Proof. Let $S:=\operatorname {\mathrm {Span}}(\Lambda _1,\dots ,\Lambda _{r})$ . If condition (A) does not hold, then $S\cap \Lambda _{d}$ is a (possibly empty) proper subset of $\Lambda _{d}$ . Suppose that L is a $(n-k)$ -plane intersecting $\Lambda _1,\dots ,\widehat {\Lambda _j},\ldots ,\Lambda _{r}$ . For any $i=1,\dots ,\widehat {j},\dots ,r$ , let us fix a point $p_i\in L\cap \Lambda _i$ and set $N:=\operatorname {\mathrm {Span}}\left(p_1,\dots ,\widehat {p_j},\dots ,p_r\right)\subset L\cap S$ .

Notice that a general point $p\in \Lambda _{d}$ is such that $p\not \in S$ , and consider $\pi _p(N)\subset \mathbb {P}^{n-1}$ . Then $\dim \pi _p(N)=\dim N\leq r-2 \leq n-k-1$ . By definition of N, the subspace $\pi _p(N)$ intersects $\Gamma _1,\dots ,\widehat {\Gamma _j},\ldots ,\Gamma _{r}$ and by condition $\operatorname {\mathrm {SP}}(n-k-1)$ , $\pi _p(N)$ must intersect $\Gamma _j$ too. Therefore, the linear subspace $\widetilde {N}:=\overline {\pi _p^{-1}\left(\pi _p(N)\right)}=\operatorname {\mathrm {Span}}(N,p)\subset \mathbb {P}^n$ intersects $\Lambda _j$ , and $\dim \widetilde {N}=\dim N+1\leq n-k$ . Moreover, since $\Lambda _j\subset S$ and $N=\widetilde {N}\cap S$ , we deduce that $N\cap \Lambda _j\neq \emptyset $ . Thus, L meets $\Lambda _j$ , and we conclude that $\Lambda _1,\dots ,\Lambda _{r}$ are $\operatorname {\mathrm {SP}}(n-k)$ , as in condition $(B)$ .

Proof of Theorem 2.5.

If $n\leq d+k-3$ , the assertion is trivial. So we assume hereafter that $n\geq d+k-2$ . We argue by induction on $d\geq 2$ .

If $d=2$ , the $(k-1)$ -planes $\Lambda _1,\Lambda _2\subset \mathbb {P}^n$ are $\operatorname {\mathrm {SP}}(n-k)$ if and only if they coincide, so the statement is true.

Then we assume $d\geq 3$ . We point out that there exists $j\in \{1,\dots ,d\}$ such that $\Lambda _j\neq \Lambda _i$ for all $i\neq j$ ; otherwise, by grouping the indices of coincident $(k-1)$ -planes, we would obtain a partition of $\{1,\dots ,d\}$ , where each part consists of at least 2 coincident $(k-1)$ -planes, which are trivially $\operatorname {\mathrm {SP}}(n-k)$ , but this is impossible as the sequence $\Lambda _1,\dots ,\Lambda _d$ is indecomposable. Without loss of generality, we assume $j=d$ .

Therefore, given a general point $p\in \Lambda _d$ , we have that $p\not \in \Lambda _i$ for any $i=1,\dots ,d-1$ . Let us consider the projection $\pi _p:\mathbb {P}^n\dashrightarrow \mathbb {P}^{n-1}$ from p, and for any $i=1,\dots ,d-1$ , let us set $\Gamma _i:=\pi _p(\Lambda _i)$ . By Lemma 2.13, the $(k-1)$ -planes $\Gamma _1,\dots ,\Gamma _{d-1}\subset \mathbb {P}^{n-1}$ are $\operatorname {\mathrm {SP}}(n-k-1)$ .

If the sequence $\Gamma _1,\dots ,\Gamma _{d-1}$ is indecomposable, we deduce by induction that $\dim \operatorname {\mathrm {Span}}\left(\Gamma _1,\dots ,\Gamma _{d-1}\right)\leq d+k-4$ . Therefore, the closure of $\pi _p^{-1}\left(\operatorname {\mathrm {Span}}\left(\Gamma _1,\dots ,\Gamma _{d-1}\right)\right)$ is a linear space $\Sigma \subset \mathbb {P}^n$ of dimension at most $d+k-3$ . Since $\Lambda _1,\dots ,\Lambda _{d-1}\subset \Sigma $ , Lemma 2.9 implies that $\Lambda _{d}\subset \Sigma $ too. Thus, we conclude that $\dim \operatorname {\mathrm {Span}}\left(\Lambda _1,\dots ,\Lambda _{d}\right)\leq d+k-3$ , as wanted.

