline{G}^r_d\Bigl(C, (p_1, \alpha^1), \ldots, (p_n, \alpha^n)\Bigr):=\bigl\{\ell\in \overline{G}^r_d(C): \alpha^{\ell}(p_i)\geq \alpha^i, \mbox{ for } i=1, \ldots, n\bigr\}, \end{align*} $$

viewed as a generalised degeneracy locus of expected dimension:

(3) $$ \begin{align} \rho(g,r,d, \alpha^1, \ldots, \ldots, \alpha^n):=g-(r+1)(g-d+r)-\sum_{i=1}^n \sum_{j=0}^r \alpha_j^i. \end{align} $$

We now consider a degeneration to the following flag curve of genus g, already considered in [Reference Eisenbud and Harris7]. Let $[C_0, x_1, \ldots , x_n]$ be the n-pointed genus g curve of compact type consisting of a rational spine $R_{\operatorname {\mathrm {sp}}}$ to which general elliptic tails $E_1,\ldots ,E_g$ are attached at general points $p_1, \ldots , p_g\in R_{\operatorname {\mathrm {sp}}}$ , respectively, and a further rational component $R_0$ also attached at a general point $x_0$ of $R_{\operatorname {\mathrm {sp}}}$ . The marked points $x_1,\ldots ,x_n$ specialise to general points of $R_0$ .

Let $\mathcal {C}\rightarrow (B, b_0)$ be the versal deformation space of $[C_0, x_1, \ldots , x_n]$ and denote by $\tau _1, \ldots , \tau _n \colon B\rightarrow \mathcal {C}$ the sections corresponding to the n marked points. We may assume that each point of B parametrises an n-pointed curve of genus g of compact type. We further consider the induced moduli map $B\rightarrow \mathcal {M}_g^{\mathrm {ct}}$ forgetting the markings and let:

$$ \begin{align*}\sigma_B\colon \widetilde{\mathcal{G}}^r_d/B:=\widetilde{\mathcal{G}}^r_d\times _{\mathcal{M}_g^{\mathrm{ct}}} B\rightarrow B\end{align*} $$

be the corresponding family of limit linear series and consider the evaluation map:

(4) $$ \begin{align} \mathrm{ev}\colon \widetilde{\mathcal{G}}^r_d/B\dashrightarrow B\times P_r^n, \ \ \bigl(C_b, \ell \bigr)\mapsto \Bigl(b, \Bigl(\phi_{\ell}(\tau_1(b)), \ldots, \phi_{\ell}(\tau_n(b))\Bigr)\Bigr), \end{align} $$

where $\phi _{\ell }$ denotes the rational map to $\mathbb P^r$ corresponding to the aspect of the limit linear series $\ell $ on the component of $C_b$ on which all the marked points $\tau _1(b), \ldots , \tau _n(b)$ lie.

Using [Reference Eisenbud and Harris8, Theorem 1.1], it follows that $\widetilde {\mathcal {G}}^r_d/B$ is smooth of dimension $3g-3+n+\rho (g,r,d)$ over B and every limit linear series on $C_0$ smooths to a linear series on a neighboring smooth curve. It follows that $\mbox {deg}(\mathrm {ev})=L_{d,g,r}$ . We will determine this degree by looking at the scheme-theoretic fibre $\mathrm {ev}^{-1}(b_0, y_1, \ldots , y_n)$ , where $y_1, \ldots , y_n$ are general points in $\mathbb {P}^r$ considered up to projective equivalence. We will show in Lemma 3.2 that every point $[C_0, \ell ]\in \mathrm {ev}^{-1}(b_0, y_1, \ldots , y_n)$ corresponds to a limit linear series $\ell \in \overline {G}^r_d(C_0)$ which is base point free at each point $x_1, \ldots , x_n$ . In particular, $\mathrm {ev}^{-1}(b_0, y_1, \ldots , y_n)$ is disjoint from the indeterminacy locus of the map $\mathrm {ev}$ defined in (4).

To that end, we wish to count limit linear series $\ell $ on $C_0$ of degree d and rank r, subject to the condition that, after twisting down base points on the $R_0$ -aspect, the points $x_1,\ldots ,x_n$ have prescribed images in $\mathbb {P}^r$ . For a limit linear series $\ell $ on $C_0$ , we denote by $\ell _{R_0}, \ell _{R_{\operatorname {\mathrm {sp}}}}$ and $\ell _{E_i}$ its corresponding aspects. By the additivity of the Brill-Noether number for $\ell $ encoded in the very definition of a limit linear series, we have the following inequality:

$$ \begin{align*}&\rho(g,r,d)\geq \rho\Bigl(\ell_{R_0}, \alpha^{\ell_{R_0}}(x_0)\Bigr) + \rho\Bigl(\ell_{R_{\mathrm{sp}}}, \alpha^{\ell_{R_{\operatorname{\mathrm{sp}}}}}(x_0), \alpha^{\ell_{R_{\mathrm{sp}}}}(p_1),\ldots, \alpha^{\ell_{R_{\operatorname{\mathrm{sp}}}}}(p_g)\Bigr)\\&\quad+ \sum_{i=1}^g \rho\Bigl(\ell_{E_i}, \alpha^{\ell_{E_i}}(p_i)\Bigr). \end{align*} $$

Since over the curve $[C_0, x_1, \ldots , x_n]$ the map $\mathrm {ev}$ evaluates the $R_0$ -aspect of each limit linear series, it follows that we must only consider the components of $\overline {G}^r_d(C_0)$ in which $\ell _{R_0}$ varies in a family of dimension $\rho (g,r, d)=\dim P_r^n$ . This happens when the remaining aspects of $\ell $ satisfy $\rho \bigl (\ell _{E_i}, \alpha ^{\ell _{E_i}}(p_i)\bigr )=0$ for $i=1, \ldots , g$ and $\rho \Bigl (\ell _{R_{\operatorname {\mathrm {sp}}}}, \alpha ^{\ell _{R_{\operatorname {\mathrm {sp}}}}}(x_0), \alpha ^{\ell _{R_{\operatorname {\mathrm {sp}}}}}(p_1),\ldots , \alpha ^{\ell _{R_{\mathrm {sp}}}}(p_g)\Bigr )=0$ .

This implies that on each elliptic tail $E_i$ , the ramification sequence at the node $p_i$ must be equal to $(d-r-1, \ldots , d-r-1,d-r)$ . Indeed, we need $\alpha ^{\ell _{E_i}}_r(p_i)=d-r$ , or else:

$$ \begin{align*} \alpha^{\ell_{E_i}}_0(p_i)+\cdots+\alpha^{\ell_{E_i}}_r(p_i)\le r(d-r-1), \end{align*} $$

but also $\alpha ^{\ell _{E_i}}_{r-1}(p_i)\le d-r-1$ , or else $E_i$ would carry a linear series of rank 1 and degree 1. We therefore have a unique choice of the $E_i$ -aspect, precisely $\ell _{E_i}=(d-r-1)p_i+\bigl |(r+1)p_i\bigr |$ , for $i=1, \ldots , g$ . By compatibility of the aspects of limit linear series, we find that $\alpha ^{\ell _{R_{\operatorname {\mathrm {sp}}}}}(p_i)=(0, 1, \ldots , 1)$ for $i=1, \ldots , g$ , that is, $\ell _{R_{\operatorname {\mathrm {sp}}}}$ has a simple cusp at each of the points $p_i, \ldots , p_g$ .

