2.1 Moduli of vector bundles and the map $\theta $

Let C be a smooth genus g algebraic curve with genus $g \geq 2$ . Let us denote by $\text {Pic}^d(C)$ the Picard variety of degree d line bundles on C. The Jacobian of C is $\text {Jac}(C) = \text {Pic}^0(C)$ . As customary, we will denote by $\Theta \subset \text {Pic}^{g-1}(C)$ the Riemann theta divisor.

The Picard group $\text {Pic}(\mathcal {SU}_C(2))$ is isomorphic to $\mathbb {Z}$ , and it is generated by the determinant line bundle $\;\mathcal {L}$ [Reference Drezet and Narasimhan15]. For every $E \in \mathcal {SU}_C(2)$ , let us define the theta divisor

$$ \begin{align*}\theta(E) := \{ L \in \text{Pic}^{g-1}(C) \ | \ h^0(C, E \otimes L) \not = 0 \}.\end{align*} $$

In the rank 2 case, $\theta (E)$ is a divisor in the linear system $|2 \Theta | \cong \mathbb {P}^{2^g - 1}$ and $|2\Theta |$ is isomorphic to the linear system $|\mathcal {L}|^*$ [Reference Beauville, Narasimhan and Ramanan3]. It is well known that we can identify the map $\mathcal {SU}_C(2) \to |\mathcal {L}|^*$ with the Theta map

$$ \begin{align*} \theta : \mathcal{SU}_C(2) & \to |2 \Theta|;\\ E & \mapsto \theta(E). \end{align*} $$

In rank 2, the map $\theta $ is a finite morphism. If C is not hyperelliptic, $\theta $ is known to be an embedding [Reference Brivio and Verra9, Reference van Geemen and Izadi30]. This is also the case in genus 2, where $\theta $ is an isomorphism onto $\mathbb {P}^3$ [Reference Narasimhan and Ramanan25]. If C is hyperelliptic of genus $g \geq 3$ , we have that $\theta $ factors through the involution

$$ \begin{align*}E \mapsto i^*E\end{align*} $$

induced by the hyperelliptic involution i, embedding the quotient $\mathcal {SU}_C(2)/i^*$ into $|2 \Theta |$ [Reference Desale and Ramanan11]. An interesting explicit description of the image of the hyperelliptic theta map is given in [Reference Desale and Ramanan11].

2.2 The classifying maps

Let D be a general degree g effective divisor on C. Let us consider isomorphism classes of extensions

$$ \begin{align*} (e) \quad 0 \to \mathcal{O}(-D) \to E_e \to \mathcal{O}(D) \to 0. \end{align*} $$

These extensions are parametrized by the $(3g - 2)$ -dimensional projective space $\mathbb {P}_D^{3g-2} := \mathbb {P} \text {Ext}^1(\mathcal {O}(D), \mathcal {O}(-D)) = |K + 2D|^*$ , where K is the canonical divisor of C. The divisor $K + 2D$ is very ample and embeds C as a degree $4g - 2$ curve in $\mathbb {P}_D^{3g-2}$ . Let us define the rational surjective classifying map which sends a semi-stable extension class $(e)$ to the vector bundle $E_e$ . The composed map

can be described in terms of polynomial maps. From [Reference Bertram4, Theorem 2] we have an isomorphism

(1) $$ \begin{align} H^0(\mathcal{SU}_C(2), \mathcal{L}) \cong H^0(\mathbb{P}_D^{3g-2}, \mathcal{I}_C^{g-1}(g)), \end{align} $$

where $\mathcal {I}_C$ is the ideal sheaf of C. In particular, we have

Let us denote by $\text {Sec}^n(C)$ the variety of $(n + 1)$ -secant n-planes on C. We have that the singular locus of $\text {Sec}^{n+1}(C)$ is the secant variety $\text {Sec}^{n}(C)$ for every n. The linear system $|\mathcal {I}_C^{g-1}(g)|$ can be alternatively characterized as follows (see e.g., [Reference Alzati and Bolognesi1, Lemma 2.5]):

2.3 The exceptional fibers of the classifying map $f_D$

Since $\dim \mathcal {SU}_C(2) = 3g - 3$ , the generic fiber of $f_D$ has dimension one. The set of stable bundles for which $\dim (f_D^{-1}(E))> 1$ is a proper subset of $\mathcal {SU}_C(2)$ . For simplicity, let us define the “Serre dual” divisor $B := K - D$ with $\deg (B) = g-2$ . As in the previous paragraphs, the isomorphism classes of extensions

$$ \begin{align*} \quad 0 \to \mathcal{O}(-B) \to E \to \mathcal{O}(B) \to 0 \end{align*} $$

are classified by the projective space

$$ \begin{align*}\mathbb{P}_B^{3g-6} := \mathbb{P} \text{Ext}^1 (\mathcal{O}(B), \mathcal{O}(-B)) = |K + 2B|^*,\end{align*} $$

which is endowed with the rational classifying map defined the same way as $f_D$ .

Proposition 2.3. Let $E \in \mathcal {SU}_C(2)$ be a stable bundle. Then

$$ \begin{align*} \dim(f_D^{-1}(E)) \geq 2 \qquad \text{if and only if} \qquad E \in \overline{f_B(\mathbb{P}_B^{3g-6})}. \end{align*} $$

Proof Let E be a stable bundle. Then, by Riemann-Roch and Serre duality theorems, the dimension of $f^{-1}_D(E)$ is given by

$$ \begin{align*} h^0(C, E \otimes \mathcal{O}(D)) &= h^0(C, E \otimes \mathcal{O}(B)) + 2g - 2(g-1) \\ &= h^0(C, E \otimes \mathcal{O}(B)) + 1 \end{align*} $$

Thus, $\dim (f_D^{-1}(E))> 2$ if and only if there exists a nonzero sheaf morphism $\mathcal {O}(-B) \to E$ . This is equivalent to $E \in \overline {f_B(\mathbb {P}_B^{3g-6})}$ .

If $g> 2$ , the divisor $|K + 2B|$ embeds C as a degree $4g - 6$ curve in $\mathbb {P}_B^{3g-6}$ (recall that $\mathbb {P} \text {Ext}^1 (\mathcal {O}(B),\mathcal {O}(-B)) = |K + 2 B|^*$ ). Again by Theorem 2.1, the map $\varphi _B$ is given by the linear system $|\mathcal {I}_C^{g-3}(g-2)|$ . Moreover, by [Reference Pareschi and Popa27, Theorem 4.1] this linear system has projective dimension $\left ( \sum _{i = 0}^{g-2} {\binom {g}{i}} \right ) - 1$ .