So we assume that $\Gamma _1,\dots ,\Gamma _{d-1}$ is a decomposable sequence (i.e., there exists a partition of $\{1,\dots ,d-1\}$ in $m\geq 2$ parts, where each part $\{i_1,\dots ,i_t\}$ is such that the corresponding $(k-1)$ -planes $\Gamma _{i_1},\dots ,\Gamma _{i_t}$ are $\operatorname {\mathrm {SP}}(n-k-1)$ ). Let

(2.10) $$ \begin{align} D_1:=\{1,\dots,d_1\}, D_2:=\{d_1+1,\dots,d_1+d_2\},\dots, D_m:=\{\sum_{i=1}^{m-1} d_i+1,\dots,d-1\} \end{align} $$

denote such a partition, where the part $D_j$ has cardinality $d_j$ and $\sum _{j=1}^md_j=d-1$ . Furthermore, up to taking a finer partition, we may assume that each part indexes an indecomposable sequence of $(k-1)$ -planes $\Gamma _i$ . We point out that the partition does not depend on the choice of the general point $p\in \Lambda _d$ because $\Lambda _d$ is irreducible and the number of partitions of $\{1,\dots ,d-1\}$ is finite. For any $j=1,\dots , m$ , let

$$ \begin{align*}S^{\prime}_j:=\operatorname{\mathrm{Span}}\left(\Gamma_i\left|i\in D_j\right.\right)\subset \mathbb{P}^{n-1} \quad\text{and}\quad S_j:=\operatorname{\mathrm{Span}}\left(\Lambda_i\left|i\in D_j\right.\right)\subset \mathbb{P}^{n}. \end{align*} $$

By induction, $\dim S^{\prime }_j\leq d_j+k-3$ for any $j=1,\dots ,m$ .

Since we assumed $n\geq d+k-2$ , we have that $2\leq d_j<d-1\leq n-k+1$ for any $j=1,\dots ,m$ . Hence, Lemma 2.14 ensures that for any $j=1,\dots ,m$ , either $\Lambda _d\subset S_j$ – as in case (A) – or the $(k-1)$ -planes $\Lambda _i$ indexed by $D_j$ are $\operatorname {\mathrm {SP}}(n-k)$ , as in case (B). Let $0\leq t\leq m$ be the largest number of parts such that $\Lambda _d\subset S_j$ and, without affecting generality, assume that this happens for $j=m-t+1,\dots ,m$ . Therefore,

(2.11) $$ \begin{align} \dim\operatorname{\mathrm{Span}}(S_{m-t+1},\dots,S_m)\leq \sum_{j=m-t+1}^m(d_j+k-3)-(t-1)(k-1)=\sum_{j=m-t+1}^m d_j -2t+k-1. \end{align} $$

If $t=m$ – that is, $\Lambda _d\subset S_j$ for any $j=1,\dots ,m$ – the previous formula gives

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}(\Lambda_{1},\dots,\Lambda_d)=\dim\operatorname{\mathrm{Span}}(S_{1},\dots,S_m)\leq \sum_{j=1}^m d_j -2m+k-1, \end{align*} $$

so the assertion follows as $ \sum _{j=1}^m d_j=d-1$ and $m\geq 2$ .

If instead $0\leq t<m$ , the first $m-t$ parts of (2.10) satisfy condition (B) of Lemma 2.14, and they do not satisfy condition (A) (i.e., for any $j=1,\dots ,m-t$ , the $(k-1)$ -planes $\Lambda _i$ indexed by $D_j$ are $\operatorname {\mathrm {SP}}(n-k)$ and $\Lambda _d\not \subset S_j$ ). In particular, $p\not \in S_j$ because $p\in \Lambda _d$ is a general point. Therefore, the $(k-1)$ -planes indexed by $D_j$ give an indecomposable sequence; otherwise, by Lemma 2.12, the corresponding sequence of $\Gamma _i=\pi _p(\Lambda _i)$ would be decomposable, but this is not the case. Then we deduce by induction that $\dim S_j\leq d_j+k-3$ for any $j=1,\ldots , m-t$ .

Finally, let us consider the partition of $\{1,\dots ,d\}$ in $m-t+1$ parts given by

(2.12) $$ \begin{align} D_1, D_2,\dots,D_{m-t}, D_{m-t+1}\cup \dots\cup D_m\cup\{d\}, \end{align} $$

where the first $m-t$ parts are those of (2.10), whereas the last part is the union of $\{d\}$ with the last t parts of (2.10). By summing up, the sequence $\Lambda _1,\dots ,\Lambda _d$ is indecomposable, and for any $j=1,\dots ,m-t$ , the part $D_j$ indexes a sequence of $(k-1)$ -planes which are $\operatorname {\mathrm {SP}}(n-k)$ , whose span is $S_j$ and satisfies $\dim S_j\leq d_j+k-3$ . Therefore, the assumptions of Lemma 2.15 hold with respect to partition (2.12).

Concerning the last part of (2.12), let B be the span of the corresponding $(k-1)$ -planes. If $t=0$ , the last part is just $\{d\}$ and $\dim B=\dim \Lambda _d =k-1$ . If instead $t>0$ , then $B=\operatorname {\mathrm {Span}}(S_{m-t+1},\dots ,S_m)$ because $\Lambda _d\subset S_j$ for any $j=m-t+1,\dots ,d$ , and $\dim B$ is bounded by (2.11). Thus, for any $t\geq 0$ , we obtain from Lemma 2.15 that

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}(\Lambda_1,\dots,\Lambda_d)\leq \dim B+ \sum_{j=1}^{m-t}d_j-(m-t+1) \leq \sum_{j=1}^m d_j +k-m-t-2. \end{align*} $$

In particular, the assertion holds since $ \sum _{j=1}^m d_j=d-1$ , $m\geq 2$ and $t\geq 0$ .

