2.1. Construction from a two-dimensional torus

In this paper, the ground field is the complex numbers. In this section, we recall the construction of the Kummer surface associated with a two-dimensional torus.

Let $T$ be a two-dimensional torus. Consider the involution $\iota \,:\,T\rightarrow T$ which sends $a\mapsto -a$ and takes the quotient surface $T/\langle \iota \rangle$ . The surface $T/\langle \iota \rangle$ is known as the singular Kummer surface of $T$ . It is well-known that this surface has 16 ordinary singularities, and by resolving them, we obtain a $K3$ surface called the Kummer surface of $T$ and denoted by $\mathrm{Km}(T)$ (see e.g., [ Reference Frías-Medina and Zamora10 , Theorem 3.4]). This procedure is called the Kummer process. Note that by construction, $\mathrm{Km}(T)$ has 16 disjoint smooth rational curves; indeed, they correspond to the singular points of the quotient surface. Nikulin proved in [ Reference Knutsen15 ] the converse:

Note that the above construction holds true for any two-dimensional torus, not necessarily a projective one. In particular, with this process it is possible to construct $K3$ surfaces that are not projective. However, there is an equivalence between the projectivity of the torus and the associated $K3$ surface (see [ Reference Birkenhake and Lange2 , Theorem 4.5.4]):

Now, if $A$ is a principally polarized abelian surface, then $A$ is one of the following (see [ Reference Birkenhake and Lange2 , Corollary 11.8.2]):

1. (a) The Jacobian of a smooth hyperelliptic curve of genus 2 or

2. (b) The canonical polarized product of two elliptic curves.

As we will see next, Case (a) is the one of our interest.

2.2. Hyperelliptic curves of genus 2 and Kummer surfaces

We are interested in $K3$ surfaces that are a smooth complete intersection of type $(2,2,2)$ in ${\mathbb{P}}^5$ , i.e. that are a complete intersection of three quadrics. Moreover, we restrict to the case in which the quadrics are diagonal. The interest of having diagonal quadrics defining the $K3$ surface is that they enable us to work with hyperelliptic curves of genus 2.

Indeed, let $C$ be the hyperelliptic curve of genus 2 given by the affine equation

(1) \begin{equation} y^2=f(x)=(x-a_0)(x-a_1)\cdots (x-a_5), \end{equation}

where $a_0,\dots,a_5\in \mathbb{C}$ and $a_i\neq a_j$ if $i\neq j$ . We can consider the jacobian surface $J(C)$ associated with $C$ , and applying the Kummer process, we obtain that the $K3$ surface $\mathrm{Km}(J(C))$ is isomorphic to the surface in ${\mathbb{P}}^5$ defined by the complete intersection of the 3 diagonal quadrics by [ Reference Nikulin16 , Theorem 2.5]:

(2) \begin{equation} Q_i=\sum _{j=0}^5 a_j^i x_j^2, \;i=0,1,2. \end{equation}

In order to obtain a hyperelliptic curve of genus 2 beginning with a smooth $K3$ surface $S$ in ${\mathbb{P}}^5$ given by the complete intersection of three diagonal quadrics, we may assume an additional hypothesis. Edge studied in [ Reference Edge6 ] the Kummer surfaces defined by (2). One of his results establishes that whenever a surface $X$ given by the intersection of three linearly independent quadrics has a common self-polar simplex $\Sigma$ in ${\mathbb{P}}^5$ and contains a line in general position, then the equations defining $X$ can be written with the form (2). Observe that this fact is equivalent to requiring that $X$ contains 16 disjoint lines; indeed, using the natural involutions of ${\mathbb{P}}^5$ one can obtain the other lines.

Then, let $S$ be a smooth $K3$ surface in ${\mathbb{P}}^5$ given by the complete intersection of the quadrics

\begin{equation*} Q_i=\sum _{j=0}^5 a_{ij} x_j^2, \;\;\; i=0, 1, 2, \end{equation*}

where $a_{ij}\in \mathbb{C}$ for $i=0,1,2$ and $j=0,\dots,5$ and assume that $S$ contains 16 disjoint lines. As a consequence, we may assume that $S$ is given by the quadrics in (2) for some $a_i\in \mathbb{C}$ where $a_i\neq a_j$ if $i\neq j$ . By [ Reference Knutsen15 , Theorem 1] there exists a unique (up to isomorphism) two-dimensional torus that gives rise to the surface $S$ . Taking the hyperelliptic curve $C$ given by the equation $y^2=f(x)=(x-a_0)(x-a_1)\cdots (x-a_5)$ , we obtain that $S$ is isomorphic to $\mathrm{Km}(J(C))$ .

From now on, we say that a Kummer surface is a smooth surface in ${\mathbb{P}}^5$ given by the complete intersection of three diagonal quadrics as in (2).