From here, on $R_{\operatorname {\mathrm {sp}}}$ , the ramification sequence of $\ell $ at the point $x_0$ :

$$ \begin{align*} \left(\alpha^{\ell_{R_{\operatorname{\mathrm{sp}}}}}_{0}(x_0),\ldots,\alpha^{\ell_{R_{\operatorname{\mathrm{sp}}}}}_{r}(x_0)\right) \end{align*} $$

must satisfy the equality:

$$ \begin{align*} \sum_{j=0}^{r}\alpha^{\ell_{R_{\operatorname{\mathrm{sp}}}}}_{j}(x_0)=(r+1)(d-r)-rg, \end{align*} $$

whereas the $R_0$ -aspect of $\ell $ satisfies:

(5) $$ \begin{align} \sum_{j=0}^r \alpha_j^{\ell_{R_0}}(x_0)=rg. \end{align} $$

Let $\mu =(\mu _0\geq \cdots \geq \mu _r):=\overline {\alpha ^{\ell _{R_{\operatorname {\mathrm {sp}}}}}_{r}(x_0)}$ , that is, we write the partition $\Big(\alpha ^{\ell _{R_{\operatorname {\mathrm {sp}}}}}_{r}(x_0),\ldots ,\alpha ^{\ell _{R_{\operatorname {\mathrm {sp}}}}}_{0}(x_0)\Big )$ , where the ramification indices are given in descending order, and let $\lambda $ be the complement of $\mu $ in $(d-r)^{r+1}$ , that is, $\lambda _j=d-r-\mu _j$ , for $j=0,1,\ldots ,r$ . Summarising the discussion so far, for each limit linear series $\ell $ on $C_0$ contributing towards the degree of the map $\mathrm {ev}$ , one has:

(6) $$ \begin{align} \alpha^{\ell_{R_0}}(x_0)=\overline{\lambda}. \end{align} $$

The number of possible aspects $\ell _{R_{\operatorname {\mathrm {sp}}}}$ on $R_{\operatorname {\mathrm {sp}}}$ with ramification sequence $\overline {\mu }$ at $x_0$ and cusps at $p_1, \ldots , p_g$ is given by:

$$ \begin{align*} \beta_{\lambda}:=\int_{\operatorname{\mathrm{Gr}}(r+1,d+1)}\sigma_{1^r}^g\cdot \sigma_{\mu}. \end{align*} $$

The transversality of the intersection follows from [Reference Eisenbud and Harris6], see also [Reference Mukhin, Tarasov and Varchenko15], that is, for a general choice of the points $p_1, \ldots , p_g$ and $x_0$ , one has precisely $\beta _{\lambda }$ distinct linear series on $R_{\operatorname {\mathrm {sp}}}$ with these property. Since $\sigma _{\lambda '}\cdot \sigma _{\mu }=0$ , for any Schubert index $\lambda '\neq \lambda $ with $|\lambda '|=rg$ , whereas $\sigma _{\lambda }\cdot \sigma _{\mu }=1$ , we can write:

(7) $$ \begin{align} \sigma_{1^r}^g=\sum_{|\lambda|=rg}\beta_{\lambda}\cdot \sigma_{\lambda}\in \operatorname{{\mathrm{CH}}}^{g}\bigl(\operatorname{\mathrm{Gr}}(r+1,d+1)\bigr). \end{align} $$

Such maps are obtained by twisting the $R_0$ -aspect of each limit linear series $\ell $ on $C_0$ by the order $\lambda _r$ of its base point $x_0$ . Our degeneration shows:

Proposition 2.3. For a general n-pointed curve $[C,x_1, \ldots , x_n]$ of genus g, the degree of the map $\mathrm {ev}_{(x_1, \ldots , x_n)}$ is given by the formula:

$$ \begin{align*} L_{g,r,d}=\sum_{|\lambda|=rg}\beta_{\lambda}L_{g,r,d,\lambda}. \end{align*} $$

3 Counting linear series with assigned incidences on $\mathbb {P}^1$

Having reduced both Theorems 1.1 and 1.2 to a question on rational curves, we use Schubert calculus to complete their proofs.

Let us first sketch the argument. The set of maps $\mathbb {P}^1\to \mathbb {P}^r$ counted by the number $L_{g,r,d,\lambda }$ naturally sits inside the projective space $\mathbb {P}^{(r+1)(d+1)-1}$ parametrising morphisms $f=[f_0, \ldots , f_r]$ of degree d, as given by the intersection of the conditions:

1. (i) $f(x_i)=y_i$ for $i=1, \ldots ,n$ ,

2. (ii) f has ramification at least $\overline {\lambda }$ at $x_0$ .

The conditions $f(x_i)=y_i$ cut out linear subspaces, while, upon summing over all $\lambda $ with the multiplicities $\beta _\lambda $ , the ramification conditions at $x_0$ cut out an intersection of g subvarieties of degree $r+1$ . The expected degree of the intersection is therefore $(r+1)^g$ , and we show in the proof of Theorem 1.1 that this intersection is indeed transverse when $d\ge rg+r$ .

In general, however, the intersection described above has many excess components. Under the conditions of Theorem 1.2, we remove these excess contributions by passing to a certain incidence correspondence dominating $\mathbb {P}^{(r+1)(d+1)-1}$ to compute $L_{g,r,d}$ .

3.1 Proof of Theorem 1.1

For a complex polynomial $u=a_0+\cdots +a_dt^d$ , we denote by $c(u)$ the column vector of its coefficients. Let $\mathbb {P}^{(r+1)(d+1)-1}$ be the projective space parametrising $(r+1)$ -tuples $(f_0,\ldots ,f_r)$ of polynomials of degree d in one variable viewed as sections of $\mathcal {O}_{\mathbb {P}^1}(d)$ , up to simultaneous scaling, and not all zero. When not all polynomials $f_i$ are zero and have no common zeroes, they define a map $f=[f_0, \ldots , f_r]$ of degree d from $\mathbb {P}^1$ to $\mathbb {P}^r$ .

We introduce the map:

(8) $$ \begin{align} \pi\colon \mathbb{P}^{(r+1)(d+1)-1}\dashrightarrow \operatorname{\mathrm{Gr}}(r+1,d+1), \end{align} $$

remembering the linear series spanned by $f_0, \ldots , f_r$ , whenever they are linearly independent. The indeterminacy locus of this map is irreducible of codimension $d-r+1$ , for an $(r+1)$ -tuple of polynomials $(f_0, \ldots , f_r)$ lies in the indeterminacy locus of $\pi $ if and only if the $(r+1)\times (d+1)$ -matrix of coefficients $\bigl (c(f_0), \ldots , c(f_r)\bigr )$ has rank at most r.