Proof of Theorem 1.1.

Let $k\in \{3,4\}$ and let C be a smooth curve of genus $g\geq k+1$ , with $\big (g,\operatorname {\mathrm {gon}}(C)\big )\neq (k+1,k)$ . In light of (3.3), we need to prove that $\operatorname {\mathrm {cov.gon}}\big (C^{(k)}\big )\geq \operatorname {\mathrm {gon}}(C)$ .

Since $C^{(k)}$ is a variety of general type, it is not covered by rational curves. It follows that $\operatorname {\mathrm {cov.gon}}\big (C^{(k)}\big )\geq 2$ , and the assertion holds when C is a hyperelliptic curve.

We assume hereafter that C is non-hyperelliptic and, in order to simplify notation, we identify the curve with its canonical model $\phi (C)\subset \mathbb {P}^{g-1}$ .

Aiming for a contradiction, we suppose that there exists a covering family $\mathcal {E}\stackrel {\pi }{\longrightarrow } T$ of d-gonal curves, with $d<\operatorname {\mathrm {gon}}(C)$ . Therefore, as $\operatorname {\mathrm {gon}}(C)\leq \left\lfloor \frac {g+3}{2}\right\rfloor $ (see, for example, [Reference Arbarello, Cornalba, Griffiths and Harris1, Theorem V.1.1]), we have

(3.6) $$ \begin{align} d\leq \operatorname{\mathrm{gon}}(C)-1\leq \left\lfloor\frac{g+1}{2}\right\rfloor \quad\text{and}\quad g\geq 2d-1. \end{align} $$

Using notation as in Definition 3.1 and (3.1), we consider a general point $(t,y)\in T\times \mathbb {P}^1$ and its fiber $\varphi _t^{-1}(t,y)=\left\{x_1,\ldots ,x_d\right\}\subset E_t$ via the d-gonal map $\varphi _t\colon E_t \longrightarrow \{t\}\times \mathbb {P}^1$ . Moreover, for any $i=1,\dots ,d$ , we define $P_i:=f_t(x_i)\in C^{(k)}$ and we set

(3.7) $$ \begin{align} P_1=p_1+\dots+p_{k} \quad P_2=p_{k+1}+\dots+p_{2k} \quad \dots\, \quad P_d=p_{(d-1)k+1}+\dots+p_{dk}. \end{align} $$

We point out that, since $(t,y)\in T\times \mathbb {P}^1$ is general, the points $P_i\in C^{(k)}$ are distinct, and they lie outside the diagonal divisor of $C^{(k)}$ (i.e. for any $i=1,\dots ,d$ , the points $p_{(i-1)k+1},\dots ,p_{ik}\in C$ are distinct). Moreover, for $i=1,\dots ,d$ , the linear space

(3.8) $$ \begin{align} \Lambda_i:=\operatorname{\mathrm{Span}}\left(p_{(i-1)k+1},\dots,p_{ik}\right)\subset \mathbb{P}^{g-1} \end{align} $$

spanned by $\operatorname {\mathrm {Supp}} (P_i)$ has dimension $k-1$ , and it is parameterized by $\gamma (P_i)\in \mathbb {G}(k-1,g-1)$ . By Theorem 3.2, the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d$ are in special position with respect to $(g-1-k)$ -planes. In particular, the dimension of the linear span of the points $p_1,\dots ,p_{dk}$ is governed by Corollary 2.6.

Let us consider the effective divisor $D=D_{(t,y)}\in \operatorname {Div}(C)$ given by

(3.9) $$ \begin{align} D:=p_1+\dots +p_{dk}=\sum_{j=1}^h n_jq_j, \end{align} $$

where the points $q_j\in \{p_1,\dots ,p_{dk}\}$ are assumed to be distinct and $n_j=\operatorname {\mathrm {mult}}_{q_j}(D)$ . We now distinguish some cases depending on the values of the integers $n_j$ .

Case A: suppose that $n_j=1$ for any $j=1,\dots ,h$ – that is, the points $p_1,\dots ,p_{dk}$ are all distinct.

Case A.1: assume in addition that $d=2$ (i.e., the map $\varphi _t\colon E_t \longrightarrow \{t\}\times \mathbb {P}^1$ is hyperelliptic). Since the $(k-1)$ -planes $\Lambda _1$ and $\Lambda _2$ are $\operatorname {\mathrm {SP}}(g-1-k)$ , they coincide – that is, $\gamma (P_1)=\gamma (P_2)$ (cf. Remark 2.3). In particular, $\dim \operatorname {\mathrm {Span}}(p_1,\dots ,p_{2k})=k-1$ , and the geometric version of the Riemann–Roch theorem (see [Reference Arbarello, Cornalba, Griffiths and Harris1, p. 12]) yields

(3.10) $$ \begin{align} \dim |D|=\deg D-1-\dim\operatorname{\mathrm{Span}}(p_1,\dots,p_{2k})=2k-1-(k-1)=k=\frac{\deg D}{2}. \end{align} $$

Since $D\neq 0$ and C is assumed to be non-hyperelliptic, Clifford’s theorem (cf. [Reference Arbarello, Cornalba, Griffiths and Harris1, p. 107]) assures that D is a canonical divisor of C, and hence, $g=k+1$ . We note that for $k=3$ , we obtain $g=4$ and $2=d<\operatorname {\mathrm {gon}}(C)\leq 3$ , but the case $\big (g,\operatorname {\mathrm {gon}}(C)\big )= (4,3)$ is excluded by assumption.