For a Kummer surface $S$ given by (2), it is possible to give the parametric form of the 32 lines contained in $S$ . Indeed, in [ Reference Edge6 ] Edge noted that the equation of a line $\ell$ contained in $S$ is given in the following parametric form:

(3) \begin{equation} \left (\frac{t+a_0}{\sqrt{f^{\prime}(a_0)}}, \frac{t+a_1}{\sqrt{f^{\prime}(a_1)}},\frac{t+a_2}{\sqrt{f^{\prime}(a_2)}},\frac{t+a_3}{\sqrt{f^{\prime}(a_3)}}, \frac{t+a_4}{\sqrt{f^{\prime}(a_4)}},\frac{t+a_5}{\sqrt{f^{\prime}(a_5)}}\right ). \end{equation}

Recall that the natural automorphisms $\sigma _i\,:\, x_i\mapsto -x_i$ of ${\mathbb{P}}^5$ act on $S$ . Denote by $E$ the group $\langle \sigma _0,\dots,\sigma _5 \rangle$ . Applying each element of $E$ to the line $\ell$ , we obtain the other 32 lines on $S$ . The identity gives the line $\ell$ and the remaining elements give the other 31 lines:

• $\ell _i\,:\!=\,\sigma _i(\ell )$ for all $i\in \{0,\dots,5\}$ ,

• $\ell _{ij}\,:\!=\,\sigma _i\sigma _j(\ell )$ for different $i,j\in \{0,\dots,5\}$

• $\ell _{ijk}\,:\!=\,\sigma _i\sigma _j\sigma _k(\ell )$ for different $i,j,k\in \{0,\dots,5\}$ .

It is well-known that a singular Kummer surface $K$ is birational to a Weddle surface $W$ (see e.g., [ Reference Shioda17 , Proposition 1]). A Weddle surface is a quartic surface in ${\mathbb{P}}^3$ with six nodes. The 32 lines on $S$ have a geometric interpretation in both $K$ and $W$ as Edge pointed in [ Reference Edge6 ]. Indeed, the projection $\pi$ of $S$ from $\ell$ is a Weddle surface $W$ and it occurs:

• $\pi (\ell _i)=k_i$ is a node on $W$ , for all $i=0,\dots,5$ ,

• $\pi (\ell _{ij})$ is the line through $k_i$ and $k_j$ , for different indices $i,j\in \{0,\dots,5\}$ ,

• $\pi (\ell _{ijh})$ is the line in the intersection of the plane generated by $k_i,k_j,k_h$ with the complementary plane, for different indices $i,j,h\in \{0,\dots,5\}$ , and

• $\pi (\ell )$ is the cubic on $W$ through the six nodes $k_i$ ’s.

On the other hand, since $S$ is the resolution of singularities of $K$ it occurs:

• The 16 nodes of $K$ correspond to the 16 lines $\ell _{i}$ and $\ell _{ijk}$ , and

• The conics of contact of $K$ with its 16 tropes correspond to the 16 lines $\ell$ and $\ell _{ij}$ .

A trope is a plane which intersects the quartic along a conic. The nodes and the tropes of a singular Kummer surface provide an interesting configuration on it.

Gonzalez-Dorrego classified in [ Reference Frías-Medina and Zamora10 ] the non-degenerate $(16,6)$ -configurations and used them to classify the singular Kummer surfaces. Given a singular Kummer surface, the nodes and the tropes establish a non-degenerate $(16,6)$ -configuration (see [ Reference Frías-Medina and Zamora10 , Corollary 2.18]), and conversely, given a $(16,6)$ -configuration, there exists a singular Kummer surface whose associated $(16,6)$ -configuration is the given one (see [ Reference Frías-Medina and Zamora10 , Theorem 2.20]).

Rosenhain tetrahedra always exist in a singular Kummer surface; in fact, there exist $80$ of them ([ Reference Frías-Medina and Zamora10 , Corollary 3.21]). Moreover, these tetrahedra are relevant because using them we can construct divisors that are linearly equivalent and whose class induces the closed embedding to ${\mathbb{P}}^5$ (see [ Reference Frías-Medina and Zamora10 , Proposition 3.22 and Remark 3.24]):

These divisors will be used in the next section to construct vanishing thetanulls on Humbert-Edge curves of type 5.

3.1. Properties and characterization

Here, we review the main properties of the Humbert-Edge curves of type 5 and present a characterization using the lines lying on a Kummer surface.

Definition 3.1. An irreducible, non-degenerate, and non-singular curve $X_5\subseteq{\mathbb{P}}^5$ is a Humbert-Edge curve of type 5 if it is the complete intersection of 4 diagonal quadrics $Q_0,\dots,Q_3$ :

\begin{equation*} Q_i=\sum _{j=0}^5 a_{ij} x_j^2, \;\;\; i=0,\dots, 3. \end{equation*}

The basic properties of a Humbert-Edge curve of type 5 are stated below.