For a Schubert variety $\Sigma _\lambda =\Sigma _{\lambda }({\textbf F})\subseteq \operatorname {\mathrm {Gr}}(r+1,d+1)$ of codimension at most $rg$ in $\operatorname {\mathrm {Gr}}(r+1,d+1)$ , let $\widetilde {\Sigma }_{\lambda }:=(\pi _1)_*\bigl (\pi _2^*(\Sigma _{\lambda })\bigr )$ be the closure of its pullback under $\mathbb {P}^{(r+1)(d+1)-1}$ . Because the codimension of $\Sigma _{\lambda }$ is lower than that of the indeterminacy locus of $\pi $ (by our assumption, $d-r+1>rg$ ), the cycle $\widetilde {\Sigma }_{\lambda }$ has the expected codimension of $|\lambda |=rg$ and defines a well-defined class $\widetilde {\sigma }_{\lambda }\in \operatorname {{\mathrm {CH}}}^{rg}\bigl (\mathbb {P}^{(r+1)(d+1)-1}\bigr )$ . Using (7) we have the formula:

$$ \begin{align*} \widetilde{\sigma}_{1^r}^{g}=\sum_{|\lambda|=rg}\beta_{\lambda}\widetilde{\sigma}_{\lambda}. \end{align*} $$

Recall that we have fixed n general points $y_1, \ldots , y_n\in \mathbb {P}^r$ . The condition on maps $f\colon \mathbb {P}^1\to \mathbb {P}^r$ that $f(x_i)=y_i$ for $i=1, \ldots , n$ impose $nr$ linear conditions on the matrix of coefficients $\bigl (c(f_0), \ldots , c(f_r)\bigr )$ . Observe that this condition is automatically satisfied for those i for which $x_i$ is a base point of f. The points $y_1, \ldots , y_n$ having been chosen to be general, these linear conditions are independent. Since $(r+1)(d+1)-1-nr=rg$ , the conditions $f(x_i)=y_i$ give rise to a linear subspace:

$$ \begin{align*} {\textbf L}\cong\mathbb{P}^{rg}\subseteq \mathbb{P}^{(r+1)(d+1)-1}. \end{align*} $$

Now, let ${\Sigma }_{\lambda }(x_0)=G^r_d\bigl (\mathbb {P}^1, (x_0, \overline {\lambda }\bigr ))$ be the Schubert variety of $\operatorname {\mathrm {Gr}}(r+1,d+1)$ parametrising linear series on $\mathbb {P}^1$ with ramification sequence at least $\overline {\lambda }$ at $x_0$ . We wish to intersect its pullback $\widetilde {\Sigma }_{\lambda }(x_0)$ with ${\textbf L}$ on $\mathbb {P}^{(r+1)(d+1)-1}$ . We call a point $[f_0,\ldots ,f_r]$ in this intersection generic if $\langle f_0,\ldots ,f_r\rangle $ is a linear series of rank r with ramification sequence exactly $\overline {\lambda }$ , and which defines a (nondegenerate) morphism $f\colon \mathbb {P}^1\to \mathbb {P}^r$ after twisting down the base points at $x_0$ with $f(x_i)=y_i$ (in particular, $\langle f_0,\ldots ,f_r\rangle $ has no base points away from $x_0$ ).

Remark 3.1. We have already seen above that the condition $d\ge rg+r$ ensures that the classes $\widetilde {\sigma }_{1^r}^g$ and $\widetilde {\sigma }_{\lambda }$ live in codimension strictly smaller than that of the indeterminacy locus of $\pi $ . However, as we will see in Lemma 3.2, the same condition $d\geq rg+r$ also ensures that ${\textbf L}$ contains no points corresponding to degenerate maps $f\colon \mathbb {P}^1\to \mathbb {P}^r$ . In fact, this is already evident in the case of constant maps; indeed, suppose instead that $d\ge n-1$ . Then, we may take the nonzero polynomials $f_0,\ldots ,f_r$ to vanish at $x_1,\ldots , x_{n-1}$ , and after twisting away all base points, the resulting map $f\colon \mathbb {P}^1\to \mathbb {P}^r$ to be the constant map with image $y_n$ . Then, $f=[f_0,\ldots , f_r]$ lies on the one hand in ${\textbf L}$ , and on the other hand in the indeterminacy locus of $\pi $ .

Lemma 3.2. The intersection points of $\widetilde {\Sigma }_{\lambda }(x_0)$ with ${\textbf L}$ are generic in the previous sense. In particular, the intersection occurs away from the indeterminacy locus of $\pi $ .

Proof. We construct the locus ${\textbf L}$ ‘relatively’, allowing the points $y_1,\ldots ,y_n$ to vary, and show that, for dimension reasons, the locus where ${\textbf L}\cap \widetilde {\Sigma }_{\lambda }(x_0)$ contains nongeneric points cannot dominate the space of choices of the $y_i$ . In particular, if the $y_i$ are chosen to be general, we obtain the desired conclusion.

More precisely, let $V\subseteq (\mathbb {P}^r)^n$ be the open subset of collections of points $y_1,\ldots ,y_n\in \mathbb {P}^r$ , where the $y_i$ are in linearly general position, that is, no m of the $y_i$ lie on a linear space of dimension $m-2$ if $2\le m\le r+1$ . Consider the product $V\times \mathbb {P}^{(r+1)(d+1)-1}$ , where the second factor parametrises maps $f=[f_0,\ldots , f_r]$ as before, and the closed subscheme $V\times \widetilde {\Sigma }_{\lambda }(x_0)$ of the expected dimension $rg$ as defined above. We then define the locus, abusively denoted ${\textbf L}\subseteq V\times \mathbb {P}^{(r+1)(d+1)-1}$ , of maps f satisfying $f(x_i)=y_i$ for $i=1,2,\ldots ,n$ , by relativising the above construction.

We have a forgetful map $\psi \colon {\textbf L}\cap \widetilde {\Sigma }_{\lambda }(x_0) \to V$ , and wish to show that the locus of nongeneric points of source does not dominate V; to do so, we show that the locus of nongeneric points has dimension strictly less than that of V.

First, consider the locus on ${\textbf L}\cap \widetilde {\Sigma }_{\lambda }(x_0)$ of nongeneric $f=[f_0, \ldots , f_r]\in {\textbf L}\cap \widetilde {\Sigma }_{\lambda }(x_0)$ away from the indeterminacy locus of $\pi $ . Suppose that f has base points of total order k away from $x_0,\ldots , x_n$ and order $k'$ on $x_1,\ldots ,x_n$ , and that $k+k'>0$ . We see upon twisting down by these base points that the locus of such f has the expected codimension $(r+1)(k+k')$ in $\widetilde {\Sigma }_{\lambda }(x_0)$ , and the incidence conditions $f(x_i)=y_i$ impose at least $(n-k')r$ additional conditions inside $V\times \widetilde {\Sigma }_{\lambda }(x_0)$ . In total, we find that the locus of possible f has codimension strictly greater than $rg+rn$ in $V\times \widetilde {\Sigma }_{\lambda }(x_0)$ , and therefore cannot dominate V. Similarly, a parameter count shows that f cannot have ramification sequence strictly more than $\overline {\lambda }$ at $x_0$ .