Thus, we only need to consider the case where $k=4$ and $g=5$ . Let $\iota \colon C^{(4)}\dashrightarrow C^{(4)}$ be the (rational) involution $y_1+\dots +y_4 \longmapsto \operatorname {\mathrm {Span}}(y_1,\dots ,y_4)|_C-(y_1+\dots +y_4)$ sending a general point to the residual of the canonical divisor $\operatorname {\mathrm {Span}}(y_1,\dots ,y_4)|_C\in \operatorname {Div}(C)$ cut out on C by the hyperplane $\operatorname {\mathrm {Span}}(y_1,\dots ,y_4)\subset \mathbb {P}^4$ . Since $\Lambda _1=\operatorname {\mathrm {Span}}(p_1,\dots ,p_4)$ is a hyperplane and $D=P_1+P_2$ is a canonical divisor, we deduce that $\operatorname {\mathrm {Span}}(p_1,\dots ,p_4)|_C=D$ , and hence, $\iota (P_1)=D-P_1=P_2$ . Therefore, as $\varphi _t^{-1}(t,y)=\left\{x_1,x_2\right\}\subset E_t$ is a general fiber of the hyperelliptic map and $P_i=f(x_i)$ , the restriction of $\iota $ to the curve $f(E_t)\subset C^{(4)}$ is the automorphism induced by the hyperelliptic involution of $E_t$ .

By Riemann’s theorem (see, for example) [Reference Arbarello, Cornalba, Griffiths and Harris1, p. 27]), the composition of the Abel-Jacobi map with a suitable translation gives a morphism $v\colon C^{(4)}\longrightarrow J(C)$ , which maps birationally to the theta divisor $\Theta \subset J(C)$ . Furthermore, the $(-1)$ –involution on the Jacobian variety $J(C)$ restricts to an involution on $\Theta $ , which satisfies $-v(P)=v(K_C-P)$ for any canonical divisor $K_C$ on C such that $K_C-P\geq 0$ (cf. [Reference Arbarello, Cornalba, Griffiths and Harris1, p. 27]). Since for general $P\in C^{(4)}$ the divisor $P+\iota (P)$ is canonical, we obtain $-v(P)=v(\iota (P))$ – that is, we have a commutative diagram

Let us consider the quotient map $\psi \colon \Theta \longrightarrow \Theta /\langle -1\rangle \subset \operatorname {\mathrm {Kum}} (C)$ , where $\operatorname {\mathrm {Kum}}(C)=J(C)/\langle -1\rangle $ is the Kummer variety. Since $\iota $ acts on $f(E_t)\subset C^{(4)}$ as the hyperelliptic involution, the fourfold $\Theta /\langle -1\rangle $ is covered by rational curves $(\psi \circ v)\left(f(E_t)\right)$ .

We recall that the morphisms $C^{(4)}\stackrel {v}{\longrightarrow }J(C)\longrightarrow \operatorname {\mathrm {Kum}}(C)$ induce a chain of isomorphisms on $(4,0)$ -forms

(3.11) $$ \begin{align} H^{4,0}\left(C^{(4)}\right)\cong H^{4,0}\left(J(C)\right)\cong \bigwedge^4 H^0(C,\omega_C) \cong H^{4,0}\left(\operatorname{\mathrm{Kum}}(C)\right), \end{align} $$

where the last isomorphism occurs because the action of the $(-1)$ –involution on $H^{1,0}\left(J(C)\right)\cong H^0(C,\omega _C)$ is anti-invariant, so the action on $\bigwedge ^4 H^0(C,\omega _C)$ is invariant (cf. e.g., [Reference Spanier24]). Let Y be a desingularization of $ \Theta /\langle -1\rangle $ , and let X be a suitable blow-up of $C^{(4)}$ such that we have a commutative diagram

(3.12)

By (3.11) and (3.12), we obtain a morphism $X\longrightarrow \operatorname {\mathrm {Kum}}(C)$ such that the map induced on $(4,0)$ -forms is an isomorphism, which factors through $H^{4,0}\left(Y\right)$ . In particular, $H^{4,0}\left(C^{(4)}\right) \cong H^{4,0}\left(X\right)= \eta ^* H^{4,0}\left(Y\right)$ . Thus, the canonical map of X factors through the canonical map of Y. Moreover, these maps are generically finite because the canonical map of $C^{(4)}$ is. Hence, Y is a variety of general type, and it cannot be covered by rational curves. So we get a contradiction, as Y is birational to $\Theta /\langle -1\rangle $ , which is covered by rational curves.