The diagonal form of the equations defining a Humbert-Edge curve $X_5$ of type 5 implies that it admits the action of the group $E$ generated by the six involutions $\sigma _i\,:\, x_i\mapsto -x_i$ acting with fixed points and whose product is the identity. Moreover, these involutions establish a relation between the Humbert-Edge curves of type 5 and the Humbert’s curves in ${\mathbb{P}}^4$ . For every $i=0,\dots,5$ , we can consider the covering $\pi _i\,:\,X_5\rightarrow X_5/\langle \sigma _i\rangle$ induced by the involution $\sigma _i$ . This is a two-to-one covering ramified at 16 points obtained as the intersection points of $X_5$ with the hyperplane $V(x_i)$ . In addition, the quotient of $X_5/\langle \sigma _i\rangle$ is a Humbert’s curve in ${\mathbb{P}}^4$ . This double covering can be interpreted geometrically as the projection of $X_5$ with center $e_i$ onto the hyperplane $V(x_i)$ .

Next result shows that a Humbert-Edge curve of type 5 is always contained in a Kummer surface. It is a consequence of the fact noted by Edge in [ Reference Edge7 ] that a Humbert-Edge curve of type $n$ can be written in a normal form.

Proof. Assume that $X_5$ is given by the equations

\begin{equation*} Q_i=\sum _{j=0}^5 a_{ij} x_j^2, \;\;\; i=0,\dots,3 \end{equation*}

where $a_{ij}\in \mathbb{C}$ . For each $j=0,\dots,5$ , consider the coefficients $a_{ij}$ as the entries of the point $p_j=\left(a_{0j}\,:\,a_{1j}\,:\,a_{2j}\,:\,a_{3j}\right)$ in the projective space ${\mathbb{P}}^3$ . We have six points $p_0,\dots,p_5$ in ${\mathbb{P}}^3$ that are in general position, so there exists a unique rational normal curve $C\subset{\mathbb{P}}^3$ through these points. Finally, take a change of coordinates of ${\mathbb{P}}^3$ such that $C$ is in the standard parametric form, then we may assume that $p_j=\left(1\,:\,a_j\,:\,a_j^2\,:\,a_j^3\right)$ for all $j=0,\dots,5$ . Therefore, we obtain that $X_5$ is given by the equations

(4) \begin{equation} Q_i=\sum _{j=0}^5 a_{j}^i x_j^2, \;\;\; i=0,\dots, 3 \end{equation}

with $a_j\in \mathbb{C}, j=0,\dots,5$ and $a_j\neq a_k \mbox{ if } j\neq k$ . The quadrics $Q_0,Q_1,Q_2$ define the Kummer surface associated with the hyperelliptic curve $y^2=\prod _{j=0}^5 (x-a_j)$ . Therefore, $X_5$ is contained in a Kummer surface.

Next we present a characterization for Humbert-Edge curves of type 5 using the lines on Kummer surfaces.

Proof. (i) $\Leftrightarrow$ (ii) We have this equivalence by [ Reference Farkas and Kra9 , Theorem 3.4].

(i) $\Rightarrow$ (iii) Assume that $X$ is Humbert-Edge curve of type 5. Proposition 3.3 implies that $X$ is contained in a Kummer surface $S$ . So, we may assume the existence of different scalars $a_0,\dots,a_5\in \mathbb{C}$ such that $X$ is given by the equations

\begin{equation*} Q_i=\sum _{j=0}^5 a_{j}^i x_j^2, \;\;\;i=0,\dots,3 \end{equation*}

and $S$ is given by the equations $Q_0,Q_1,Q_2$ . Consider the line $\ell$ in parametric form as in (3). A direct computation shows that for every $t\in \mathbb{C}$ ,

\begin{equation*} Q_3\left (\frac {t+a_0}{\sqrt {f^{\prime}(a_0)}}, \frac {t+a_1}{\sqrt {f^{\prime}(a_1)}},\frac {t+a_2}{\sqrt {f^{\prime}(a_2)}},\frac {t+a_3}{\sqrt {f^{\prime}(a_3)}}, \frac {t+a_4}{\sqrt {f^{\prime}(a_4)}},\frac {t+a_5}{\sqrt {f^{\prime}(a_5)}}\right )=1. \end{equation*}

Therefore, $X$ does not intersect the line $\ell$ and the diagonal form of the third equation implies that $X$ also does not intersect the lines $\ell _{ij}$ for every $i,j\in \{0,\dots,5\}$ with $i\neq j$ .

(iii) $\Rightarrow$ (i) Assume that $X$ is contained in a Kummer surface $S$ defined by the equations

\begin{equation*} Q_i= \sum _{j=0}^5 a_j^i x_j^2, \;\;\; i=0,1, 2, \end{equation*}

where $a_j\neq a_k$ if $j\neq k$ , and such that $X$ does not intersect the lines $\ell$ , $\ell _{jk}$ in two different points for all $j,k\in \{0,\dots,5\}$ with $j\neq k$ . We denote $f(x)=\prod _{j=0}^5 (x-a_j)$ . Since $S$ is a $K3$ surface of type $(2,2,2)$ in ${\mathbb{P}}^5$ containing $X$ , our situation should be one of the cases determined by Knutsen in [ Reference Hidalgo, Reyes-Carocca and Valdés14 , Theorem 6.1 (3)]. In fact, we are in Case a) of the latter, in Knutsen’s notation we have $n=4$ , $d=16$ , $g=17$ , and $g=d^2/16 +1$ . Moreover, such a result implies that $X$ is the complete intersection of $S$ and a hypersurface of degree $d/8$ , i.e. $X$ is the complete intersection of four quadrics. Denote the fourth quadric by

\begin{equation*} Q=\sum _{j=0}^5 d_j x_j^2 + \sum _{0\leq k \lt j\leq 5} d_{jk} x_jx_k. \end{equation*}