Consider now a point of ${\textbf L}\cap \widetilde {\Sigma }_{\lambda }(x_0)$ , for which $\mbox {dim} \langle f_0, \ldots , f_r\rangle \leq r$ . We show again by counting parameters that no such f can exist. By twisting away base points at $x_0$ (which decreases the number of moduli and the number of conditions by the same amount), we may assume that f is base point free at $x_0$ . We may also assume that f has no base points away from $x_1,\ldots , x_n$ . Suppose now that f has k (simple) base points among these $x_i$ , we label them as $x_{n-k+1}, \ldots , x_n$ ; we twist down our linear series to have degree $d-k$ , and lose the corresponding k linear conditions. Note that in this case, the ramification condition at $x_0$ can no longer be imposed in terms of $f_0, \ldots , f_r$ alone, since by assumption, the resulting map $f\colon \mathbb {P}^1\to \mathbb {P}^r$ is degenerate, that is, the corresponding linear series has dimension $r'<r$ . Note, however, that if the remaining $y_i$ do not themselves live in a linear subspace of $\mathbb {P}^r$ of dimension $r'$ , then this is impossible; we therefore need $n-k\le r'+1$ .

Then, it must be true that if $x_1,\ldots ,x_{n-k}$ are general points of $\mathbb {P}^1$ , there exists a map $f\colon \mathbb {P}^1\to \mathbb {P}^{r'}$ of degree $d-k$ with $f(x_i)=y_i$ for $i=1,\ldots , n-k$ . Therefore, we have:

$$ \begin{align*} (d-k+1)(r'+1)-1\ge r'(n-k). \end{align*} $$

Rearranging yields:

$$ \begin{align*} k\le d-r'(n-d-1). \end{align*} $$

On the other hand, because $n-k\le r'+1$ , we find:

$$ \begin{align*} (d-n+1)\ge r'(n-d-2). \end{align*} $$

However, by assumption, we have $n\ge d+2$ and $r'\ge 0$ , so we have reached a contradiction.

Lemma 3.3. For a general choice of the points $x_1, \ldots , x_n\in \mathbb {P}^1$ and $y_1, \ldots , y_n\in \mathbb {P}^r$ , the intersection of $\widetilde {\Sigma }_{\lambda }(x_0)$ and ${\textbf L}$ is transverse.

Proof. Let $\mathcal {M}_{n,d,r}$ be the open subscheme of the space $\operatorname {\mathrm {Hom}}_d(\mathbb {P}^1,\mathbb {P}^r)\times (\mathbb {P}^1)^n$ parametrising elements $([f\colon \mathbb {P}^1\to \mathbb {P}^r],x_1,\ldots ,x_n)$ , where f is a nondegenerate morphism of degree d and the $x_i$ are pairwise distinct points that in addition are distinct from a fixed point $x_0\in \mathbb {P}^1$ .

One may construct $\mathcal {M}_{n,d,r}$ as an open subset of a $(\mathbb {P}^1)^{n}$ -bundle over $\mathbb {P}^{(r+1)(d+1)-1}$ . We have a smooth, regular map $\chi \colon \mathcal {M}_{n,d,r}\to \operatorname {\mathrm {Gr}}(r+1,d+1)$ , from which we can pull back the smooth, open Schubert cycle of linear series with ramification exactly $\overline {\lambda }$ at $x_0$ to obtain the smooth subscheme $Y_{n,d,r}$ parametrising the morphisms we wish to count. Finally, the projection $\phi \colon Y_{n,d,r}\to (\mathbb {P}^1)^{n+1}\times (\mathbb {P}^r)^n$ remembering the marked points and their images on the source is generically unramified of finite degree.

By construction, any nonzero tangent vector to the intersection $\widetilde {\Sigma }_{\lambda }(x_0)$ and ${\textbf L}$ in the generic locus yields a nonzero relative tangent vector of $\phi $ . Thus, when the points $x_i, y_i$ are general, there are no such tangent vectors, and the intersection is transverse.

We are now in a position to complete the proof of Theorem 1.1.

Proof of Theorem 1.1

By the above discussion summarised in Proposition 2.3, it suffices to intersect $nr$ linear conditions with $\widetilde {\sigma }_{1^r}^{g}$ on $\mathbb {P}^{(r+1)(d+1)-1}$ and compute the degree, that is:

$$ \begin{align*} L_{g,r,d}=\sum_{|\lambda|=rg} \beta_{\lambda} {\textbf L}\cdot \widetilde{\sigma}_{\lambda}={\textbf L}\cdot \widetilde{\sigma}_{1^r}^g=\mathrm{deg}\bigl(\widetilde{\sigma}_{1^r}^g\bigr)=\mathrm{deg}\bigl(\sigma_{1^r}^g\bigr), \end{align*} $$

where the last two degrees are computed on $\operatorname {\mathrm {Gr}}(r+1,d+1)$ and on $\mathbb {P}^{(r+1)(d+1)-1}$ , respectively.

Theorem 1.1 then follows from the fact that the degree of $\widetilde {\sigma }_{1^r}$ is $r+1$ . To see this, note that on $\operatorname {\mathrm {Gr}}(r+1,d+1)$ , the Schubert cycle $\Sigma _{1^r}$ is the locus of $(r+1)$ -planes intersecting a fixed codimension 2 subspace $P\subseteq H^0(\mathbb {P}^1,\mathcal {O}(d))$ in a subspace of dimension at least r. Identifying $\mathbb {P}^{(r+1)(d+1)-1}$ with the space of $(r+1)\times (d+1)$ matrices, whose entries are taken up to simultaneous scaling, the pullback of $\Sigma _{1^r}$ may be identified with the determinantal locus of matrices, such that the $(r+1)\times 2$ submatrix formed by the first two columns has rank 1. This, in turn, is the pullback under linear projection from $\mathbb {P}^{(r+1)(d+1)-1}$ of the Segre embedding $\mathbb {P}^{1}\times \mathbb {P}^{r}\to \mathbb {P}^{2r+1}$ . Denoting by $h_1$ and $h_2$ the pullbacks to $\mathbb {P}^1\times \mathbb {P}^{r+1}$ of the hyperplane classes of $\mathbb {P}^1$ and $\mathbb {P}^{r+1}$ , observe that:

$$ \begin{align*} \mbox{deg}\bigl(\mathbb{P}^1\times \mathbb{P}^{r}\bigr)=(h_1+h_2)^{r+1}={r+1\choose 1}h_1 h_2^r=r+1. \end{align*} $$

This completes the proof.

Remark 3.4. The inequality $d\ge rg+r$ in Theorem 1.1 is sharp. Indeed, the largest possible value of d outside of this range is $d=rg$ , corresponding to $d=n-1$ . In this case, following the proof of Theorem 1.1 shows that our intersection of cycles inside $\mathbb {P}^{(r+1)(d+1)-1}$ contains an additional zero-dimensional locus of constant maps $[f_0, \ldots , f_r]$ , where each $f_i$ is a constant multiple of the degree $d=n-1$ polynomial vanishing at all of the points $x_1,\ldots ,x_n$ except one, $x_i$ , and the image of f is the point $y_i$ . There is one such map for each of the marked points $x_i$ , so we find that $L_{g,r,d}=(r+1)^g-n=(r+1)^g-(d+1)$ .

Remark 3.5. It is interesting to observe that on a smooth curve C of genus g, a general stable vector bundle E of rank $r+1$ and degree d has precisely $(r+1)^g$ line subbundles of maximal degree $d'$ , where $d-(r+1)d'=r(g-1)$ (see [Reference Okonek and Teleman16] or [Reference Oxbury17]). The reinterpretation of the numbers $L_{g,r,d}$ from this point of view will be pursued elsewhere.