Let us denote by $\mathbb {P}_c$ the linear span of $\theta (\overline {f_B(\mathbb {P}_B^{3g-6})})$ in $|2 \Theta |$ . Hence $\mathbb {P}_c$ has dimension $\left ( \sum _{i = 0}^{g-2} {\binom {g}{i}} \right ) - 1$ , and Proposition 2.3 also applies to $\varphi _D$ : the fibers of $\varphi _D$ with dimension $\geq 2$ are those over $\mathbb {P}_c$ .

4.1 Linear systems in $\mathbb {P}_N^{2g-2}$ contracting rational normal curves

Recall that the secant variety $\text {Sec}^{g - 2}(C)$ is the base locus for $\varphi _D$ (see Theorem 2.1 and Proposition 2.2). Hence we will distinguish the following two secant varieties in $\mathbb {P}_N^{2g-2}$ :

$$ \begin{align*} \text{Sec}^N &:= \text{Sec}^{g-2}(C) \cap \mathbb{P}_N^{2g-2} \ , \\ \text{Sec}^{g-2}(N) &:= \bigcup_{\substack{M \subset N \\ \#M = g - 1}} \text{span} \{M \} \ . \end{align*} $$

Note that, since the points of N are already in $\mathbb {P}_N^{2g-2}$ , we have the inclusion $\text {Sec}^{g-2}(N) \subset \text {Sec}^N$ . We will also need to consider the linear systems on $\mathbb {P}_N^{2g-2}$

$$ \begin{align*} \mathcal{I}_{\text{Sec}^N}(g), \quad \mathrm{and}\ \quad \mathcal{I}_{\text{Sec}^{g-2}(N)}(g) \end{align*} $$

of forms of degree g vanishing on the corresponding secant varieties. The previous inclusion of secant varieties implies that $\mathcal {I}_{\text {Sec}^N}(g)$ is in general a linear subsystem of $\mathcal {I}_{\text {Sec}^{g-2}(N)}(g)$ .

4.2 Moduli spaces of pointed rational curves

In this section, we will outline the relation between the restricted map $\varphi _{D,N}$ and the moduli spaces of pointed rational curves.

4.2.1 Two compactifications of $\mathcal {M}_{0,n}$

The moduli space $\mathcal {M}_{0,n}$ of ordered configurations of n distinct points on the projective line is not compact. We will consider two compactifications of $\mathcal {M}_{0,n}$ . The first one is the GIT quotient

$$ \begin{align*}\mathcal{M}_{0,n}^{\text{GIT}} := (\mathbb{P}^1)^n // \text{PGL}(2, \mathbb{C})\end{align*} $$

of $(\mathbb {P}^1)^n$ by the diagonal action of $G=\text {PGL}(2, \mathbb {C})$ for the natural G-linearization of the line bundle (see [Reference Dolgachev and Ortland12, Reference Dolgachev13]). The quotient $\mathcal {M}_{0,n}^{\text {GIT}}$ is naturally embedded in the projective space $\mathbb {P} ( H^0 ((\mathbb {P}^1)^n, L )^{G} )$ of invariant sections.

The second one is the Mumford–Knudsen compactification $\overline {\mathcal {M}}_{0,n}$ [Reference Knudsen20]. The points of $\overline {\mathcal {M}}_{0,n}$ represent isomorphism classes of stable curves. More details on these constructions can be found in [Reference Kapranov18] and [Reference Knudsen20].

Both $\mathcal {M}_{0,2g}^{\text {GIT}}$ and $\overline {\mathcal {M}}_{0,n}$ contain $\mathcal {M}_{0,n}$ as an open set. However, the Mumford–Knudsen compactification is finer on the boundary: there exists a contraction dominant morphism

$$ \begin{align*}c_n: \overline{\mathcal{M}}_{0,n} \to \mathcal{M}_{0,n}^{\text{GIT}}\end{align*} $$

contracting some components of the boundary of $\overline {\mathcal {M}}_{0,n}$ , that restricts to the identity on $\mathcal {M}_{0,n}$ [Reference Bolognesi7]. Moreover, we will denote by

$$ \begin{align*}\lambda_k: \overline{\mathcal{M}}_{0,n} \to \overline{\mathcal{M}}_{0,n-1},\end{align*} $$

for $k = 1,\ldots ,n$ , the forgetful morphism that forgets the labeling of the kth point.

4.2.4 The Cremona inversion

Let $e_1, \ldots e_{n - 1} \in \mathbb {P}^{n - 3}$ in general position. Without loss of generality, we may assume $e_k = [0: \cdots : 1 : \cdots : 0]$ for $k = 1, \ldots , n - 2$ ; and $e_{n - 1} = [1: \cdots : 1]$ . The Cremona inversion with respect to $e_{n-1}$ is the birational map

On $\mathbb {P}^2$ the Cremona inversion is given by the linear system of quadrics passing through $e_1,\ e_2$ , and $e_3$ , on $\mathbb {P}^3$ by the cubics that vanish on the six secant lines of $e_1,\ e_2,\ e_3$ , and $e_4$ , and so on. The Cremona inversion has the following property: any nondegenerate rational normal curve passing through the points $e_1, \ldots , e_{n - 1}$ is transformed into a line passing by the point $\text {Cr}_{n - 1}(e_{n - 1})$ . Let be the linear projection with center $\text {Cr}_{n - 1}(e_{n - 1})$ . From the previous property, we obtain that the composition $\tau _{n - 1} \circ \text {Cr}_{n - 1}$ contracts nondegenerate rational normal curves passing through $e_1, \ldots , e_{n - 1}$ .

Let $k \in \{1,\ldots ,n-1\}$ . It is straightforward to see that one can let $e_k$ play the role of $e_{n - 1}$ in the definition of $\text {Cr}_{n - 1}$ , and define similarly the Cremona inversion $\text {Cr}_k$ . Let be the linear projection with center $\text {Cr}_k(e_{k})$ .

Lemma 4.3. Let $e_1, \ldots e_{n - 1} \in \mathbb {P}^{n - 3}$ in general position. Then, the composition $\tau _k \circ \text {Cr}_k$ contracts the nondegenerate rational normal curves passing through $e_1, \ldots , e_{n-1}$ .