Now, when we evaluate the quadric $Q$ in the parametric form of $\ell$ we obtain a quadratic equation with parameter $t$ with leading coefficient

\begin{equation*} \frac {1}{\prod _{j=0}^5 f^{\prime}(a_j)} \left ( \sum _{j=0}^5 f^{\prime}(a_0)\cdots \widehat {f^{\prime}(a_j)} \cdots f^{\prime}(a_5) d_j + \sum _{0\leq k\lt j\leq 5} f^{\prime}(a_0)\cdots \sqrt {f^{\prime}(a_k)}\sqrt {f^{\prime}(a_j)} \cdots f^{\prime}(a_5) d_{jk} \right ). \end{equation*}

The hypothesis that $Q$ does not intersect the line $\ell$ in two different points implies that such coefficient vanishes (the coefficient of $t$ could vanish but in such case the constant term must be different from zero). This also occurs for all of the 15 lines $\ell _{jk}$ by hypothesis, and then, we have 16 imposed conditions. The leading coefficient for the line $\ell _{jk}$ can be deduced from the above one, in fact, since the line $\ell _{jk}$ is obtained from $\ell$ by the application of $\sigma _j\sigma _k$ , it is enough to add a negative sign to the coefficient of the terms $d_{rs}$ whenever $r$ or $s$ are equal to $j$ or $k$ . Solving the linear system in the variables $d_j$ ’s and $d_{jk}$ ’s, we obtain that all the $d_{jk}$ ’s are equal to zero, that $d_0,d_1,d_2,d_3$ , and $d_4$ are free parameters and

(5) \begin{equation} d_5= - f^{\prime}(a_5)\left (\frac{d_0}{f^{\prime}(0)}+\frac{d_1}{f^{\prime}(a_1)}+\frac{d_2}{f^{\prime}(a_2)}+\frac{d_3}{f^{\prime}(a_3)}+\frac{d_4}{f^{\prime}(a_4)} \right ). \end{equation}

Therefore, $X$ is the complete intersection of four diagonal quadrics in ${\mathbb{P}}^5$ and we conclude that it is a Humbert-Edge curve of type 5.

(iii) $\Leftrightarrow$ (iv) As above, assuming that $X$ is contained in a Kummer surface $S$ defined by the equations

\begin{equation*} Q_i= \sum _{j=0}^5 a_j^i x_j^2, \;\;\; i=0, 1, 2, \end{equation*}

where $a_j\neq a_k$ if $j\neq k$ , by [ Reference Hidalgo, Reyes-Carocca and Valdés14 , Theorem 6.1 (3)] we ensure that $X$ is the complete intersection of $S$ and a hypersurface of degree 2. Under the hypothesis of (iii) or (iv), when we solve the system of equations in the parameter $t$ as we previously did, we obtain that the coefficient of every mixed term vanishes and $d_5$ has the form of (5). The remaining conditions, (iv) or (iii), respectively, do not impose new conditions on the coefficients.

3.2. Theta characteristics

In this section, we use the coverings given by the subgroups generated by involutions $\sigma _i$ ’s and the Rosenhain tetrahedra of singular Kummer surfaces to construct theta characteristics on a Humbert-Edge curve of type 5. We recall the definition of a theta characteristic and a vanishing thetanull.

Recall that given a Humbert-Edge curve $X_5$ of type 5, for every $i=0,\dots,5$ the double covering $\pi _i\,:\,X_5\rightarrow X_5/\langle \sigma _i\rangle$ is ramified at 16 points obtained as the intersection points of $X_5$ with the hyperplane $V(x_i)$ . Denote by $R_i=p_{i1}+\cdots +p_{i16}$ the ramification divisor for every $i=0,\dots,5$ .

Proof. For distinct $i,j\in \{0,\dots,5\}$ , consider the subgroup generated by the involutions $ \sigma _i$ and $\sigma _j$ and take the induced covering of degree four $\pi _{ij}\,:\,X_5 \rightarrow X_5/\langle \sigma _i,\sigma _j\rangle$ . This is a simply ramified covering with the 32 ramified points $p_{i1},\dots,p_{i16},p_{j1},\dots,p_{j16}$ . Since $X_5/\langle \sigma _i,\sigma _j \rangle =E_{ij}$ is an elliptic curve, it follows that

\begin{equation*} K_{X_5} \sim \pi _{ij}^*(K_{E_{ij}})+R_i+R_j = R_i+R_j. \end{equation*}

So, $K_{X_5}\sim R_i+R_j$ for all $i,j\in \{0,\dots,5\}$ . Fix and index $i\in \{0,\dots,5\}$ and take $j,k\in \{0,\dots,5\}\backslash \{i\}$ with $j\neq k$ . Using the fact that

\begin{equation*} R_i+R_k \sim K_{X_5} \sim R_k+R_j, \end{equation*}

we have that $R_i\sim R_j$ . Thus, $K_{X_5}\sim R_i+R_j\sim 2 R_i$ and $R_i$ is a theta characteristic.