3.2 Proof of Theorem 1.2

We recast the calculation of the previous section in the following light: we consider the incidence correspondence on $\mathbb {P}^{(r+1)(d+1)-1}\times \operatorname {\mathrm {Gr}}(r+1,d+1)$ of $(r+1)$ -tuples of degree d polynomials, spanning a $r+1$ -dimensional subspace of $H^0(\mathbb {P}^1, \mathcal {O}_{\mathbb {P}^1}(d))$ , then pullback Schubert cycle conditions on the Grassmannian side and linear conditions on the projective space side. This incidence correspondence is defined by pulling back the diagonal under the map:

$$ \begin{align*} (\pi,\operatorname{\mathrm{id}})\colon \mathbb{P}^{(r+1)(d+1)-1}\times\operatorname{\mathrm{Gr}}(r+1,d+1)\dashrightarrow\operatorname{\mathrm{Gr}}(r+1,d+1)\times\operatorname{\mathrm{Gr}}(r+1,d+1), \end{align*} $$

and the condition $d\ge rg+r$ is needed in order to prevent the indeterminacy locus from being too large. In this section, we obtain formulas for $L_{g,r,d}$ in the cases $r=1$ and $g\ge 1$ by shrinking this base locus.

More precisely, for $j=0,1,\ldots ,r$ , let $\rho _{j}\colon \mathbb {P}^{(r+1)(d+1)-1}\dashrightarrow \mathbb {P}^{d}$ be the linear projection remembering $f_j\in H^0(\mathcal {O}_{\mathbb {P}^1}(d))$ , where we recall that $(f_0,\ldots ,f_r)$ is the $(r+1)$ -tuple of polynomials, whose coefficients are parametrised by $\mathbb {P}^{(r+1)(d+1)-1}$ . We now consider the following incidence correspondence:

If $\mathcal {Q}$ denotes the rank $d-r$ tautological quotient bundle on $\operatorname {\mathrm {Gr}}(r+1,d+1)$ , then Z can be realised as the degeneracy locus of the composition:

$$ \begin{align*} \pi_1^*\bigl(\mathcal{O}_{\mathbb{P}^d}(-1)\bigr) \longrightarrow \mathcal{O}_{\mathbb{P}^d\times \operatorname{\mathrm{Gr}}(r+1,d+1)}^{d+1}\longrightarrow \pi_2^*\bigl(\mathcal{Q}\bigr), \end{align*} $$

and thus has class:

$$ \begin{align*} \Bigl\{c\bigl(\pi_2^*\mathcal{Q}\bigr)\cdot c\bigl(\pi_1^*\mathcal{O}_{\mathbb{P}^d}(1)\bigr)\Bigr\}_{d-r}=\sum_{i+j=d-r}\pi_2^*\bigl(\sigma_i\bigr)\cdot \pi_1^*\bigl(H^j\bigr)\in \operatorname{{\mathrm{CH}}}^{d-r}\bigl(\mathbb{P}^d\times\operatorname{\mathrm{Gr}}(r+1,d+1)\bigr), \end{align*} $$

where H is the hyperplane class on $\mathbb {P}^d$ , and where we have also used that $c_j(\mathcal {Q})=\sigma _j$ .

Because the codimension of the base locus of $\rho _j$ is $d+1>d-r$ , the closure:

$$ \begin{align*} Z_j:=\bigl(\rho_j\times \mathrm{id}_{\operatorname{\mathrm{Gr}}(r+1,d+1)}\bigr)^{-1}(Z) \end{align*} $$

of the pullback of the correspondence Z has the same class, that is, $\sum _{i+j=d-r} \pi _2^*\bigl (\sigma _i\bigr )\cdot \pi _1^*\bigl (H^j\bigr )$ , where this time $\pi _2\colon \mathbb {P}^{(r+1)(d+1)-1}\times \operatorname {\mathrm {Gr}}(r+1,d+1)\rightarrow \operatorname {\mathrm {Gr}}(r+1,d+1)$ denotes the second projection.

Proof of Theorem 1.2

We wish to compute the intersection inside $\mathbb {P}^{(r+1)(d+1)-1}\times \operatorname {\mathrm {Gr}}(r+1,d+1)$ of the $nr$ linear conditions pulled back from $\mathbb {P}^{(r+1)(d+1)-1}$ given by the equations $f(x_i)=y_i$ for $i=1, \ldots , n$ , the pullback under $\pi _2$ of the Schubert cycles $\sigma _\lambda $ , where $\lambda $ is a Schubert index with $|\lambda |=rg$ , and the classes of the cycles $Z_0,\ldots , Z_r$ defined above. We proceed as in Lemmas 3.2 and 3.3.

First, we introduce the incidence correspondence:

We first claim that the intersection $\pi _2^*(\sigma _{\lambda })\cdot \pi _1^*({\textbf L})$ in question is supported away from the locus of $(f,\Lambda )\in \mathcal {X}$ , where f defines a degenerate map $f\colon \mathbb {P}^{1}\to \mathbb {P}^{r'}$ , for some $r'<r$ . When $d\ge rg+r$ , the same proof as in Lemma 3.2 applies.

Suppose that either $r=1$ or $n=d+2$ , there is such a $(f,\Lambda )\in Z$ in our intersection, and that k of the points $x_1,\ldots ,x_n$ are base points of the $r'$ -dimensional linear system $\Lambda _f$ spanned by $f_0,\ldots ,f_r$ . As in the proof of Lemma 3.2, it must be the case that $k\ge n-r'-1$ .

Denote the total ramification of $\Lambda _f$ at $x_0$ by t. Then,

$$ \begin{align*} t&\le \dim\operatorname{\mathrm{Gr}}(r'+1,d-k+1)\\ &\le \dim\operatorname{\mathrm{Gr}}(r'+1,d-n+r'+2)\\ &=(r'+1)(d-n+1). \end{align*} $$

Thus,

$$ \begin{align*} d&\ge \frac{t}{r'+1}+n-1\\ &=\frac{t}{r'+1}+d+\frac{d}{r}-g, \end{align*} $$

whence

$$ \begin{align*} d\le rg-\frac{rt}{r'+1}. \end{align*} $$

On the other hand, we require that $\Lambda \in \sigma _\lambda $ , where $|\lambda |=rg$ , and $\Lambda _f\subseteq \Lambda $ . Such a $\Lambda $ can only exist if:

$$ \begin{align*} t+(r-r')(d-r)\ge rg, \end{align*} $$

as $\dim (\Lambda /\Lambda _f)=r-r'$ , and each dimension can contribute at most $d-r$ to the ramification of $\Lambda $ at $x_0$ . Since $r>r'$ , we obtain:

$$ \begin{align*} t+(r-r')\left(rg-\frac{rt}{r'+1}-r\right)&\ge rg\\ t\left(1-\frac{(r-r')r}{r'+1}\right)+(r-r')(rg-r)&\ge rg. \end{align*} $$

When $r=1$ , and thus $r'=0$ , we obtain a contradiction. It remains to consider the case $n=r+2$ , in which case:

$$ \begin{align*} rg=(d-r)(r+1). \end{align*} $$

Then, comparing the inequalities:

$$ \begin{align*} t&\le (r'+1)(d-n+1)=(r'+1)(d-r-1)\\ t&\ge rg-(r-r')(d-r)=(d-r)(r+1)-(r-r')(d-r) \end{align*} $$

also yields a contradiction.