Let us denote $H_t$ , for $t\neq k$ , the hyperplane in $\mathbb {P}^{n-3}$ spanned the points $e_i$ , with $i\neq k,t.$ There are $n-2$ such hyperplanes and each one is contracted to a point by $\text {Cr}_k$ .

Proposition 4.4. [Reference Kapranov18]

Let $e_1, \ldots e_{n - 1} \in \mathbb {P}^{n - 3}$ in general position. Let $k \in \{1,\ldots ,n - 1\}$ . Then, the following diagram is commutative:

Here, the map $b_k$ is the Kapranov blow-up map centered in the images of the hyperplanes $H_t$ , for $1\leq t \leq n-1$ and $t \neq k$ , by $\tau _k \circ \text {Cr}_k$ .

Remark. We observe here a little subtlety. We only get here $n-1$ forgetful maps through Cremona transformations, because we are tacitly assuming that Kapranov’s blow-up construction of $\overline {\mathcal {M}}_{0,n}$ labels with integers from 1 to $n-1$ the points $e_1,\dots ,e_{n-1}$ of the projective base of $\mathbb {P}^{n-3}$ , and labels as n the last point (which is free to move inside the $\mathbb {P}^{n-3}$ , which is a birational model of $\overline {\mathcal {M}}_{0,n}$ ). This is due to this small asymmetric aspect of Kapranov’s construction, but it is clear that one could assume that the last, freely moving point is labeled with any $k\in \{ 1, \dots , n-1\}$ , and obtain other forgetful maps.

4.2.5 Rationalizations of $\mathcal {M}^{\text {GIT}}_{0,2g}$

Let $g \geq 3$ , and let $e_1, \ldots , e_{2g - 1} \in \mathbb {P}^{2g - 3}$ in general position. Let $\Omega $ be the linear system of $(g-1)$ -forms in $\mathbb {P}^{2g-3}$ vanishing with multiplicity $g-2$ in $e_1, \ldots , e_{2g - 1} \in \mathbb {P}^{2g - 3}$ .

Theorem 4.5. ([Reference Kumar21])

Let $g \geq 3$ , and let $e_1, \ldots , e_{2g - 1} \in \mathbb {P}^{2g - 3}$ be in general position. Then, the rational map

induced by the linear system $\Omega $ maps $\mathbb {P}^{2g-3}$ birationally onto $\mathcal {M}^{\text {GIT}}_{0,2g}$ .

We also observe that the contraction map $c_{2n}$ can also be described in terms of Kumar’s linear system $\Omega $ :

Lemma 4.6. Let $g \geq 3$ , and let $e_1, \ldots , e_{2g - 1} \in \mathbb {P}^{2g - 3}$ in general position. Then, the following diagram is commutative:

Here, the map b is the Kapranov blow-up map centered in $p_1, \ldots , p_{2g - 1}$ .

Let $e_0 \in \mathbb {P}^{2g - 3}$ such that $w = i_{\Omega }(e_0)$ lies in the open set $\mathcal {M}_{0,2g} \subset \mathcal {M}^{\text {GIT}}_{0,2g}$ . The point w can represent a hyperelliptic genus $(g - 1)$ curve $C_w$ (namely the double cover of $\mathbb {P}^1$ ramifying in the $2g$ marked points) together with an ordering of the Weierstrass points. Let $\mathcal {SU}_{C_w}(2)^{inv}$ be the moduli space of rank 2 semistable vector bundles with trivial determinant over the curve $C_w$ , that are invariant with respect to the hyperelliptic involution.

Consider the partial linear subsystem $\Lambda $ of $\Omega $ consisting of the $(g - 1)$ -forms in $\mathbb {P}^{2g-3}$ vanishing with multiplicity $g - 2$ in all the points $e_0, e_1, \ldots , e_{2g - 1}$ . Let be the rational projection induced by the linear system $\Lambda $ .

Theorem 4.7. ([Reference Kumar21])

Let $g \geq 3$ , and let $e_1, \ldots , e_{2g - 1} \in \mathbb {P}^{2g - 3}$ be $2g - 1$ points in general position. Let $e_0 \in \mathbb {P}^{2g - 3}$ such that $w = i_{\Omega }(e_0)$ lies in the open set $\mathcal {M}_{0,2g} \subset \mathcal {M}^{\text {GIT}}_{0,2g}$ . Then, the map $\kappa $ induced by the linear system $\Lambda $ is a degree 2 osculating projection onto a connected component of the moduli space $\mathcal {SU}_{C_w}(2)^{inv}$ . Furthermore, the map $\kappa $ ramifies along the Kummer variety $\text {Kum}(C_w) \subset \mathcal {SU}_{C_w}(2)^{inv}$ .

In fact, one can talk of osculating projection (and this means $\mathcal {M}^{\text {GIT}}_{0,2g}$ is embedded in some known projective space) since $i_\Omega $ embeds $\mathcal {M}^{\text {GIT}}_{0,2g}$ in its natural projective space of invariants, defined by the democratic polarization with weights vector $(1,1,\dots ,1)$ (see also Section 5.2.3).

4.3 Forgetful linear systems and $\mathcal {M}^{\text {GIT}}_{0,2g}$

Let C be a smooth genus $g \geq 3$ curve (not necessarily hyperelliptic). Let D be a general degree g effective divisor on C. Let $N = p_1 + \cdots + p_{2g}$ be a general reduced divisor in the linear system $|2D|$ . Consider the span $\mathbb {P}_N^{2g-2}$ in $\mathbb {P}_D^{3g-2}$ of the $2g$ marked points $p_1, \ldots , p_{2g}$ .

We will now apply the discussion of Section 4.2 to the general points $p_1, \ldots , p_{2g}$ in the projective space $\mathbb {P}_N^{2g-2}$ , taking $n = 2g + 1$ in the notation therein. For every $k = 1, \ldots , 2g$ , we can compose Proposition 4.4 and Lemma 4.6 and get a commutative diagram

(3)

where $\Omega $ is the linear system of $(g-1)$ -forms in $\mathbb {P}^{2g-3}$ vanishing with multiplicity $g-2$ at the $2g - 1$ points $\tau _{k} \circ \text {Cr}_{k}(H_i)$ , with $i\neq k$ and $1 \leq i \leq 2g$ . Let us define the rational map

This is due to the fact that the linear system $|\mathcal {I}_{\text {Sec}^{g-2}(N)}(g)|$ is invariant with respect to the action of the symmetric group $\Sigma _{2g}$ that operates on $\mathbb {P}_N^{2g-2}$ by linear automorphisms. Let us put together the results of Lemma 4.3, Theorem 4.5 and Proposition 4.8 in the following Proposition:

This is why these maps where dubbed “forgetful linear systems.” In fact the rational normal curves passing through the $2g$ points make up the universal curve over an open subset of $\mathcal {M}_{0,2g}^{\text {GIT}}$ .