Now, let $i,j,k\in \{0,\dots,5\}$ be different indices. The covering of degree eight $\pi _{ijk}\,:\,X_5\rightarrow X_5/\langle \sigma _i,\sigma _j,\sigma _k\rangle$ is a simply ramified covering in the 48 points $p_{i1},\dots,p_{i16}, p_{j1},\dots,p_{j16}, p_{k1},\dots,p_{k16}$ . Note that $X_5/\langle \sigma _i,\sigma _j,\sigma _k\rangle \cong{\mathbb{P}}^1$ . Then,

\begin{equation*} K_{X_5} \sim \pi _{ijk}^*(K_{{\mathbb {P}}^1})+R_i+R_j+R_k\sim \pi _{ijk}^*(K_{{\mathbb {P}}^1})+R_i+K_{X_5}. \end{equation*}

Therefore, we have that $\pi _{ijk}^*(-K_{{\mathbb{P}}^1})\sim R_i$ and we conclude that $\pi _{ijk}^*(\mathcal{O}_{{\mathbb{P}}^1}(2))$ is a theta characteristic of $X_5$ .

Next step is to compute $h^0(\pi _{ijk}^*(\mathcal{O}_{{\mathbb{P}}^1}(2))$ . To do so, we will use the fact that $h^0(\pi _{ijk}^*(\mathcal{O}_{{\mathbb{P}}^1}(2)))=h^0({\pi _{ijk}}_*\pi _{ijk}^*(\mathcal{O}_{{\mathbb{P}}^1}(2)))$ . The covering $\pi _{ijk}$ is determined by a line bundle $\mathcal{L}$ on ${\mathbb{P}}^1$ such that $\mathcal{L}^8=\mathcal{O}_{{\mathbb{P}}^1}({\pi _{ijk}}_*(Ri+R_j+R_k))$ , and in addition, we have that ${\pi _{ijk}}_*\mathcal{O}_{X_5}=\mathcal{O}_{{\mathbb{P}}^1}\oplus \mathcal{L}^{-1}\oplus \cdots \oplus \mathcal{L}^{-7}$ . By the projection formula:

\begin{align*}{\pi _{ijk}}_*\pi _{ijk}^*(\mathcal{O}_{{\mathbb{P}}^1}(2)) & = \mathcal{O}_{{\mathbb{P}}^1}(2) \otimes{\pi _{ijk}}_*\mathcal{O}_{X_5} \\ & = \mathcal{O}_{{\mathbb{P}}^1}(2) \otimes ( \mathcal{O}_{{\mathbb{P}}^1}\oplus \mathcal{L}^{-1}\oplus \cdots \oplus \mathcal{L}^{-7} ) \\ & = (\mathcal{O}_{{\mathbb{P}}^1}(2) \otimes \mathcal{O}_{{\mathbb{P}}^1}) \oplus (\mathcal{O}_{{\mathbb{P}}^1}(2) \otimes \mathcal{L}^{-1} ) \oplus \cdots \oplus (\mathcal{O}_{{\mathbb{P}}^1}(2) \otimes \mathcal{L}^{-7}). \end{align*}

From the equality $\mathcal{L}^8=\mathcal{O}_{{\mathbb{P}}^1}({\pi _{ijk}}_*(R_i+R_j+R_k))$ , we get that the degree of $\mathcal{L}$ is equal to 6, and this implies that $\mathcal{O}_{{\mathbb{P}}^1}(2) \otimes \mathcal{L}^{-n}$ has no sections for every $n=1,\dots, 7$ . Therefore, ${\pi _{ijk}}_*\pi _{ijk}^*(\mathcal{O}_{{\mathbb{P}}^1}(2))=\mathcal{O}_{{\mathbb{P}}^1}(2)$ , and it follows that $h^0(\pi _{ijk}^*(\mathcal{O}_{{\mathbb{P}}^1}(2)))=3$ . Finally, the line bundle $\mathcal{O}_{X_5}(R_i)$ has 3 sections since $R_i\sim \pi _{ijk}^*(-K_{{\mathbb{P}}^1})$ .

Using the geometry of the Kummer surface, we can determine some vanishing thetanulls on a Humbert-Edge curve of type 5.