Therefore, we are back in the situation of Lemma 3.2, in which all intersection points occur where f is nondegenerate, and in particular, $(f,\Lambda )$ lies away from the indeterminacy of the $\rho _j$ . The same parameter counts show that f indeed defines a map $f\colon \mathbb {P}^1\to \mathbb {P}^1$ of degree d with vanishing exactly $\overline {\lambda }$ at $x_0$ .

Furthermore, the intersection in question is transverse by exactly the same argument as in Lemma 3.3, so it suffices to compute the degree of the intersection cycle on $\mathbb {P}^{(r+1)(d+1)-1}\times \operatorname {\mathrm {Gr}}(r+1,d+1)$ . This equals:

$$ \begin{align*} \int_{\mathbb{P}^{(r+1)(d+1)-1}\times\operatorname{\mathrm{Gr}}(r+1,d+1)}\pi_2^*\bigl(\sigma_{1^r}^g\bigr) \cdot \pi_1^*\bigl(H^{nr}\bigr) \left(\sum_{i+j=d-r}\pi_2^*\bigl(\sigma_i\bigr)\cdot \pi_1^*\bigl(H^j\bigr)\right)^{r+1} \end{align*} $$

$$ \begin{align*} =\int_{\operatorname{\mathrm{Gr}}(r+1,d+1)}\sigma_{1^r}^g\cdot\left[ \sum_{\alpha_0+\cdots+\alpha_{r}=(r+1)(d-r)-rg}\left(\prod_{i=0}^{r}\sigma_{\alpha_i}\right)\right], \end{align*} $$

as desired.

Remark 3.6. While the proof of Theorem 1.2 shows that our refined incidence correspondence avoids the constant maps of Remark 3.1 when $r=1$ or $n=r+2$ , this is not the case in general. Indeed, suppose that $r\ge 2$ , $n\ge r+3$ and $d\ge n-1$ . Then, take $f_0,\ldots ,f_r$ to have simple zeroes at $x_1,\ldots ,x_{n-1}$ and an order $d-(n-1)$ zero at $x_0$ , such that $f=[f_0, \ldots , f_r]$ defines the constant map with image $y_n\in \mathbb {P}^r$ . If $\lambda $ is a Schubert index with $|\lambda |=rg$ , then the condition that $f_0,\ldots ,f_r\in \Lambda $ , where $\Lambda \in \Sigma _\lambda (x_0)\subseteq \operatorname {\mathrm {Gr}}(r+1,d+1)$ , may be satisfied as long as $rg\le (d-n+1)+\dim \operatorname {\mathrm {Gr}}(r,d)=(d-n+1)+r(d-r)$ . Substituting $rg=dr+r+d-rn$ , this is equivalent to $n\ge \frac {r^2+r-1}{r-1}=r+2+\frac {1}{r-1}$ . When $r\ge 2$ and $n\ge r+3$ , this is immediate.

For $r=1$ , Theorem 1.2 can be used to recover (via simple manipulations) the explicit formulas in terms of binomial coefficients for the degrees $L_{g,1,d}$ . These numbers are also determined in [Reference Cela, Pandharipande and Schmitt5] using excess intersection on Hurwitz spaces of admissible covers.

Proposition 3.7. For $d\geq \frac {g+2}{2}$ , we have:

$$ \begin{align*} L_{g,1,d}&=\sum_{\alpha_0+\alpha_1=2d-2-g}\int_{\operatorname{\mathrm{Gr}}(2,d+1)} \sigma_1^g\cdot \sigma_{\alpha_0}\cdot \sigma_{\alpha_1}\\ &= \sum_{i=0}^{\bigl \lfloor \frac{2d-g-2}{2}\bigr \rfloor} \frac{(2d-g-2i-1)^2}{g+1} {g+1\choose d-i}\\ &=2^g-2\sum_{i=0}^{g-d-1}\binom{g}{i}+(g-d-1)\binom{g}{g-d}+(d-g-1)\binom{g}{g-d+1}, \end{align*} $$

where, in the last line, we take $\binom {g}{j}=0$ when $j<0$ .

Proof. We use Giambelli’s formula $\sigma _{a,b}=\sigma _a\cdot \sigma _b-\sigma _{a+1}\cdot \sigma _{b-1}\in \operatorname {{\mathrm {CH}}}^{a+b}\bigl (\operatorname {\mathrm {Gr}}(2,g+1)\bigr )$ for $a\geq b$ , coupled with the formula (see e.g. [Reference Griffiths and Harris11, p. 269]

$$ \begin{align*} \int_{\operatorname{\mathrm{Gr}}(2,d+1)}\sigma_{a,b}\cdot \sigma_1^g&=\frac{a-b+1}{g+1}\cdot\binom{g+1}{d-b}\\ &=\binom{g}{d-b-1}-\binom{g}{d-b} \end{align*} $$

for all $a\geq b$ with $a+b=2d-2-g$ . Substituting in the formula provided by Theorem 1.2 yields the claims.

3.3 Degrees of determinantal Schubert cycles

We note here that comparison of the incidence correspondences given above in the proofs of Theorems 1.1 and 1.2 allows one to compute the degrees of pullbacks of Schubert cycles of low codimension on $\operatorname {\mathrm {Gr}}(r+1,d+1)$ to $\mathbb {P}^{(r+1)(d+1)-1}$ .

Proposition 3.8. Let $\Sigma _{\lambda }$ be a Schubert cycle of codimension $|\lambda |\le d-r$ in $\operatorname {\mathrm {Gr}}(r+1,d+1)$ , and let $\widetilde {\Sigma }_{\lambda }$ be the closure of its pullback under the rational map $\pi \colon \mathbb {P}^{(r+1)(d+1)-1}\dashrightarrow \operatorname {\mathrm {Gr}}(r+1,d+1)$ . Then, the degree of $\widetilde {\Sigma }_{\lambda }$ is:

$$ \begin{align*} \int_{\operatorname{\mathrm{Gr}}(r+1,d+1)}\sigma_\lambda\cdot\left[ \sum_{\alpha_0+\cdots+\alpha_{r}-|\lambda|=(r+1)(d-r)-rg}\left(\prod_{i=0}^{r}\sigma_{\alpha_i}\right)\right]. \end{align*} $$

Proof. Let $N=(r+1)(d+1)-1-|\lambda |$ . Recall that the codimension of $\Sigma _{\lambda }$ is strictly smaller that the codimension of the indeterminacy locus of $\pi $ . Accordingly, adopting the notation of the previous two sections, we have:

$$ \begin{align*} \deg(\widetilde{\Sigma}_{\lambda})&=\int_{\mathbb{P}^{(r+1)(d+1)-1}}H^{N}\cdot[\widetilde{\Sigma}_{\lambda}]\\ &=\int_{\mathbb{P}^{(r+1)(d+1)-1}\times\operatorname{\mathrm{Gr}}(r+1,d+1)}\pi_{1}^{*}\bigl(H^{N}\cdot[\widetilde{\Sigma}_{\lambda}]\bigr)\cdot[Z_0]\cdots[Z_r]\\ &=\int_{\mathbb{P}^{(r+1)(d+1)-1}\times\operatorname{\mathrm{Gr}}(r+1,d+1)}\pi_{1}^{*}\bigl(H^{N}\bigr)\cdot\pi_2^{*}(\Sigma_{\lambda})\cdot[Z_0]\cdots[Z_r]\\ &=\int_{\mathbb{P}^{(r+1)(d+1)-1}\times\operatorname{\mathrm{Gr}}(r+1,d+1)}\pi_{1}^{*}\bigl(H^{N}\bigr)\cdot\pi_2^{*}(\Sigma_{\lambda})\cdot\left(\sum_{i+j=d-r}\pi_2^*\bigl(\sigma_i\bigr) \cdot \pi_1^*\bigl(H^j\bigr)\right)^{r+1}\\ &=\int_{\operatorname{\mathrm{Gr}}(r+1,d+1)}\sigma_\lambda\cdot\left[ \sum_{\alpha_0+\cdots+\alpha_{r}=(r+1)(d-r)-rg-|\lambda|}\left(\prod_{i=0}^{r}\sigma_{\alpha_i}\right)\right], \end{align*} $$