Case A.2: assume that $d\geq 3$ . Consider the sequence $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{g-1}$ of $(k-1)$ -planes satisfying $\operatorname {\mathrm {SP}}(g-1-k)$ , together with a partition of the set of indices $\left\{1,\dots ,d\right\}$ such that each part $\{i_1,\dots ,i_t\}$ corresponds to an indecomposable sequence $\Lambda _{i_1},\dots ,\Lambda _{i_t}$ of $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(g-1-k)$ , as in the assumption of Corollary 2.6.

We point out that each part consists of at least $3$ elements. Indeed, if we had two planes $\Lambda _{i_1}$ and $\Lambda _{i_2}$ in special position, we could argue as in Case A.1 and we would deduce that $g=k+1$ . However, this is impossible, because we know that $3\leq d\leq \operatorname {\mathrm {gon}}(C)-1\leq \left\lfloor \frac {g+1}{2}\right\rfloor $ , with $k\in \{3,4\}$ and $\big (k,g,\operatorname {\mathrm {gon}}(C)\big )\neq (4,5,4)$ by assumption.

Let m be the number of parts. Hence, $m\leq \left\lfloor \frac {d}{3}\right\rfloor $ and Corollary 2.6 implies

$$ \begin{align*} \dim\operatorname{\mathrm{Span}}(p_1,\dots,p_{dk}) & =\dim\operatorname{\mathrm{Span}}\left(\Lambda_1,\dots,\Lambda_d\right)\leq d+k-3 +(m-1)(k-2) \notag \\ &\leq d+k-3+\left(\left\lfloor\frac{d}{3}\right\rfloor-1\right)(k-2). \end{align*} $$

By the geometric version of the Riemann–Roch theorem, we obtain

(3.13) $$ \begin{align} \dim |D|&=\deg D-1-\dim\operatorname{\mathrm{Span}}(p_1,\dots,p_{dk})\geq dk-1-\left(d+k-3+\left(\left\lfloor\frac{d}{3}\right\rfloor-1\right)(k-2)\right) \notag\\ & = (k-1)d -(k-2)\left\lfloor\frac{d}{3}\right\rfloor>\frac{kd}{2}=\frac{\deg D}{2}, \end{align} $$

where the last inequality holds because $k\geq 3$ . Hence, Clifford’s theorem assures that $\deg D\geq 2g$ , so that $\dim |D|=\deg D-g=kd-g$ by the Riemann–Roch theorem. Combining this fact and (3.13), we obtain $kd-g\geq (k-1)d -(k-2)\frac {d}{3}$ , which gives $3g\leq d(k+1)$ . Then we conclude by (3.6) that $(k,g,d)=(4,5,3)$ and, in particular, $\operatorname {\mathrm {gon}}(C)=4$ . However, $\big (k,g,\operatorname {\mathrm {gon}}(C)\big )\neq (4,5,4)$ by assumption, so we get a contradiction.

Case B: suppose that the points $p_1,\dots ,p_{kd}$ are not distinct (i.e., the integers $n_j$ are not all equal to  $1$ ). Let us denote by

(3.14) $$ \begin{align} N_{\alpha}:=\left\{\left. q_j\in\operatorname{\mathrm{Supp}}(D)\right|n_j=\alpha\right\} \end{align} $$

the set of the points supporting D such that $\operatorname {\mathrm {mult}}_{q_j}D=\alpha $ . Given some $\alpha \geq 1$ such that $N_{\alpha }\neq \emptyset $ , we may assume $N_{\alpha }=\{q_1,\dots ,q_s\}$ , with $s\leq \left\lfloor \frac {\deg D}{\alpha }\right\rfloor =\left\lfloor \frac {kd}{\alpha }\right\rfloor $ .

Since $T\times \mathbb {P}^1$ is irreducible, there exists a suitable open subset $U\subset T\times \mathbb {P}^1$ such that, as we vary $(t,y)\in U$ , the multiplicities $n_j$ of the points supporting the divisor $D=D_{(t,y)}$ given by (3.9) do not vary (i.e., the cardinality of each $N_{\alpha }$ is constant on U). Thus, we can define a rational map $\xi _{\alpha }\colon T\times \mathbb {P}^1\dashrightarrow C^{(s)}$ sending a general point $(t,y)\in U$ to the effective divisor $Q:=q_1+\dots +q_s\in C^{(s)}$ such that $N_{\alpha }=\{q_1,\dots ,q_s\}$ .

For a general $t\in T$ , let $\xi _{\alpha ,t}\colon \{t\}\times \mathbb {P}^1\dashrightarrow C^{(s)}$ be the restriction of $\xi _{\alpha }$ to $\{t\}\times \mathbb {P}^1$ , and let us consider the composition with the Abel–Jacobi map u,

As the Jacobian variety $J(C)$ does not contain rational curves, the map $u\circ \xi _{\alpha ,t}$ must be constant.