Proof. By Proposition 3.3, $X_5$ is contained in a Kummer surface $S$ . Denote by $K$ the singular Kummer surface associated with $S$ . For a Rosenhain tetrahedron on $K$ , let $D$ be the associated divisor in $S$ (see Proposition 2.5). By [ Reference Hidalgo, Reyes-Carocca and Valdés14 , Proposition 3.1], we have that $X_5$ and $D$ are dependent in $\mathrm{Pic}(S)$ ; in fact, $X_5$ appears as an element in the linear system $|2D|$ . Using the fact that the canonical bundle of $S$ is trivial and that $X_5\in |2D|$ , the adjunction formula implies that

\begin{equation*} K_{X_5}=(K_S+X_5)|_{X_5}=(2D)|_{X_5}=2 D|_{X_5}. \end{equation*}

Thus, the divisor $D|_{X_5}$ is a theta characteristic. Only the calculation of $h^0(D|_{X_5})$ remains. Twisting the exact sequence of sheaves

\begin{equation*} 0 \rightarrow \mathcal {O}_S(-X_5) \rightarrow \mathcal {O}_S \rightarrow \mathcal {O}_{X_5} \rightarrow 0 \end{equation*}

by $\mathcal{O}_S(D)$ , we obtain the exact sequence

\begin{equation*} 0 \rightarrow \mathcal {O}_S(-D) \rightarrow \mathcal {O}_S(D) \rightarrow \mathcal {O}_{X_5}(D) \rightarrow 0, \end{equation*}

and therefore, we obtain the exact sequence in cohomology

\begin{equation*} 0 \rightarrow H^0(S,-D) \rightarrow H^0(S, D) \rightarrow H^0(X_5, D|_{X_5}) \rightarrow H^1(S,-D). \end{equation*}

We have $H^0(S,-D)=0$ and since $D$ is very ample, by Mumford vanishing theorem we obtain that $H^1(S,-D)=0$ . Then, $H^0(S, D) = H^0(X_5, D|_{X_5})$ and from $\mathrm{dim}|D|=5$ it follows that $h^0(X_5, D|_{X_5})=6$ . We conclude the proof recalling that there are 80 Rosenhain tetrahedra associated with a singular Kummer surface (see Proposition 2.5).

We conclude this subsection with the following remarks:

• The way to construct the vanishing thetanulls for a Humbert-Edge curve of type 5 using the geometry of a Kummer surface differs completely from the classical case: a Humbert’s curve $X$ admits exactly 10 vanishing thetanulls and can be constructed by taking the quotients of $X$ by the subgroup generated by two involutions (see the proof of [ Reference Farkas and Kra9 , Theorem 2.2]), the geometry of the Del Pezzo surface containing $X$ is not involved in such process.

• The procedure used in Proposition 3.8 to construct the vanishing thetanulls holds true for every smooth curve on degree 16 and genus 17 on a Kummer surface $S$ . Indeed, if $Y$ is any smooth curve of degree 16 and genus 17 contained in $S$ , then using again [ Reference Hidalgo, Reyes-Carocca and Valdés14 , Proposition 3.1] we have that $Y$ and $D$ are dependent on $\mathrm{Pic}(S)$ and the previous argument holds.

3.3. Moduli space of Humbert-Edge curves of type 5

As we mention in Remark 3.4, given a Humbert-Edge curve of type 5 it is always possible to associate a hyperelliptic curve of genus 2 and vice versa. Here, we discuss about this fact, and using the results of Carocca, Gónzalez-Aguilera, Hidalgo, and Rodríguez [ Reference Carocca, González-Aguilera, Hidalgo and Rodríguez3 ], we prove that the moduli space of Humbert-Edge curves of type 5 is isomorphic to the moduli space of hyperelliptic curves of genus 2.

Given a Humbert-Edge curve $X_5$ of type 5, using Edge’s idea of considering the coefficients as points in ${\mathbb{P}}^3$ and the unique rational normal curve in ${\mathbb{P}}^3$ through them, one is able to write down the equations of $X_5$ in normal form (see the proof of Proposition 4):

(6) \begin{equation} Q_i=\sum _{j=0}^5 a_{j}^i x_j^2, \;\;\; i=0,\dots, 3, \end{equation}

where $a_j\in \mathbb{C}$ for $j=0,\dots,5$ and $a_j\neq a_k$ if $j\neq k$ . Note that since the rational normal curve that we are considering is constructed via the Veronese map $\nu \,:\,{\mathbb{P}}^1\rightarrow{\mathbb{P}}^3$ , one can fix three points in ${\mathbb{P}}^1$ , and therefore, $X_5$ depends only on three different complex numbers. Thus, we can assume that we are fixing the points $0,1$ , and $\infty$ , and then, we have three free different parameters $\lambda _1, \lambda _2$ , and $\lambda _3$ defining the curve $X_5$ . Since this idea can be carried out in the general case of Humbert-Edge curves of type $n$ , with this fact in mind in [ Reference Carocca, González-Aguilera, Hidalgo and Rodríguez3 , Section 4.1] the authors found an embedding for Humbert-Edge curves of type $n$ in ${\mathbb{P}}^n$ in such way that the equations depend on $n-2$ different parameters. In the particular case of the Humbert-Edge curve $X_5$ of type 5, the equations take the form