where we have used the equality:

$$ \begin{align*} \pi_{1}^{*}(\widetilde{\Sigma}_{\lambda})\cap Z_0\cap\cdots\cap Z_r=\pi_{2}^{*}(\Sigma_{\lambda})\cap Z_0\cap\cdots\cap Z_r \end{align*} $$

as subschemes of the incidence correspondence $\mathcal {X}$ .

4 Young tableaux interpretation

Comparison of Theorems 1.1 and 1.2 yields the following purely combinatorial statement.

Proposition 4.1. Suppose that $g\ge 0,r\ge 1,d\ge rg+r$ , and d is divisible by r. Then,

$$ \begin{align*} \int_{\operatorname{\mathrm{Gr}}(r+1,d+1)}\sigma_{1^r}^g\cdot\left[ \sum_{\alpha_0+\cdots+\alpha_{r}=(r+1)(d-r)-rg}\left(\prod_{i=0}^{r}\sigma_{\alpha_i}\right)\right]=(r+1)^g. \end{align*} $$

Indeed, both sides are equal to $L_{g,d,r}$ whenever $n=d-g+1+\frac {d}{r}$ is an integer. However, when $d\ge g+r$ , both sides are independent of d; for the left-hand side, this can be seen in terms of Schubert calculus but will also be made transparent in the combinatorial interpretation that follows. In particular, Proposition 4.1 holds under the weaker inequality $d\ge g+r$ with no condition on the divisibility by r.

We give a combinatorial interpretation of the left-hand side in terms of a Young tableaux. Consider a filling of the boxes of a $(r+1)\times (d-r)$ grid with:

• • $rg$ red integers among , with each appearing exactly r times, and

• • $(r+1)(d-r)-rg$ blue integers among , with each appearing any number of times,

subject to the following conditions:

• • the red integers are top- and left- justified, i.e. they appear above blue integers in the same column and to the left of blue integers in the same row,

• • the red integers are strictly increasing across rows and weakly increasing down columns, and

• • the blue integers are weakly increasing across rows and strictly increasing down columns.

An example filling is given in the case $(g,d,r)=(6,15,2)$ below.

Note that the rightmost $d-r-g$ columns must be filled with the blue integers $0,1,\ldots ,r$ in order, so a filling as above is determined by the leftmost g columns, which are those that may contain red integers. In particular, the number of such fillings is independent of d when $d\ge r+g$ . Now, we claim that this number of fillings is given exactly by the intersection number on the left-hand side of Proposition 4.1. Indeed, by the Pieri rule, the term $\sigma _{1^r}$ corresponds to the transposed semistandard Young tableau given by the red integers, and the broken strips formed by the blue entries equal to i correspond to the Schubert cycle $\sigma _{\alpha _i}$ .

Proposition 4.1 therefore implies:

A combinatorial proof of Proposition 4.2 via the Robinson-Schrensted-Knuth (RSK) algorithm has been given by Gillespie, Reimer and Berg [Reference Gillespie and Reimer-Berg10].

5 Variants

5.1 Linear series with fixed incidences and secancy conditions

We briefly explain how our methods also recover the more general Tevelev degrees of [Reference Cela, Pandharipande and Schmitt5], where some of the points $x_i$ are constrained to lie in the same fibre of f. Recall from §1 that, if $1\le k\le n$ , we defined $L^{\prime }_{g,d,k}$ to be the number of morphisms $f\colon C\to \mathbb {P}^1$ of degree d sending general points $x_1,\ldots ,x_n\in C$ to points $y_1,\ldots ,y_n\in \mathbb {P}^1$ , where $y_1=y_2=\cdots =y_k$ , but the $y_i$ are otherwise general.

More generally, we may fix integers $0\leq a\leq k\leq d$ , a general n-pointed curve $(C, x_1, \ldots , x_k, x_{k+1},\ldots , x_n)$ of genus g, where n is given by (2), and consider the variety:

$$ \begin{align*} G^{r,k-a}_{d,k}(C,x_1, \ldots, x_k):=\Bigl\{\ell\in G^r_d(C): \mathrm{dim }\ \ell(-x_1-\cdots-x_{k})\geq r-k+a \Bigr\}, \end{align*} $$

parametrising linear systems $\ell $ whose induced map $\phi _{\ell } \colon C\dashrightarrow \mathbb {P}^r$ has the property that:

$$ \begin{align*} \bigl\langle \phi_{\ell}(x_1), \ldots, \phi_{\ell}(x_k)\bigr\rangle \cong \mathbb{P}^{k-a-1}. \end{align*} $$

Then $G^{r,k-a}_{d,k}(C,x_1, \ldots , x_k)$ is a determinantal variety of dimension:

$$ \begin{align*} \rho(g,r,d)-a(r+1-k+a). \end{align*} $$

Fixing points $y_1, \ldots , y_n\in \mathbb {P}^r$ general with the property that:

$$ \begin{align*} \mbox{dim } \langle y_1, \ldots, y_k\rangle =k-1-a, \end{align*} $$

one can ask for the number of maps $f\colon C\rightarrow \mathbb {P}^r$ of degree d, such that $f(x_i)=y_i$ for $i=1,\ldots , n$ . For any such map, the corresponding linear series $\ell :=f^*\bigl |\mathcal {O}_{\mathbb {P}^r}(1)\bigr |$ lies in $G^{r,k-a}_{d,k}(C,x_1, \ldots , x_k)$ .

In the interest of simplicity, we deal only with the case:

$$ \begin{align*} r=1, \ a=k-1, \end{align*} $$

in which case, this number equals $L_{g,d,k}'$ . We only sketch the proof; we refer the reader to [Reference Cela and Lian4, §6] for detailed proofs and more general statements.

Proof of Theorem 1.3

Consider a linear series V on our general curve C satisfying the needed incidence conditions. We employ a further degeneration after that of §2, allowing $x_1,\ldots ,x_k$ to coalesce onto a bubbled rational component $R_k$ , attached to $R_0$ at x, and consider the resulting limit $V_0$ on this bubbled curve.Footnote ⁴ We find that the $R_k$ -aspect of $V_0$ must have ramification sequence $(d-k,d-1)$ at x, and sends $x_1,\ldots ,x_k$ to the same point after twisting down the base points at x.