Case B.1: suppose that for any $\alpha $ such that $N_{\alpha }\neq \emptyset $ , the map $\xi _{\alpha ,t}\colon \{t\}\times \mathbb {P}^1\dashrightarrow C^{(s)}$ is non-constant. It follows from Abel’s theorem that $|q_1+\dots +q_s|$ is a $\mathfrak {g}^r_s$ containing $\xi _{\alpha ,t}\left(\{t\}\times {\mathbb {P}}^1\right)$ , so that $r\geq 1$ and $s=|N_{\alpha }|\geq \operatorname {\mathrm {gon}}(C)$ . Hence, $\alpha \leq k-1$ because, otherwise, we would have $s\leq \left\lfloor \frac {kd}{\alpha }\right\rfloor \leq d<\operatorname {\mathrm {gon}}(C)$ .

We also point out that $\alpha |N_{\alpha }|=\alpha s$ must be a multiple of d. To see this fact, let $i=1,\dots ,d$ and consider the points $P_i=p_{(i-1)k+1}+\dots +p_{ik}$ in (3.7). Then the number of points $p_j\in N_{\alpha }\cap \operatorname {\mathrm {Supp}}(P_i)$ does not depend on i; otherwise, by varying $(t,y)\in \{t\}\times \mathbb {P}^1$ , the points $P_i$ would describe different irreducible components of $E_t$ , but such a curve is irreducible. Hence, $\alpha |N_{\alpha }|$ – which is the number of points among $p_1,\dots ,p_{dk}$ contained in $N_{\alpha }$ – must be a multiple of d.

We claim that $N_1=\emptyset $ and there exists a unique $2\leq \alpha \leq k-1$ such that $N_{\alpha }\neq \emptyset $ , with in particular $|N_{\alpha }|=\frac {kd}{\alpha }$ . Indeed, if $k=3$ , we have that $3d=\deg (D)=|N_1|+2|N_2|$ . Then there exists $j\in \{1,2\}$ such that $|N_j|\leq d$ , but we observed that if $N_j\neq \emptyset $ , then $|N_j|\geq \operatorname {\mathrm {gon}}(C)>d$ . By assumption of Case B, we have $N_2\neq \emptyset $ , so we conclude that $N_1$ is empty and $|N_2|=\frac {3d}{2}$ . Similarly, if $k=4$ and $N_3\neq \emptyset $ , then $|N_3|>d$ and $4d=\deg D=|N_1|+2|N_2|+3|N_3|$ , so that $N_1=N_2=\emptyset $ and $|N_3|=\frac {4d}{3}$ . Finally, if $k=4$ and $N_3= \emptyset $ , then $N_2\neq \emptyset $ and $4d=\deg D=|N_1|+2|N_2|$ . Since $|N_2|>d$ and $2|N_2|$ is a multiple of d, we conclude that $N_1=\emptyset $ and $|N_2|=2d$ .

Accordingly, we have to discuss the following three cases:

1. (i) $k=3$ and $N_2\neq \emptyset $ ;

2. (ii) $k=4$ and $N_3\neq \emptyset $ ;

3. (iii) $k=4$ and $N_2\neq \emptyset $ .

To this aim, we use the following fact.

(i) If $k=3$ and $N_2\neq \emptyset $ , the divisor D has the form $D=2(q_1+\dots +q_s)$ , with $s=\frac {3d}{2}$ . We note that $d\neq 2$ ; otherwise, we would have $P_1=P_2=q_1+q_2+q_3$ . As s is an integer, we may set $d=2c$ and $s=3c$ , for some integer $c\geq 2$ .

Suppose that $c\neq 2$ . Given a point $x\in \{q_1,\dots ,q_s\}\subset C$ , it belongs to the support of two points of $C^{(3)}$ , say $P_1$ and $P_2$ . Then $\operatorname {\mathrm {Supp}}(P_1)\cup \operatorname {\mathrm {Supp}}(P_2)$ consists of at most 5 distinct points. As $s=3c\geq 9$ , there exists a point $y\in \{q_1,\dots ,q_s\}\setminus \{x\}$ belonging to the support of two other points of $C^{(3)}$ , say $P_3$ and $P_4$ . Now, we can choose $n\leq d-5$ distinct points $a_1,\dots ,a_n\in \{q_1,\dots ,q_s\}\setminus \{x,y\}$ , such that the set $A:=\left\{x,y,a_1,\dots ,a_n\right\}$ contains a point in the support of $P_i$ for any $i=1,\dots ,d-1$ and $A\cap \operatorname {\mathrm {Supp}}(P_d)=\emptyset $ (given x and y, it suffices to choose – at most – one point in the support of any $P_5,\ldots , P_{d-1}$ , avoiding the points of $\operatorname {\mathrm {Supp}}(P_d)$ ). Thus, the set A satisfies the assumption of Claim 3.6, but $|A|\leq d-3<\operatorname {\mathrm {gon}}(C)-3$ , a contradiction.

However, suppose that $c=2$ . Since the curve $E_t$ is irreducible, the points $P_i$ in (3.7) must be indistinguishable. Then it is easy to check that, up to reordering indices, there are only two admissible configurations:

1. 1. $P_1=q_1+q_2+q_3$ , $P_2=q_1+q_4+q_5$ , $P_3=q_2+q_4+q_6$ , $P_4=q_3+q_5+q_6$ ;

2. 2. $P_1=q_1+q_2+q_3$ , $P_2=q_1+q_2+q_4$ , $P_3=q_5+q_6+q_3$ , $P_4=q_5+q_6+q_4$ .