Remark. Since $\mathcal {R}$ is a linear subsystem of $|\mathcal {I}_{\text {Sec}^{g-2}(N)}(g)|$ by Lemma 4.1, we have that $\varphi _{D,N}$ factors through $h_N$ :

(4)

4.3.1 A comparison of base loci

For future use, we need to compare the locus $\text {Sec}^{g-2}(N)$ and the more intricate locus $\text {Sec}^N$ obtained by intersecting the base locus of $\varphi _D$ with $\mathbb {P}_N^{2g-2}$ . This section is devoted to this comparison.

By definition, the points in $\text {Sec}^N$ are given by the intersections $\langle L_{g-1} \rangle \cap \mathbb {P}_N^{2g-2} $ , where $L_{g-1}$ is an effective divisor of degree $g-1$ and $\langle L_{g-1} \rangle $ is its linear span in $\mathbb {P}_D^{3g-2}$ . If $L_{g-1}$ is contained in N, it is clear that $\langle L_{g-1} \rangle \subset \text {Sec}^{g-2}(N) \subset \mathbb {P}_N^{2g-2}$ .

Lemma 4.10. Let $L_{g-1}$ be an effective divisor on C of degree $g - 1$ , not contained in N. Then,

$$ \begin{align*} \langle L_{g-1} \rangle \cap \mathbb{P}_N^{2g-2} \not = \phi \quad \text{if and only if} \quad \dim|L_{g-1}| \geq 1. \end{align*} $$

Moreover, if the intersection is nonempty, we have that

$$ \begin{align*}\dim (\langle L_{g-1} \rangle \cap \mathbb{P}_N^{2g-2}) = \dim |L_{g-1}| - 1.\end{align*} $$

Proof First, let us suppose that $L_{g-1}$ and N have no points in common. The vector space $V := H^0(C, 2D + K - L_{g-1})$ is the annihilator of the span $\langle L_{g-1} \rangle $ in $\mathbb {P}_D^{3g-2}$ . By the Riemann-Roch theorem, we see that V has dimension $2g$ , hence

$$ \begin{align*}\dim \langle L_{g-1} \rangle = (3g - 2) - 2g = g - 2.\end{align*} $$

Let d be the dimension of the span $\langle L_{g-1}, N \rangle $ of the points of $L_{g-1}$ and N. Since the dimension of $\mathbb {P}_N^{2g-2} = \langle N \rangle $ is $2g-2$ , we have that $d \leq (g-2) + (2g - 2) + 1 = 3g-3$ , where the equality holds iff $\langle L_{g-1} \rangle \cap \mathbb {P}_N^{2g-2}$ is empty.

In particular, this intersection is nonempty iff $d \leq 3g -4$ . Since $\dim |K + 2D|^* = \dim \mathbb {P}_D^{3g-2} = 3g - 2$ , this is equivalent to the annihilator space

$$ \begin{align*}W := H^0(C, 2D + K - L_{g-1} - N) = H^0(C, K - L_{g-1})\end{align*} $$

being of dimension $\geq 2$ . By Riemann-Roch and Serre duality, we obtain that this condition is equivalent to $\dim |L_{g-1}| \geq 1$ .

More precisely, let us suppose that $\langle L_{g-1} \rangle \cap \mathbb {P}_N^{2g-2}$ is nonempty and let $e := \dim (\langle L_{g-1} \rangle \cap \mathbb {P}_N^{2g-2})$ . Then, we have that

$$ \begin{align*}d = 3g - 3 - (e + 1),\end{align*} $$

and the annihilator space W is of dimension $2 + e$ . Again by a Riemann-Roch computation, we conclude that $e = \dim |L_{g-1}| - 1$ .

Finally, if $L_{g-1}$ and N have some points in common, we have to count them only once when defining the vector space W to avoid requiring higher vanishing multiplicity to the sections.

From this Lemma, we conclude that $\text {Sec}^{g-2}(N)$ is a proper subset of $\text {Sec}^N$ if and only if there exists a divisor $L_{g-1}$ not contained in N with $\dim |L_{g-1}| \geq 1$ . By the Existence Theorem of Brill–Noether theory (see [Reference Arbarello, Cornalba, Griffiths and Harris2, Theorem 1.1, page 206]), this is possible only if $g \geq 4$ in the nonhyperelliptic case, whereas such a linear system may exist also when $g=3$ when C is hyperelliptic. We will discuss the first low genera cases in Section 7.

5.1 A rational normal curve coming from involution invariant secant lines

In the hyperelliptic case, we have an additional base locus for every $g \geq 3$ , which appears due to the hyperelliptic nature of the curve. This locus arises as follows. For each pair $\{ p, i(p) \}$ of involution-conjugate points in C, consider the secant line l in $\mathbb {P}_D^{3g-2}$ passing through the points p and $i(p)$ . Let $Q_p$ be the intersection of the line l with $\mathbb {P}_N^{2g-2}$ . Let us define $\Gamma \subset \mathbb {P}_N^{2g-2}$ as the locus of intersection points $Q_p$ when p moves inside C.

Figure 1 The situation in genus $3$ . The curves $\Gamma $ and C intersect along the divisor D, of degree 6. The secant lines l cutting out the hyperelliptic pencil define the curve $\Gamma $ .

Proof Let us start by showing that the intersection $Q_p$ is nonempty for every line $l=\overline {p,i(p)}$ , with $p\in C$ . Since $\dim |p + i(p)| = \dim |h| = 1$ , the intersection $l \cap \mathbb {P}_N^{2g-2}$ is nonempty by Lemma 4.10.