It now suffices to count linear series on $R_0$ with the aggregate ramification condition $\sigma _1^g$ at $x_0$ , the new ramification condition $\sigma _{k-1}$ at x, an additional linear incidence condition at x (with image $y_1=\cdots =y_k$ ) and linear incidence conditions at $x_{k+1},\ldots ,x_n$ . The computation of §3.2 yields the count:

$$ \begin{align*} \int_{\operatorname{\mathrm{Gr}}(2,d+1)}\sigma_1^g\sigma_{k-1}\cdot\left[\sum_{i+j=2(d-1)-g-(k-1)}\sigma_{i}\sigma_j\right]. \end{align*} $$

However, we find the following extraneous solutions: if the linear series in question has a base point at x, then we twist down, so that the new ramification sequence is $(0,k-2)$ , and d decreases by 1; in addition, we lose the linear incidence condition at x. Therefore, we see a (zero-dimensional) excess contribution of:

$$ \begin{align*} \int_{\operatorname{\mathrm{Gr}}(2,d)}\sigma_1^g\sigma_{k-2}\cdot\left[\sum_{i+j=2(d-2)-g-(k-2)}\sigma_{i}\sigma_j\right]. \end{align*} $$

Subtracting the above yields the formula for $L^{\prime }_{g,d,k}$ . One needs to check that there are no additional degenerate contributions, and that the intersections are transverse as before, but we omit the details.

Applying the Pieri rule to the formula of Theorem 1.3 yields the following recursions, recovering [Reference Cela, Pandharipande and Schmitt5, Proposition 7] after the change of coordinates $\operatorname {\mathrm {Tev}}_{g,\ell ,r}=L^{\prime }_{g,g+\ell +1,r}$ . These recursions are then used in [Reference Cela, Pandharipande and Schmitt5] to obtain explicit formulas in terms of binomial coefficients.

Corollary 5.1. We have:

$$ \begin{align*} L^{\prime}_{g,d,1}=L^{\prime}_{g-1,d-1,1}+L^{\prime}_{g-1,d,2} \end{align*} $$

and

$$ \begin{align*} L^{\prime}_{g,d,k}=L^{\prime}_{g-1,d-1,k-1}+L^{\prime}_{g-1,d,k+1} \end{align*} $$

for $k>1$ .

Remark 5.2. The proof of Theorem 1.1 may also be employed to show that $L^{\prime }_{g,d,k}=2^g$ whenever $n\ge d+k+1$ . For general r, the number of linear series in question is $(r+1)^g$ whenever $n\ge d+a+2$ .

However, even when $r=1$ , the proof of Theorem 1.2 breaks down as soon as $k>1$ , as we will see contributions from constant maps with value $y_1=\cdots =y_k$ and base points at $x_{k+1}=\cdots =x_n$ . Thus, the additional degeneration as above is needed to obtain the general formula for $L^{\prime }_{g,d,k}$ .

5.2 Linear series with imposed incidences and prescribed ramification

We fix a general pointed curve $[C, p_1,\ldots , p_m, x_1,\ldots ,x_n]\in \mathcal {M}_{g,m+n}$ , general points $y_1,\ldots ,y_n\in \mathbb {P}^r$ , as well as m partitions $\lambda _1, \ldots , \lambda _m$ of length $r+1$ . We may consider morphisms $f\colon C\to \mathbb {P}^r$ of degree d satisfying $f(x_i)=y_i$ for $i=1, \ldots , n$ , and f has ramification sequence at least $\overline {\lambda _j}$ at $p_j$ for $j=1, \ldots , m$ . Suppose, for simplicity, that the $(r+1)$ -st part of each $\lambda _j$ is zero, so that f has no base points. Equivalently, like in (1) we can consider the evaluation map:

(9) $$ \begin{align} \mathrm{ev}_{(x_1, \ldots, x_n)}\colon G^r_d\Bigl(C, \bigl(p_1, \overline{\lambda}_1\bigr), \ldots, \bigl(p_m, \overline{\lambda}_m\bigr)\Bigr)\dashrightarrow P_r^n, \end{align} $$

and ask for its degree when the dimension of the two varieties in question are equal. Using (3), we expect a finite number of such maps $f\colon C\rightarrow \mathbb {P}^r$ whenever $\rho (g,r,d, \overline {\lambda }_1, \ldots , \overline {\lambda }_m)=rn-(r^2+2r)$ , that is, when:

(10) $$ \begin{align} n=\frac{dr+d+r-\lambda_{\operatorname{\mathrm{tot}}}-gr}{r}, \end{align} $$

where $\lambda _{\operatorname {\mathrm {tot}}}:=|\lambda _1|+\cdots +|\lambda _m|$ is the total size of the partitions $\lambda _j$ . Let $L_{g,r,d}^{\lambda _1,\ldots ,\lambda _m}$ be this number, that is, the degree of the map given by (9).

Degenerating the general genus g curve C to a flag curve as in §2 so that the points $p_1, \ldots , p_m$ specialise to general points on the component $R_{\operatorname {\mathrm {sp}}}$ , whereas $x_1, \ldots , x_n$ specialise, as before, to general points of the rational component $R_0$ , we reduce the computation to the numbers $L_{g,r,d,\lambda }$ , as defined in Definition 2.2, where now $|\lambda |=rg+\lambda _{\operatorname {\mathrm {tot}}}$ . Following the proof of Theorem 1.1, we obtain the following result.

Proposition 5.3. Suppose that $d\ge rg+r+\lambda _{\operatorname {\mathrm {tot}}}$ , or equivalently, $n\ge d+2$ . Then,

$$ \begin{align*} L_{g,r,d}^{\lambda_1,\ldots,\lambda_m}=(r+1)^g\cdot \prod_{j=1}^{m}\deg(\widetilde{\Sigma}_{\lambda_j}), \end{align*} $$

where $\deg (\widetilde {\Sigma }_{\lambda _j})$ is the degree of the cycle $\widetilde {\Sigma }_{\lambda _j}$ on $\mathbb {P}^{(r+1)(d+1)-1}$ obtained by taking the closure of the pullback of $\Sigma _{\lambda _j}(x_0)\subseteq \operatorname {\mathrm {Gr}}(r+1,d+1)$ under the rational map $\pi \colon \mathbb {P}^{(r+1)(d+1)-1}\dashrightarrow \operatorname {\mathrm {Gr}}(r+1,d+1)$ (see Proposition 3.8).

Similarly, closely following the proof of Theorem 1.2, we obtain:

Proposition 5.4. Suppose that:

• • $d\ge rg+r+\lambda _{\operatorname {\mathrm {tot}}}$ ,

• • $n=r+2$ , or

• • $r=1$ .

Then,

$$ \begin{align*} L_{g,r,d}^{\lambda_1,\ldots,\lambda_m}=\int_{\operatorname{\mathrm{Gr}}(r+1,d+1)}\sigma_{1^r}^{g}\cdot\prod_{j=1}^{m}\sigma_{\lambda_j}\cdot\left[ \sum_{\alpha_0+\cdots+\alpha_{r}=(r+1)(d-r)-rg-\lambda_{\operatorname{\mathrm{tot}}}}\left(\prod_{i=0}^{r}\sigma_{\alpha_i}\right)\right]. \end{align*} $$

Indeed, in both results, the only significant modification is that the total ramification imposed at $x_0$ after degeneration is $rg+\lambda _{\operatorname {\mathrm {tot}}}$ , instead of $rg$ . However, this number is equal to $dr+d+r-nr$ in both cases, and from here, the proofs go through without change.