In both cases, we have that the line $\operatorname {\mathrm {Span}}(q_4,q_5)$ meets the planes $\Lambda _2$ , $\Lambda _3$ , and $\Lambda _4$ . Since $\Lambda _1,\dots ,\Lambda _4$ are $\operatorname {\mathrm {SP}}(g-4)$ , we deduce that $\Lambda _1\cap \operatorname {\mathrm {Span}}(q_4,q_5)\neq \emptyset $ , so that $\operatorname {\mathrm {Span}}(\Lambda _1,q_4,q_5)= \operatorname {\mathrm {Span}}(q_1,\dots ,q_5)\cong \mathbb {P}^3$ . Analogously, by considering the line $\operatorname {\mathrm {Span}}(q_4,q_6)$ , we deduce that $\operatorname {\mathrm {Span}}(q_1,\dots ,q_4,q_6)\cong \mathbb {P}^3$ . Therefore, either $\operatorname {\mathrm {Span}}(q_1,\dots ,q_4)\cong \mathbb {P}^2$ or $\operatorname {\mathrm {Span}}(q_1,\dots ,q_6)\cong \mathbb {P}^3$ . In the former case, $|q_1+\dots +q_4|$ is a $\mathfrak {g}^1_4$ on C, which is impossible as $\operatorname {\mathrm {gon}}(C)>d=2c=4$ .

In the latter case, we set $L:=q_1+\dots +q_6$ , and $|L|$ turns out to be a $\mathfrak {g}^2_6$ on C. Since $\operatorname {\mathrm {gon}}(C)>4$ , $\dim |L-p-q|=0$ for any $p,q\in C$ , so that $|L|$ is very ample. Hence, $|L|$ is the unique $\mathfrak {g}^2_6$ on C (see, for example, [Reference Ciliberto10, Teorema 3.14]). Therefore, by retracing our construction, we have that if $P'\in C^{(3)}$ is a general point, then there exists an effective divisor $L'=q^{\prime }_1+\dots +q^{\prime }_s$ such that $|L'|=|L|$ and $P'\leq L'$ . Thus, we obtain a contradiction as $C^{(3)}$ is a threefold, but $\dim |L|=2$ .

In conclusion, case (i) does not occur.

(ii) If $k=4$ and $N_3\neq \emptyset $ , we have $D=3(q_1+\dots +q_s)$ , with $s=\frac {4d}{3}$ . In particular, each $q_j$ belongs to the support of three distinct points among $P_1,\dots ,P_d\in C^{(4)}$ . We argue as above, and we set $d=3c$ and $s=4c$ , for some integer $c\geq 2$ (if c equaled 1, then we would have $P_1=P_2=P_3=q_1+\dots +q_4$ ).

Suppose that $c\neq 2$ . Given a point $x\in \{q_1,\dots ,q_s\}$ , it belongs to the support of three points of $C^{(4)}$ , say $P_1,P_2,P_3$ . Then $\operatorname {\mathrm {Supp}}(P_1)\cup \operatorname {\mathrm {Supp}}(P_2)\cup \operatorname {\mathrm {Supp}}(P_3)$ consists of at most 10 distinct points. As $s=4c\geq 12$ , there exists a point $y\in \{q_1,\dots ,q_s\}\setminus \{x\}$ belonging to the support of three other points of $C^{(4)}$ , say $P_4,P_5,P_6$ . Hence, we can choose $n\leq d-7$ distinct points $a_1,\dots ,a_n\in \{q_1,\dots ,q_s\}\setminus \{x,y\}$ , such that the set $A:=\left\{x,y,a_1,\dots ,a_n\right\}$ contains a point in the support of $P_i$ for any $i=1,\dots ,d-1$ and $A\cap \operatorname {\mathrm {Supp}}(P_d)=\emptyset $ . Therefore, A satisfies the assumption of Claim 3.6, with $|A|\leq d-5<\operatorname {\mathrm {gon}}(C)-4$ , a contradiction.