Let us show that this intersection is a point, that is that the line l is not contained in $\mathbb {P}_N^{2g-2}$ . Recall that $\mathbb {P}_D^{3g-2} = |2D + K|^*$ ; we will discuss three cases.

i) If the points p and $i(p)$ are both not contained in the divisor N, the vector space

$$ \begin{align*}V := H^0(C,2D + K - N - (p + i(p))) = H^0(C,2D + K - N - h)\end{align*} $$

is exactly the annihilator of the span $\langle l, \mathbb {P}_N^{2g-2} \rangle $ in $\mathbb {P}_D^{3g-2}$ . In particular, the codimension of $\langle l, \mathbb {P}_N^{2g-2} \rangle $ in $\mathbb {P}_D^{3g-2}$ is the dimension of V. By Riemann-Roch and Serre duality, we get that $\dim V = g - 2$ , thus $\dim \langle l, \mathbb {P}_N^{2g-2} \rangle = 3g - 2 - (g - 2) = 2g$ . This means that the intersection $l \cap \mathbb {P}_N^{2g-2}$ is a point.

ii) For the case $p \in N$ and $i(p) \not \in N$ , let us remark that the the annihilator of the span $\langle l, \mathbb {P}_N^{2g-2} \rangle $ is now the vector space $H^0(C,2D + K - N - i(p))$ . Since

$$ \begin{align*}h^0(C,2D + K - N - i(p)) < h^0(C,2D + K - N),\end{align*} $$

we conclude that the line l is not contained in $\mathbb {P}_N^{2g-2}$ .

iii) The case $\{p, i(p) \} \subset N$ is excluded by our genericity hypotheses on N. Hence we deduce that the locus $\Gamma $ is a curve in $\mathbb {P}_N^{2g-2}$ .

Let q be a point of $N \subset \mathbb {P}_N^{2g-2}$ . The line $\overline {q,i(q)}$ intersects the plane $\mathbb {P}_N^{2g-2}$ at q. Thus $\Gamma $ passes through all the points of N. Moreover, it is clear that N is the only intersection of $\Gamma $ and C, that is $\Gamma \cap C = N$ .

Let us prove now that $\Gamma $ is a rational normal curve. Since $\Gamma $ is defined by the hyperelliptic pencil, it is straightforward to see that $\Gamma $ is rational. Moreover, since the divisor D is general, the span of any subset of $2g - 1$ points of D is $\mathbb {P}_N^{2g-2}$ . Thus, it suffices to show that the degree of $\Gamma \subset \mathbb {P}_N^{2g-2}$ is precisely $2g - 2$ .

Let us set $N = q_1 + \cdots + q_{2g}$ with $q_1, \ldots , q_{2g} \in C$ . By the previous paragraph, $\Gamma $ passes through these $2g$ points. Let us consider a hyperplane H of $\mathbb {P}_N^{2g-2}$ spanned by $2g - 2$ points of N. Without loss of generality, we can suppose that these points are the first $2g - 2$ points $q_1, \ldots , q_{2g - 2}$ . To show that the degree of $\Gamma $ is $2g - 2$ , we have to show that the intersection of $\Gamma $ with H consists exactly only of these points.

Let l be the secant line $\overline {q,i(q)}$ , $q\in C$ . The intersection $l \cap H$ is empty if and only if the linear span $\langle l, H \rangle $ of l and H in $\mathbb {P}_D^{3g-2}$ is of maximal dimension $2g - 1$ , that is of codimension $g - 1$ in $\mathbb {P}_D^{3g-2}$ . Consider the divisors

$$ \begin{align*} D_H= q_1 + \cdots + q_{2g - 2} \qquad \mbox{and} \qquad D_l = q + i(q). \end{align*} $$

As before, if $\{q, i(q)\} \cap \{q_1, \ldots , q_{2g - 2} \}$ is empty, the vector space $W = H^0(C,2D + K - D_H - D_l)$ is the annihilator of the span $\langle l, H \rangle $ in $\mathbb {P}_D^{3g-2}$ . In particular, the codimension of $\langle l, H \rangle $ in $\mathbb {P}_D^{3g-2}$ is given by the dimension of W. Again by Riemann-Roch and Serre duality theorems, we can check that

$$ \begin{align*}\dim W = h^0(C,-2D + D_H + D_l) + g - 1.\end{align*} $$

Thus, the codimension of $\langle l, H \rangle $ in $\mathbb {P}_D^{3g-2}$ is greater than $g - 1$ if and only if $h^0(C,-2D + D_H + D_l)> 0$ . Since $\deg (-2D + D_H + D_l) = 0$ , this is equivalent to $-2D + D_H + D_l \sim 0$ . Since $N = q_1 + \cdots + q_{2g} \sim 2D$ , we have that

$$ \begin{align*} -2D + D_H + D_l \sim 0 &\iff q + i(q) \sim q_{2g - 1} + q_{2g} \\ &\iff h \sim q_{2g - 1} + q_{2g} \\ &\iff i(q_{2g - 1}) = q_{2g}. \end{align*} $$

By our genericity hypothesis on N, the last condition is not satisfied. Consequently, we conclude that the line l intersects the hyperplane H iff $\{q, i(q)\} \cap \{q_1, \ldots , q_{2g - 2} \}$ is nonempty, that is iff q or $i(q)$ is one of the $q_k$ for $k = 1, \ldots , 2g - 2$ . In particular,

$$ \begin{align*}\Gamma \cap H = \{q_1, \ldots, q_{2g - 2} \}\end{align*} $$

as we wanted to show.

Hence, the curve $\Gamma $ is contracted by the map $h_N$ to a point $w \in \mathcal {M}_{0,2g}^{\text {GIT}}$ by Proposition 4.9. The point w represents a hyperelliptic curve $C_{w}$ of genus $g-1$ together with an ordering of the Weierstrass points that correspond to the points of N on the rational curve $\Gamma $ .

5.2 The restriction of the theta map to $\mathcal {M}_{0,2g}^{\text {GIT}}$

Let us set once again $N = p_1 + \cdots + p_{2g}$ , a general divisor in the linear system $|2D|$ , and consider the span $\mathbb {P}_N^{2g-2}$ in $\mathbb {P}_D^{3g-2}$ of the $2g$ marked points $p_1, \ldots , p_{2g}$ .

In this section, we describe the interplay between $\theta $ and the linear systems presented in Section 4.

5.2.1 The factorization of the map $\varphi _D$

Recall that the base locus of the map $\varphi _D$ is the secant variety $\text {Sec}^{g-2}(C)$ by Proposition 2.2. As in [Reference Bertram4], one can construct a resolution $\widetilde {\varphi _D}$ of the map $\varphi _D$ via a sequence of blow-ups

along the secant varieties

$$ \begin{align*}C = \text{Sec}^0(C) \subset \text{Sec}^1(C) \subset \cdots \subset \text{Sec}^{g-1}(C) \subset \mathbb{P}_D^{3g-2}.\end{align*} $$

This chain of morphisms is defined inductively as follows: the center of the first blow-up $\text {Bl}_1$ is the curve $C = \text {Sec}^0(C)$ . For $k = 2, \ldots , g-1$ , the center of the blow-up $\text {Bl}_k$ is the strict transform of the secant variety $\text {Sec}^{k-1}(C)$ under the blow-up $\text {Bl}_{k-1}$ .

The map $\varphi _D$ is, by definition, the composition of the classifying map $f_D$ and the degree 2 map $\theta $ . Thus, the map $f_D$ lifts to a morphism $\widetilde {f_D}$ which makes the following diagram commute:

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5.2.2 Osculating projections

We recall here a generalization of linear projections that will allow us to describe the map p in higher genus. For a more complete reference, see for example [Reference Massarenti and Rischter24]. Let $X \subset \mathbb {P}^N$ be an integral projective variety of dimension n, and $p \in X$ a smooth point. Let

$$ \begin{align*} \phi: \mathcal{U} \subset \mathbb{C}^n &\longrightarrow \mathbb{C}^N \\ (t_1, \ldots, t_n) &\longmapsto \phi(t_1, \ldots, t_n) \end{align*} $$

be a local parametrization of X in a neighborhood of $p = \phi (0) \in X$ . For $m \geq 0$ , let $O^m_p$ be the affine subspace of $\mathbb {C}^N$ passing through $p \in X$ and generated by the vectors $\phi _I(0)$ , where $\phi _I$ is a partial derivative of $\phi $ of order $\leq m$ .

By definition, the m-osculating space $\;\,T_p^m X$ of X at p is the projective closure in $\mathbb {P}^N$ of $O^m_p$ . The m-osculating projection

is the corresponding linear projection with center $T_p^m$ .

5.2.3 Osculating projections of $\mathcal {M}_{0,2g}^{\text {GIT}}$

In this section, we show how the map $\varphi _{D,N}$ induces an osculating projection on the copies of $\mathcal {M}_{0,2g}^{\text {GIT}}$ that appear as factors of the map in Diagram 4.

Lemma 5.2. Let Q be a r-form in $\mathbb {P}^n$ vanishing at the points $P_1$ and $P_2$ with multiplicity $l_1$ and $l_2$ respectively. Then, Q vanishes on the line passing through $P_1$ and $P_2$ with multiplicity at least $l_1 + l_2 - r$ .

Proof See, for example, [Reference Kumar22, page 2].

Let us now consider the linear system $|\mathcal {I}_{\text {Sec}^N}(g)|$ on $\mathbb {P}_N^{2g-2}$ (see Section 4). The forms in $|\mathcal {I}_{\text {Sec}^N}(g)|$ vanish with multiplicity $g - 1$ along the points of C (see Lemma 2.2). By Lemma 5.2, these forms vanish then with multiplicity $(g-1) + (g-1) - g = g-2$ along the secant lines l cutting out the hyperelliptic pencil. Thus, these forms vanish with multiplicity $g-2$ on the curve $\Gamma $ .

Let us consider the linear system $|\mathcal {I}_{\text {Sec}^{g-2}(N)}(g)|$ . Let $\mathcal {I}(\Gamma ) \subset |\mathcal {I}_{\text {Sec}^{g-2}(N)}(g)|$ be the partial linear system of forms vanishing (with multiplicity 1) along $\text {Sec}^{g-2}(N)$ , and with multiplicity $g - 2$ along $\Gamma $ . By our previous observation and Lemma 4.1, we have the following inclusions of linear systems:

$$ \begin{align*} \mathcal{R} \subset |\mathcal{I}_{\text{Sec}^N}(g)| \subset \mathcal{I}(\Gamma) \subset |\mathcal{I}_{\text{Sec}^{g-2}(N)}(g)|. \end{align*} $$

Recall that the map $\varphi _{D,N}$ is induced by the linear system $\mathcal {R}$ . The above sequence of inclusions yields the following factorization of maps

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The first map $h_N$ is the one defined in Section 4.3, its image is the GIT quotient $\mathcal {M}_{0,2g}^{\text {GIT}}$ . According to Proposition 4.9, this map contracts the curve $\Gamma $ to a point $h_N(\Gamma )$ .

Proposition 5.3. The map $\pi _N$ is the $(g-3)$ -osculating projection $\Pi ^{g-3}_{P}$ with center the point $w = h_N(\Gamma )$ .

Proof From the geometric description of the linear systems $\mathcal {I}(\Gamma )$ and $|\mathcal {I}_{\text {Sec}^{g-2}(N)}(g)|$ (Propositions 4.8 and 4.9), the base locus of the map $\pi _N$ is the point $w = h_N(\Gamma )$ , with multiplicity $g-2$ . In particular, since the forms in $\mathcal {I}(\Gamma )$ vanish with multiplicity $g - 2$ along $\Gamma $ , the order the projection $\pi _N$ is $g-3$ .

According to Proposition 4.9, the map $h_N$ contracts the curve $\Gamma $ to a point w in $\mathcal {M}_{0,2g}^{\text {GIT}}$ representing an ordered configuration of the $2g$ marked points N. This point in turn corresponds to a hyperelliptic genus $(g - 1)$ curve $C_w$ together with an ordering of the Weierstrass points. Now recall from Section 4.3 that the lower level composed map of Diagram 3 is the map $h_N$ . The rational normal curve $\Gamma \subset \mathbb {P}_N^{2g-2}$ is contracted to a point $e_0\in \mathbb {P}^{2g-3}$ s.t. $w=i_\Omega (e_0)$ . Recall, once again from Section 4.3 that $\mathbb {P}^{2g-3}$ also contains the $2g-1$ points $\tau _k\circ \text {Cr}_k(H_i)$ , images of the hyperplanes $H_i \subset \mathbb {P}_N^{2g-2}$ , with $i\neq k$ and $1\leq i \leq 2g$ . Let us label them $e_1, \dots , e_{2g-1}$ . Let now $\Lambda $ be the partial linear system of $\Omega $ consisting of the $(g - 1)$ -forms in $\mathbb {P}^{2g-3}$ vanishing with multiplicity $g - 2$ in all the points $e_0, e_1, \ldots , e_{2g - 1}$ . As it is explained in Theorem 4.7, the rational map induced by the linear subsystem $\Lambda $ factors through $\mathcal {M}_{0,2g}^{\text {GIT}}$ , and the second map is the osculating projection from the image of $e_0$ inside $\mathcal {M}_{0,2g}^{\text {GIT}}$ .

Theorem 5.4. The map $\pi _N$ coincides with the map $\kappa $ . In particular, the map $\pi _N$ is of degree 2.

Proof Consider the GIT quotient $\mathcal {M}_{0,2g}^{\text {GIT}}$ embedded in $|\Omega |^*$ as we have seen in Theorem 4.5. The osculating projection $\pi _N$ is given by the linear system $|H - (g - 2) w|$ of hyperplanes vanishing in w with multiplicity $g-2$ . By definition of $\Omega $ , this linear system pulls back via $i_{\Omega }$ to the linear system of $(g - 1)$ -forms in $\mathbb {P}^{2g - 3}$ vanishing with multiplicity $g - 2$ in $e_1, \ldots , e_{2g - 1}$ , and also with multiplicity $g - 2$ in $e_0$ , which is precisely $\Lambda $ . Hence, the map $\pi _N$ is the map induced by the same linear system as $\kappa $ (see Theorem 4.7).

We will show in the next Section that the map $l_N$ from Diagram 6 is actually birational, and that the map $\pi _N$ (birationally) coincides with the restriction of the map $\theta $ .

5.3 The hyperelliptic theta map and rational involutions on $\mathcal {M}_{0,2g}^{\text {GIT}}$ and $\mathcal {SU}_C(2)$

The resolution $\widetilde {\varphi _D}$ of $\varphi _D$ factors through the degree 2 map $\theta $ as shown in Diagram 5. In the preceding section, we have shown that, when we restrict $\varphi _{D,N}$ to $\mathbb {P}_N^{2g-2}$ , it factors through the degree 2 map $\pi _N$ . Now, we link these two factorizations. The identification of maps in the following claim must be intended as rational maps, since for example $\pi _N$ is not everywhere defined.

Proof Let us place ourselves on the open set $\widehat {\mathcal {SU}}_C(2) \subset \mathcal {SU}_C(2)$ of general stable bundles. First we remark that the factorization $\widetilde {\varphi _D} =\theta \circ \widetilde {f_D}$ of Diagram 5 is the Stein factorization of the map $\widetilde {\varphi _D}$ along $\widetilde {\mathbb {P}_D^{3g-2}}$ . Indeed, the map $\theta $ is of degree 2 as explained in Section 1. Moreover, the preimage of a general stable bundle E by the map $f_D$ is the $\mathbb {P}^1$ arising as the projectivization of the space of extensions of the form

$$ \begin{align*} \quad 0 \to \mathcal{O}(-D)) \to E \to \mathcal{O}(D) \to 0. \end{align*} $$

In particular, the fibers of $\widetilde {f_D}$ over $\widehat {\mathcal {SU}}_C(2)$ are connected.

The restriction of $\varphi _D$ to $\mathbb {P}_N^{2g-2}$ factors through the maps $h_N$ and $\pi _N$ (see Diagram 6), followed by the map $l_N$ . According to Proposition 4.9, the fibers of $h_N$ are rational normal curves, thus connected. Moreover, the map $\pi _N$ is degree 2 by Theorem 5.4. By unicity of the Stein factorization, we have our result.

Comparing with the factorization $\widetilde {\varphi _D} =\theta \circ \widetilde {f_D}$ , we see that $l_N$ cannot have relative dimension $> 0$ . Hence, $l_N$ is a finite map. Since the degree of the map $\theta $ in the Stein factorization is 2, which is equal to the degree of $\pi _N$ , we have that $l_N$ cannot have degree $> 1$ . In particular, we have that the map $l_N$ is a birational map.

We are now in the position to show the following.

The results proved and collected in the preceding sections put us now in the position to claim the following theorem.

Proof The proof of this theorem is a collection of the results that we have developed so far. We will refer to Figure 2 as a guiding line. As we have observed (see Proposition 5.6 (2)—top right corner of Figure 2), the generic fiber of the projection $p_D$ is birational to $\mathcal {M}_{0,2g}^{\text {GIT}}$ , and on an open subset of the fiber the map $\theta $ coincides with the degree 2 map $\pi _N$ (Theorem 5.5—vertical map on the RHS of the figure). In turn, the map $\pi _N$ coincides with the osculating projection $\kappa $ (Theorem 5.4) defined on $\mathcal {M}_{0,2g}^{\text {GIT}}$ . Hence, by Kumar’s beautiful description (see [Reference Kumar21] and Theorem 4.7—bottom right corner of the figure) of the Kummer variety as the ramification locus of $\kappa $ , we have the claim. Let us also underline that we are also tacitly showing that the map $p_D$ factors through the map $\theta $ (this is the full RHS of the diagram in Figure 2).

Figure 2 An intuitive view of our construction.

Results from [Reference Zelaci31, App. E] imply that the ramification locus is in fact nonirreducible.

6.1 The restriction to $\mathbb {P}_N^4$

The base locus of the restricted map $\varphi _{D,N} = \varphi _D|_{\mathbb {P}_N^{4}}$ contains $\text {Sec}^N = \text {Sec}^1(C) \cap \mathbb {P}^4_N$ by Lemma 4.1. The secant variety $\text {Sec}^1(N) \subset \text {Sec}^N$ is the union of the 15 lines passing through pairs of the 6 points of N. According to Lemma 4.10, the further base locus $\text {Sec}^N \setminus \text {Sec}^1(N)$ is given by the intersections of $\mathbb {P}^4_N$ with the lines spanned by degree 2 divisors $L_2$ on C not contained in N satisfying $\dim |L_2| \geq 1$ . By Brill–Noether theory, there is only one linear system of such divisors on a genus 3 curve, namely the hyperelliptic linear system $|h|$ (see, e.g., [Reference Arbarello, Cornalba, Griffiths and Harris2], Chapter V). We will review these ideas in Section 7. This linear system traces the curve $\Gamma $ that we introduced in Section 5 as the set of intersection points of $\mathbb {P}_N^{2g-2}$ with the lines spanned by the divisors in the hyperelliptic pencil. Hence, we have that $\text {Sec}^N = \{15 \text { lines} \} \cup \Gamma $ , and the restricted map $\varphi _{D,N}$ factors as

where $h_N$ is the map defined by the complete linear system $|\mathcal {I}_{\text {Sec}^1(N)}(3)|$ of cubics vanishing along the 15 lines defined by the points of N, and $\pi _N$ is the projection with center the image via $h_N$ of the rational normal curve $\Gamma $ .

The image of $\varphi _{D,N}$ is a $\mathbb {P}^3$ . Indeed, this image cannot have higher dimension, since the map factors through the projection from a point of $\mathcal {M}_{0,6}^{\text {GIT}} \subset \mathbb {P}^4$ . Also, it cannot have dimension strictly smaller than 3 since otherwise the relative dimension of $\varphi _{D,N}$ would be bigger than 1, or equivalently the global map $\varphi _D$ would not surjet onto $\mathcal {SU}_C(2)$ . Hence, in this case the map $\varphi _{D,N}$ is defined exactly by the system of cubics in $\mathbb {P}^4_N$ vanishing on $\text {Sec}^N$ .

According to Proposition 4.9, the image of $h_N$ is the GIT moduli space $\mathcal {M}_{0,6}^{\text {GIT}}$ if N is general and reduced. It is a classical result that this GIT quotient is embedded in $\mathbb {P}^4$ as the Segre cubic $S_3$ (see for instance [Reference Dolgachev and Ortland12] or [Reference Dolgachev14]). This three-fold arises by considering the linear system of quadrics in $\mathbb {P}^3$ that pass through five points in general position, thus it is isomorphic to the blow-up of $\mathbb {P}^3$ at these points, followed by the blow-down of all lines joining any two points. The composition of this map with the projection off a smooth point of $S_3$ gives a $2:1$ rational map whose ramification locus is a Weddle surface [Reference Kumar21, Reference Bolognesi5]. The curve $\Gamma \subset \mathbb {P}^4_N$ is a rational normal curve by Lemma 5.1, hence $\Gamma $ is contracted to a point w by $h_N$ again by Proposition 4.9.

By [Reference Bertram4] and Lemma 4.1, the linear system $|\mathcal {O}_{S_3}(1)|$ of hyperplanes in $S_3$ is pulled back by $h_N$ to $|\mathcal {I}_{\text {Sec}^1(N)}(3)|$ on $\mathbb {P}_N^{4}$ . The linear system $|\mathcal {O}_{S_3}(1) - w|$ of hyperplanes in $S_3$ passing through w is pulled back to the complete linear system $|\mathcal {I}_{\text {Sec}^N}(3)|$ defining $\varphi _{D,N}$ . Hence, the map $\pi _N$ is the linear projection with center w. Since $S_3$ is a cubic, the projection $\pi _N$ is a degree 2 map. We will see in the next Section that this will be also the case for higher genus. The following proposition resumes what we have seen so far in this section.

The point w in $\mathcal {M}_{0,6}^{\text {GIT}}$ represents a rational curve with six marked points. Let $C'$ be the hyperelliptic genus 2 curve constructed as the double cover of this rational curve ramified in these six points. According to Theorem 4.2 of [Reference Kumar21], the Kummer variety $\text {Kum}(C')$ is contained in the image of $\pi _N$ , and it is precisely its ramification locus. Recall that, when $g=3$ , the linear system $|2D|$ is a $\mathbb {P}^3$ . By Proposition 3.2, the image of $\mathbb {P}_N^{4}$ by $\varphi _D$ is the closure of the fiber $p_{\mathbb {P}_c}^{-1}(N)$ . For each point N in $|2D|$ , this image is $\mathbb {P}^3 = |\mathcal {I}^2_{\text {Sec}^N}(3)|^*$ , which is the image of the Segre variety $\mathcal {M}_{0,6}^{\text {GIT}}$ under the projection with center P. Thus, the image of the global map $\varphi _D$ is birational to a $\mathbb {P}^3$ -bundle over $|2D| = \mathbb {P}^3$ . Of course this is also the case since the image of the theta map is a quadric hypersurface in $\mathbb {P}^7$ [Reference Desale and Ramanan11].

Case $g = 4$

In this case, the divisor N is of degree 8 and the map is given by the restriction of the linear system $|\mathcal {I}_C^3(4)|$ to $\mathbb {P}_N^6$ . This map factors through the map $\pi _N$ which coincides with the one-osculating projection $\Pi ^1_w$ , where $w = h_N(\Gamma )$ .

We are looking for degree 3 divisors $L_3$ with $\dim |L_3| \geq 1$ . These satisfy all $\dim |L_3| = 1$ and are of the form

$$ \begin{align*} L_3 = h + q \qquad \text{for } q \in C, \end{align*} $$

where h is the hyperelliptic divisor. Let p be a point of C. Then $L_3 = p + i(p) + q$ . Since $\dim |L_3| = 1$ , the secant plane $\mathbb {P}^2_{L_3}$ in $\mathbb {P}_D^{10}$ spanned by p, $i(p)$ and q intersects $\mathbb {P}^6_N$ in a point. But this point necessarily lies on $\Gamma $ , since the line passing through p and $i(p)$ is already contained in this plane. Hence, we do not obtain any additional locus.

Case $g = 5$

In this case, the divisors $L_4$ of degree 4 are all of the form

$$ \begin{align*} L_4 = h + q + r \qquad \text{for } q, r \in C, \end{align*} $$

and satisfy $\dim |L_4| = 1$ . Thus, the corresponding secant $\mathbb {P}^3_{L_4}$ spanned by p, $i(p)$ , q and r intersects $\mathbb {P}^{8}_N$ in a point. As before, this point lies on $\Gamma $ , thus we do not obtain any additional locus. The upshot is the following

Case $g = 6$

Here we have, as in the genus 5 case, the divisors of the form

$$ \begin{align*} L_3 = h + q \quad \text{for } q \in C, \end{align*} $$

which do not give rise to any additional base locus. But there is a new family of divisors

$$ \begin{align*} L_5 = 2h + r \quad \text{for } r \in C. \end{align*} $$

These divisors satisfy $\dim |L_5| = 2$ . In particular, the intersection of the $\mathbb {P}^4_{L_5}$ spanned by p, $i(p)$ , q, $i(q)$ , and r, for $p, q \in C$ , with $\mathbb {P}^{10}_N$ is a line m in $\mathbb {P}^{10}_N$ . The line $l_1$ (resp. $l_2$ ) spanned by p and $i(p)$ (resp. q, $i(q)$ ) intersects $\Gamma $ in a point $\widetilde {p}$ (resp. $\widetilde {q}$ ). In particular, the line m is secant to $\Gamma $ and passes through $\widetilde {p}$ and $\widetilde {q}$ . Since every point of $\Gamma $ comes as an intersection of a secant line in C with $\mathbb {P}^{10}_N$ , we obtain the following description of the base locus of $\varphi _{D,N}$ :