\begin{align*} x_0^2 + x_1^2+ x_2^2 & =0 \\ \lambda _1 x_0^2 + x_1^2 + x_3^2 & =0 \\ \lambda _2 x_0^2 + x_1^2+ x_4^2 & =0 \\ \lambda _3 x_0^2 + x_1^1 + x_5^2 &=0, \end{align*}

where $\lambda _1,\lambda _2,\lambda _3\in \mathbb{C}\backslash \{0,1\}$ are different complex numbers. To emphasize the dependence on the parameters $\lambda _1,\lambda _2,\lambda _3$ and considering that we are fixing $0,1,\infty$ , we denote this curve as $X_5(\lambda _1,\lambda _2,\lambda _3)$ . Also, note that if we consider the degree 32 map given by

\begin{align*} \pi _{(\lambda _1,\lambda _2,\lambda _3)}\,:\,X_5(\lambda _1,\lambda _2,\lambda _3) \rightarrow & \hspace{7mm}{\mathbb{P}}^1 \\ (x_0\,:\,\dots\, :\,x_5) \mapsto & -\left (\frac{x_1}{x_0}\right )^2, \end{align*}

then

(7) \begin{equation} \{0, 1,\infty, \lambda _1, \lambda _2, \lambda _3\} \end{equation}

is the branch locus of $\pi _{(\lambda _1,\lambda _2,\lambda _3)}$ .

On the other hand, if $C$ is a hyperelliptic curve of genus 2, then we can write the equation which defines $C$ as

(8) \begin{equation} y^2=(x-a_0)(x-a_1)\cdots (x-a_5), \end{equation}

where $a_j\in \mathbb{C}$ for $j=0,\dots,5$ and $a_j\neq a_k$ if $j\neq k$ . Since there always exists an automorphism of ${\mathbb{P}}^1$ which carries a tuple of different complex numbers $(a_0,a_1,a_2)$ to $(0,1,\infty )$ , we may assume that $C$ is given by the equation

(9) \begin{equation} y^2=x(x-1)(x-\lambda _1)(x-\lambda _2)(x-\lambda _3), \end{equation}

where $\lambda _1,\lambda _2,\lambda _3\in \mathbb{C}\backslash \{0,1\}$ are different. Similarly as before, we denote by $C(\lambda _1,\lambda _2,\lambda _3)$ the hyperelliptic curve of genus 2 with parameters $0,1,\infty,\lambda _1,\lambda _2$ and $\lambda _3$ . In addition, since $C(\lambda _1,\lambda _2,\lambda _3)$ is a hyperelliptic curve of genus 2, there exists a degree 2 map $\rho _{(\lambda _1,\lambda _2,\lambda _3)}\,:\,C(\lambda _1,\lambda _2,\lambda _3)\rightarrow{\mathbb{P}}^1$ whose branch locus is precisely given by (7).

In both cases of Humbert-Edge curves of type 5 and hyperelliptic curves of genus 2, given such a curve we have a map to ${\mathbb{P}}^1$ with a specific branch locus. In fact, the branch locus determines the curve modulo isomorphism. Indeed, Hidalgo, Reyes-Carocca, and Valdés determined in [ Reference Hidalgo13 , Section 2.3] when two generalized Fermat curves are isomorphic, in particular, the following result can be deduced:

In the case of hyperelliptic curves of genus 2, we have an analogous result (see [ Reference Edge8 , Section III.7.3]):

The above results can be summarized in the following commuting diagram:

Now, we briefly discuss the construction of the moduli space of Humbert-Edge curves of type $5$ . To construct such moduli space, we follow Section 4.2 of [ Reference Carocca, González-Aguilera, Hidalgo and Rodríguez3 ]. Let $HE_5$ be the set of all Humbert-Edge curves of type 5. Since any Humbert-Edge curve of type 5 has genus 17, we have a map $r_5\,:\, HE_5\to\mathcal{M}_{17}$ defined by $r_5(X_5(\lambda_1,\lambda_2,\lambda_3))=[X_5(\lambda_1,\lambda_2,\lambda_3)]$ , where $\mathcal{M}_{17}$ is the moduli space of curves of genus 17. By Proposition 3.9, we can consider the isomorphism class of a Humbert-Edge curve of type 5, and this gives an equivalence relation on $HE_5$ . Let $\mathcal{HE}_5$ be the set obtained from this equivalence relation. We have a projection map $p_5\,:\,HE_5 \to\mathcal{HE}_5$ , and we have a well-defined map $q_5\,:\,\mathcal{HE}_5\to\mathcal{M}_{17}$ so that $r_5=q_5\circ p_5$ . We have that $\mathcal{HE}_5$ is a moduli space, and we call it the moduli space of Humbert-Edge curves of type 5, that is the set of Humbert-Edge curves of type 5 modulo isomorphism.