If instead $c=2$ , we have $d=6$ and $s=8$ . In this setting, we can choose two points $x,y\in \{q_1,\dots ,q_8\}$ such that $x+y\leq P_i$ for only one $i\in \{1,\dots ,d\}$ . To see this fact, consider the point $q_1$ and, without loss of generality, suppose that $q_1$ belongs to the support of $P_1,P_2,P_3$ and $\operatorname {\mathrm {Supp}}(P_1+P_2+P_3)=\left\{q_1,\dots ,q_r\right\}$ for some $5\leq r\leq 8$ . If there exists $j\in \{2,\dots ,r\}$ such that $q_j$ lies on the support of only one point among $P_1,P_2,P_3$ , then $q_1+q_j\leq P_i$ for only one $i\in \{1,2,3\}$ , and we may set $(x,y)=(q_1,q_j)$ . If instead any $q_j$ belongs to the support of two points among $P_1,P_2,P_3$ , then it is easy to check that $\operatorname {\mathrm {Supp}}(P_1+P_2+P_3)=\left\{q_1,\dots ,q_5\right\}$ because the divisor $P:=P_1+P_2+P_3$ has degree 12, where $\operatorname {\mathrm {mult}}_{p_1}P=3$ and $\operatorname {\mathrm {mult}}_{p_j}P\geq 2$ for any $j=2,\dots , r$ . Then we consider the point $q_6$ , and, without loss of generality, we suppose that $q_6\in \operatorname {\mathrm {Supp}}(P_4)$ . As $s=8$ , $\operatorname {\mathrm {Supp}}(P_4)$ contains at least an element of $\left\{q_2,\dots ,q_5\right\}$ , say $q_2$ . Since $q_2$ belongs also to the support of two points among $P_1,P_2,P_3$ , we conclude that $q_2+q_6\leq P_i$ only for $i=4$ , and we may set $(x,y)=(q_2,q_6)$ .

We point out that $A:=\{x,y\}$ intersects all the sets $\operatorname {\mathrm {Supp}}(P_1),\dots ,\operatorname {\mathrm {Supp}}(P_d)$ but one because x belongs to the support $P_i$ and two other points of $C^{(4)}$ – say $P_2$ and $P_3$ – and $y\in \operatorname {\mathrm {Supp}}(P_i)$ belongs to the support of two points other than $P_2$ and $P_3$ . Hence, A satisfies the assumption of Claim 3.6, with $|A|=2= d-4<\operatorname {\mathrm {gon}}(C)-4$ , a contradiction.

It follows that case (ii) does not occur.

(iii) Finally, let $k=4$ and $N_2\neq \emptyset $ . Then the divisor D has the form $D=2(q_1+\dots +q_s)$ , with $s=2d$ . In particular, each $q_j$ belongs to the support of two distinct points among $P_1,\dots ,P_d\in C^{(4)}$ . We note that $d\neq 2$ ; otherwise, we would have $P_1=P_2=q_1+\dots +q_4$ .

If $d\geq 8$ , we argue as in cases (i) and (ii). Namely, we consider a point $x\in \{q_1,\dots ,q_s\}$ belonging to the support of two points of $C^{(4)}$ , say $P_1$ and $P_2$ , where $\operatorname {\mathrm {Supp}}(P_1)\cup \operatorname {\mathrm {Supp}}(P_2)$ consists of at most 7 elements of $\{q_1,\dots ,q_s\}$ . Moreover, as $s=2d\geq 16$ , up to reorder indices, we may consider two distinct points $y,z\in \{q_1,\dots ,q_s\}\setminus \{x\}$ such that $y\in \operatorname {\mathrm {Supp}}(P_3)\cup \operatorname {\mathrm {Supp}}(P_4)$ and $z\in \operatorname {\mathrm {Supp}}(P_5)\cup \operatorname {\mathrm {Supp}}(P_6)$ . Then we can choose $n\leq d-7$ distinct points $a_1,\dots ,a_n\in \{q_1,\dots ,q_s\}\setminus \{x,y,z\}$ , such that the set $A:=\left\{x,y,z,a_1,\dots ,a_n\right\}$ contains a point in the support of $P_i$ for any $i=1,\dots ,d-1$ and $A\cap \operatorname {\mathrm {Supp}}(P_d)=\emptyset $ . Thus, A satisfies the assumption of Claim 3.6, but $|A|\leq d-4<\operatorname {\mathrm {gon}}(C)-4$ , a contradiction.

If $3\leq d\leq 7$ , we consider the divisor $L:=D_{\text {red}}=q_1+\dots +q_{2d}\in \operatorname {Div}(C)$ . When $d= 3$ , we have that $\operatorname {\mathrm {Span}}(q_1,\dots ,q_{6})$ has dimension 3. Indeed, any point $q_j$ lies on two $3$ -planes among $\Lambda _1,\Lambda _2,\Lambda _3$ , and by special position property, it must belong to the third one too. Thus, $q_1,\dots ,q_{6}$ span a $3$ -plane and L is a complete $\mathfrak {g}^2_6$ . Hence, $|L-q_6|$ is a complete $\mathfrak {g}^r_5$ , where $1\leq r\leq 2$ and $L-q_6=P_1+q_5$ . We note further that $\dim W^1_5\leq 2$ and $\dim W^2_5\leq 0$ by Martens’ theorem (see [Reference Arbarello, Cornalba, Griffiths and Harris1, Theorem IV.5.1]). Then we argue similarly to case (i). Namely, by retracing our construction, we have that if $P'\in C^{(4)}$ is a general point, then there exists and effective divisor $L'=q^{\prime }_1+\dots +q^{\prime }_{5}$ such that $P'\leq L'$ and $|L'|$ is a complete $\mathfrak {g}^{r}_{5}$ , with $1\leq r\leq 2$ . Therefore, we get a contradiction since $\dim C^{(4)}=4$ , whereas the locus of divisors $L'\in C^{(5)}$ as above has dimension $\dim |L'|\leq 3$ .

As far as the remaining values of d are concerned, we need the following.