We denote by $\mathcal{H}_2$ the moduli space of hyperelliptic curves of genus 2. By Propositions 3.10 and 3.9, we have a well-defined map

\begin{align*} f_5\,:\,\mathcal{H}_2 & \rightarrow \mathcal{HE}_5 \\ [C(\lambda _1,\lambda _2,\lambda _3)] & \mapsto [X_5(\lambda _1,\lambda _2,\lambda _3)] \end{align*}

By construction, it is immediate that $f_5$ is a surjective map. The following proposition deals with the injectivity.

Proof. Let $C(\lambda _1,\lambda _2,\lambda _3)$ and $C(\mu _1,\mu _2,\mu _3)$ be hyperelliptic curves of genus 2. Assume that $f_5([C(\lambda _1,\lambda _2,\lambda _3)])=f_5([C(\mu _1,\mu _2,\mu _3)])$ ; that is $[X_5(\lambda _1,\lambda _2,\lambda _3)]=[X_5(\mu _1,\mu _2,\mu _3)]$ . Then, there exists an isomorphism between $X_5(\lambda _1,\lambda _2,\lambda _3)$ and $X_5(\mu _1,\mu _2,\mu _3)$ . By Proposition 3.9, there exists a Möbius transformation $M$ such that

\begin{equation*} \{M(0),M(1),M(\infty ),M(\lambda _1),M(\lambda _2),M(\lambda _3)\}=\{0,1,\infty,\mu _1,\mu _2,\mu _3\}. \end{equation*}

Thus, by Proposition 3.10 we conclude that the hyperelliptic curves $C(\lambda _1,\lambda _2,\lambda _3)$ and $C(\mu _1,\mu _2,\mu _3)$ are isomorphic.

On the other hand, in [ Reference Carocca, González-Aguilera, Hidalgo and Rodríguez3 , Proposition 4.3] the authors proved that the map $q_5\,:\,\mathcal{HE}_5\to \mathcal{M}_{17}$ is injective. Thus, using the isomorphism $f_5$ we obtain the following:

We conclude this paper noting that in the general case, the moduli space of Humbert-Edge curves of type $n$ , where $n\geq 5$ is an odd number, is isomorphic to the moduli space of hyperelliptic curves of genus $\frac{n-1}{2}$ . In general, a Humbert-Edge curve $X_n(\lambda _1,\dots,\lambda _{n-2})$ of type $n$ in ${\mathbb{P}}^n$ can be written as

\begin{align*} x_0^2 + x_1^2+ x_2^2 & =0 \\ \lambda _1 x_0^2 + x_1^2 + x_3^2 & =0 \\ \lambda _2 x_0^2 + x_1^2+ x_4^2 & =0 \\ & \vdots \\ \lambda _{n-2} x_0^2 + x_1^1 + x_n^2 &=0, \end{align*}

where $\lambda _1,\dots,\lambda _{n-2}\in \mathbb{C}\backslash \{0,1\}$ are different (see [ Reference Carocca, González-Aguilera, Hidalgo and Rodríguez3 , Section 4.1]).

Since we have an analogous of Proposition 3.9 for the general case (see [ Reference Hidalgo13 , Section 2.3]), then the argument to construct the moduli space of Humbert-Edge curves of type $5$ in fact holds true for the general case of Humbert-Edge curves of type $n$ (see Section 4.2 of [ Reference Carocca, González-Aguilera, Hidalgo and Rodríguez3 ]). Therefore, we can consider the moduli space $\mathcal{HE}_n$ of Humbert-Edge curves of type $n$ , and we have a well-defined map $q_n\,:\,\mathcal{HE}_n\to \mathcal{M}_{g_n}$ where $\mathcal{M}_{g_n}$ is the moduli space of curves of genus $g_n=2^{n-2}(n-3)+1$ .

On the other hand, a hyperelliptic curve $C(\lambda _1,\dots,\lambda _{2g-1})$ of genus $g$ is given by the equation

\begin{equation*} y^2=x(x-1)(x-\lambda _1)\cdots (x-\lambda _{2g-1}), \end{equation*}

where $\lambda _1,\dots,\lambda _{2g-1}\in \mathbb{C}\backslash \{0,1\}$ are different. Since also Proposition 3.10 can be generalized in this general context (see [ Reference Edge8 , Section III.7.3]), under the assumptions that $n\geq 5$ is odd and $n-2=2g-1=d$ , we can define a surjective map

\begin{align*} f_d\,:\,\mathcal{H}_g & \rightarrow \mathcal{HE}_n \\ [C(\lambda _1,\dots,\lambda _d)] & \mapsto [X_n(\lambda _1,\dots,\lambda _d)], \end{align*}

where $\mathcal{H}_g$ denotes the moduli space of hyperelliptic curves of genus $g=\frac{n-1}{2}$ . Finally, applying the argument of Proposition 3.11 and using the fact that the natural map $q_n\,:\,\mathcal{HE}_n\rightarrow \mathcal{M}_{g_n}$ is injective (see [Reference Carocca, González-Aguilera, Hidalgo and Rodríguez3, Proposition 4.3]) we conclude the following: