# Extensions of Gorenstein weighted projective 3-spaces and characterization
of the primitive curves of their surface sections.

Bruno Dewer

###### Abstract

We investigate the Gorenstein weighted projective spaces of dimension 3. Given
such a space $\mathbf{P}$, our first focus is its maximal extension in its
anticanonical model $\mathbf{P}\subset\mathbf{P}^{g+1}$, i.e., the variety
$Y\subset\mathbf{P}^{g+1+r}$ of largest dimension such that $Y$ is not a cone
and $\mathbf{P}$ is a linear section of $Y$. In [DS21] Thomas Dedieu and
Edoardo Sernesi have computed the dimension of $Y$ by cohomological
computations on the canonical curves inside $\mathbf{P}$. We give an explicit
description of $Y$ in the cases where it was not known. Next, we examine the
general anticanonical divisors of $\mathbf{P}$. These are K3 surfaces, not
necessarily primitively polarized. We give a geometric characterization of the
curve sections in their primitive polarization.

##  1 Introduction

The notion of extendability for projective varieties consists in the
following.

######  Definition 1.1.

A projective variety $X\subset\mathbf{P}^{N}$ is _extendable_ if there exists
$X_{1}\subset\mathbf{P}^{N+1}$ which is not a cone, such that $X$ is a
hyperplane section of $X_{1}$.

Moreover, if $X_{r}\subset\mathbf{P}^{N+r}$ is not a cone and $X$ can be
obtained as the intersection of $X_{r}$ with a linear subspace of dimension
$N$, we say that $X_{r}$ is an $r-$extension of $X$, or also that $X$ has been
extended $r$ times. If moreover there exists no extension of $X$ of larger
dimension, we say that $X_{r}$ is maximal.

The topic of this paper is the extendability of weighted projective spaces,
more precisely those of dimension $3$ that are Gorenstein.

Given four coprime positive integers $a_{0},a_{2},a_{2}$ and $a_{3}$, the
weighted projective space $\mathbf{P}=\mathbf{P}(a_{0},a_{2},a_{2},a_{3})$ is
$\mathrm{Proj}(R)$ where $R=\mathbf{C}[x_{0},x_{1},x_{2},x_{3}]$ endowed with
the grading $\deg x_{i}=a_{i}$. By definition, it is Gorenstein if its
anticanonical divisor class is Cartier, which holds if and only if all the
$a_{i}$’s divide their sum (see for instance Theorem 3.3.4 in [Do81]). As
mentioned in [Pr04], among all the weighted projective spaces of dimension
$3$, there are exactly $14$ which are Gorenstein.

Assume $\mathbf{P}\subset\mathbf{P}^{g+1}$ is Gorenstein and embedded by its
anticanonical linear system, then its general hyperplane section $S$ is a K3
surface with canonical singularities. The induced polarization
$(S,-K_{\mathbf{P}}{}|_{S})$ is of genus $g$, meaning that the general member
$\Gamma$ of $-K_{\mathbf{P}}{}|_{S}$ is a canonical curve of genus $g$. As a
consequence of this, any extension of $\mathbf{P}$ is also an extension of
$\Gamma\subset\mathbf{P}^{g-1}$.

Consider then a smooth canonical curve $\Gamma\subset\mathbf{P}^{g-1}$
obtained as a linear section of $\mathbf{P}\subset\mathbf{P}^{g+1}$ and the
number $\alpha(\Gamma,K_{\Gamma})$ introduced in [Lvo92], which can be
computed as the corank of the Gauß-Wahl application of the polarization
$(\Gamma,K_{\Gamma})$ (see Definition 2.1). In this situation, it follows from
[BM87] and [Wah87] that $\alpha(\Gamma,K_{\Gamma})$ is nonzero. Then by
Theorem 2.1 and Theorem 2.17 in [CDS20] we have $\dim
Y=1+\alpha(\Gamma,K_{\Gamma})$, for $Y$ a maximal extension of $\Gamma$. This
allows us to know exactly how many times $\mathbf{P}$ can be extended, by
Corollary 6.4 in [DS21].

The full list of the Gorenstein weighted projective $3-$spaces, as well as the
maximal extensions which are known from [DS21], is given below.

Notice that the space $\mathbf{P}(1,1,1,3,5,5,5)$ is denoted by
$\mathbf{P}(1^{3},3,5^{3})$, as the weights $1$ and $5$ each appear three
times, and the weight $3$ appears once. From now on, we will adopt this
notation for brevity. We also adopt the following convention: when
$\mathbf{P}$ is not extendable, we say that its unique maximal extension is
itself.

$\begin{array}[]{|l|l|l|}\hline\cr\mathbf{P}&\text{extendable?}&\text{maximal
extension}\\\ \hline\cr\mathbf{P}(1,1,1,1)&\text{no}&\text{itself}\\\
\mathbf{P}(1,1,1,3)&\text{no}&\text{itself}\\\
\mathbf{P}(1,1,4,6)&\text{no}&\text{itself}\\\
\mathbf{P}(1,2,2,5)&\text{yes}&\text{sextic hypersurface of
}\mathbf{P}(1^{3},3,5^{3})\\\ \mathbf{P}(1,1,2,4)&\text{no}&\text{itself}\\\
\mathbf{P}(1,3,4,4)&\text{yes}&\text{quartic hypersurface of
}\mathbf{P}(1^{4},3^{4})\\\ \mathbf{P}(1,1,2,2)&\text{no}&\text{itself}\\\
\mathbf{P}(2,3,3,4)&\text{yes}&\text{cubic hypersurface of
}\mathbf{P}(1^{5},2^{5})\\\
\hdashline[8pt/3pt]\mathbf{P}(1,4,5,10)&\text{yes}&\text{was not known}\\\
\mathbf{P}(1,2,6,9)&\text{yes}&\text{was not known}\\\
\mathbf{P}(1,2,3,6)&\text{no}&\text{itself}\\\
\mathbf{P}(1,3,8,12)&\text{yes}&\text{was not known}\\\
\mathbf{P}(1,6,14,21)&\text{yes}&\text{was not known}\\\
\mathbf{P}(2,3,10,15)&\text{yes}&\text{was not known}\\\ \hline\cr\end{array}$

######  Definition 1.2.

Let $\mathcal{K}_{g}^{i}$ be the moduli space of the polarized surfaces
$(S,L)$ with $S$ a K3 surface and $L$ an ample Cartier divisor on $S$ such
that the general members of $L$ are genus $g$ curves and the index of $L$ in
$\mathrm{Pic}(S)$ is equal to $i$. In other words, $i$ is the largest positive
integer $r$ such that $\frac{1}{r}L\in\mathrm{Pic}(S)$.

For all $g$ and $i$ we consider the function
$\alpha:\mathcal{K}_{g}^{i}\to\mathbf{Z}$ given by
$\alpha(S,L)=\alpha(\Gamma^{\prime},K_{\Gamma^{\prime}})-1$, where
$\Gamma^{\prime}$ is a general member of $L$. Given $\mathbf{P}$ a Gorenstein
weighted projective space of dimension $3$ and $S$ a general anticanonical
divisor of $\mathbf{P}$, let $g$ and $i_{S}$ respectively denote the genus and
the index of the induced polzarization $(S,-K_{\mathbf{P}}|_{S})$. This
polarization is then a member of the moduli space $\mathcal{K}_{g}^{i_{S}}$.
T.Dedieu and E.Sernesi have computed and stated in Proposition 6.2 of [DS21]
that in each of the 14 cases, there is a constant $\alpha_{g}^{i_{S}}$ such
that $\alpha$ takes the value $\alpha_{g}^{i_{S}}$ on a dense open subset of
$\mathcal{K}_{g}^{i_{S}}$. The first 8 cases of the list above are those for
which $\alpha(S,-K_{\mathbf{P}}|_{S})=\alpha_{g}^{i_{S}}$, and we are going to
examine the ones for which this equality doesn’t hold.

The core results of this paper, stated and proved in Section 4 and Section 5,
are summarized in the two following theorems.

######  Theorem 1.3.

Assume that the polarization $(S,-K_{\mathbf{P}}|_{S})$ is not general in
$\mathcal{K}_{g}^{i_{S}}$, in the sense that

$\alpha(S,-K_{\mathbf{P}}|_{S})>\alpha^{i_{S}}_{g}.$

Then $\mathbf{P}$ is one of the last six items of the list given above. Each
of them admits a maximal extension $Y$ which has a description as follows.

$\begin{array}[]{|l|l|l|}\hline\cr\mathbf{P}&Y&\textsl{dim}(Y)\\\
\hline\cr&&\\\\[-10.0pt] \mathbf{P}(1,4,5,10)&\textsl{nongeneral quintic of
}\mathbf{P}(1^{3},2,4^{3})&5\\\ \mathbf{P}(1,2,6,9)&\textsl{nongeneral
}10-\textsl{ic of }\mathbf{P}(1^{2},3,5,9^{2})&4\\\
\mathbf{P}(1,2,3,6)&\mathbf{P}(1,2,3,6)&3\\\
\mathbf{P}(1,3,8,12)&\textsl{nongeneral }9-\textsl{ic of
}\mathbf{P}(1^{2},3,4,8^{2})&4\\\ \mathbf{P}(1,6,14,21)&\textsl{nongeneral
heptic of }\mathbf{P}(1^{2},2,3,6^{2})&4\\\
\mathbf{P}(2,3,10,15)&\textsl{codim. }2\textsl{ complete intersection in
a}&5\\\ &\mathbf{P}(1^{2},2,3,5^{3})-\textsl{bundle over }\mathbf{P}^{1}&\\\
\hline\cr\end{array}$

Next, we consider $C$ a general member of the primitive divisor class
$-\frac{1}{i_{S}}K_{\mathbf{P}}|_{S}$. We focus on the same cases as in
Theorem 1.3, and to give an insight on the geometry of $S$, we give a
geometric characterization of $C$.

######  Theorem 1.4.

Let $\mathbf{P}$ be a Gorenstein weighted projective space and $S$ a general
anticanonical divisor of $\mathbf{P}$ such that
$\alpha(S,-K_{\mathbf{P}}|_{S})>\alpha^{i_{S}}_{g}$. If $C$ is a general
member of $-\frac{1}{i_{S}}K_{\mathbf{P}}|_{S}$ then it is as follows.

$\begin{array}[]{|l|l|}\hline\cr\mathbf{P}&C\\\ \hline\cr&\\\\[-10.0pt]
\mathbf{P}(1,4,5,10)&\textsl{plane quintic with a total inflection point}\\\
\mathbf{P}(1,2,6,9)&\textsl{smooth hyperelliptic curve of genus }4\\\
\mathbf{P}(1,2,3,6)&\textsl{normalization of a plane sextic with an
oscnode}\\\ \mathbf{P}(1,3,8,12)&\textsl{trigonal curve of genus }7\textsl{
with a total ramification point}\\\ \mathbf{P}(1,6,14,21)&\textsl{blowup of a
plane }21-\textsl{ic curve at 8 heptuple points}\\\
\mathbf{P}(2,3,10,15)&\textsl{normalization of a nodal }6-\textsl{gonal curve
of genus }16\textsl{ such that the}\\\ &g_{6}^{1}\textsl{ has two members of
the form }6p\textsl{ and }2p_{1}+2p_{2}+2p_{3}\textsl{ respectively}\\\
\hline\cr\end{array}$

Conversely, for all items on the list except $\mathbf{P}(1,2,3,6)$, any curve
with the given description is a member of
$-\frac{1}{i_{S}}K_{\mathbf{P}}|_{S}$.

The organization of the article is as follows. In Section 2 we go over some
information and definitions about the $3-$dimensional Gorenstein weighted
projective spaces. In Section 3 we introduce a birational model of
$\mathbf{P}$ which realizes $S$ as a nongeneral anticanonical divisor of
another weighted projective $3-$space. This allows us in Section 4 to express
$\Gamma$ as a complete intersection of two surfaces of different degrees and
to construct $Y$ as a hypersurface in a larger weighted projective space in
all cases but one, which requires additional work. In Section 5 we consider
the primitive polarization of $S$ and give a geometric characterization of the
general curve $C$ in the linear system
$|-\frac{1}{i_{S}}K_{\mathbf{P}}{}|_{S}|$.

##  2 The Gorenstein weighted projective spaces of dimension $3$

We refer to §5 and §6 in [Ia00] as a reference for basic facts about weighted
projective spaces.

Let $\mathbf{P}=\mathbf{P}(a_{0},a_{1},a_{2},a_{3})$ be a weighted projective
space. Its anticanonical divisor class is that of degree
$a_{0}+a_{1}+a_{2}+a_{3}$ surfaces and its Picard group is generated by
$[\mathcal{O}_{\mathbf{P}}(l)]$ with
$l=\mathrm{lcm}(a_{0},a_{1},a_{2},a_{3})$. Hence $-K_{\mathbf{P}}$ is Cartier
if and only if $a_{i}$ is a divisor of $a_{0}+a_{1}+a_{2}+a_{3}$ for all $i$.
Assume that this conditions holds, making $\mathbf{P}$ Gorenstein. We will
observe that $\mathbf{P}$ is embedded in $\mathbf{P}^{g+1}$ by its
anticanonical linear system. Let $\Gamma\subset\mathbf{P}^{g-1}$ be a curve
section of $\mathbf{P}$ cut out by two general hyperplanes. The adjunction
formula yields $K_{\Gamma}=-K_{\mathbf{P}}{}|_{\Gamma}.$ Hence $\Gamma$ in
$\mathbf{P}^{g-1}$ is a canonical curve, and $\mathbf{P}$ can only be extended
finitely many times, by Theorem 2.3.

We list all the Gorenstein weighted projective spaces of dimension $3$ in
Table 1 below, together with the following information. If $a_{i}$ are the
weights of $\mathbf{P}$, $l=\mathrm{lcm}(a_{0},a_{1},a_{2},a_{3})$ and
$\sigma=a_{0}+a_{1}+a_{2}+a_{3}$, then the index $i$ of $-K_{\mathbf{P}}$ in
$\mathrm{Pic}(\mathbf{P})$ is equal to $\frac{\sigma}{l}$.

$\hypertarget{Table
1}{}\begin{array}[]{|l|l|l|l|l|}\hline\cr\\#&\mathbf{P}&l&\sigma&i\\\
\hline\cr&&&&\\\\[-10.0pt] 1&\mathbf{P}(1,1,1,1)&1&4&4\\\
2&\mathbf{P}(1,1,1,3)&3&6&2\\\ 3&\mathbf{P}(1,1,4,6)&12&12&1\\\
4&\mathbf{P}(1,2,2,5)&10&10&1\\\ 5&\mathbf{P}(1,1,2,4)&4&8&2\\\
6&\mathbf{P}(1,3,4,4)&12&12&1\\\ 7&\mathbf{P}(1,1,2,2)&2&6&3\\\
8&\mathbf{P}(2,3,3,4)&12&12&1\\\ 9&\mathbf{P}(1,4,5,10)&20&20&1\\\
10&\mathbf{P}(1,2,6,9)&18&18&1\\\ 11&\mathbf{P}(1,2,3,6)&6&12&2\\\
12&\mathbf{P}(1,3,8,12)&24&24&1\\\ 13&\mathbf{P}(1,6,14,21)&42&42&1\\\
14&\mathbf{P}(2,3,10,15)&30&30&1\\\ \hline\cr\end{array}$

Table 1

###  2.1 Extendability of $\Gamma$ and $\mathbf{P}$

For each $\mathbf{P}$ in Table 1, if $S$ is a general anticanonical divisor,
then the couple $(S,-K_{\mathbf{P}}|_{S})$ represents an element of the moduli
space $\mathcal{K}_{g}^{i_{S}}$ (see Definition 1.2). We focus here on the
last $6$ examples ($\\#9$ to $\\#14$) which are mentioned in Theorem 1.3. A
maximal extension of $\mathbf{P}$ in these cases was not known so far.

######  Definition 2.1.

Let $X\subset\mathbf{P}^{N}$ be a projective variety, and
$L=\mathcal{O}_{\mathbf{P}^{N}}(1)|_{X}$. we introduce

$\alpha(X,L)=h^{0}(N_{X/\mathbf{P}^{N}}\otimes L^{-1})-N-1.$

So that, if $X=\Gamma$ is a canonical curve in $\mathbf{P}^{g-1}$, one has
$\alpha(\Gamma,L)=\alpha(\Gamma,K_{\Gamma})$.

######  Lemma 2.2.

In the case where $\Gamma$ is a canonical curve, it holds that
$\alpha(\Gamma,K_{\Gamma})=\mathrm{cork}(\Phi_{K_{\Gamma}})$ where
$\Phi_{K_{\Gamma}}$ is the Gauß-Wahl map of the polarization
$(\Gamma,K_{\Gamma})$.

We refer to §2 in [BM87], and [Wah87] for the definition of the Gauß-Wahl map
of a polarized curve and to Lemma 3.2 in [CDS20] for a proof of this lemma.
The value of $\alpha(\Gamma,K_{\Gamma})$ for $\Gamma$ a general curve linear
section of any Gorenstein weighted projective $3-$space have been computed by
T.Dedieu and E.Sernesi and we display these values in the relevant cases in
Table 2 below. This allows us to know the dimension of any maximal extension
of $\Gamma$ by the following theorem.

######  Theorem 2.3 ([CDS20], Theorems 2.1, 2.17).

Let $\Gamma\subset\mathbf{P}^{g-1}$ be a smooth canonical curve with $g\geq
11$ and such that $\mathrm{Cliff}(\Gamma)>2$. Then $\Gamma$ is extendable only
$\alpha(\Gamma,K_{\Gamma})$ times, i.e., there exists $Y$ in
$\mathbf{P}^{g-1+\alpha(\Gamma,K_{\Gamma})}$ such that
$\dim(Y)=1+\alpha(\Gamma,K_{\Gamma})$, which is a maximal extension of
$\Gamma$.

In addition, there exists a maximal extension $Y$ of $\Gamma$ which is
universal, meaning that for each surface extension $S$ of $\Gamma$, there is a
unique $g-$plane $\Lambda\subset\mathbf{P}^{g-1+\alpha(\Gamma,K_{\Gamma})}$
such that $\Gamma\subset\Lambda$ and $S=Y\cap\Lambda$.

Let us consider $\Gamma$ a general linear curve section of
$\mathbf{P}\subset\mathbf{P}^{g+1}$ where $\mathbf{P}$ is a Gorenstein
weighted projective $3-$space. Hence, $\Gamma$ is a general hyperplane section
of a K3 surface $S\subset\mathbf{P}$. Such a surface only has isolated
singularities, so by Bertini’s Theorem, $\Gamma$ is smooth. The values of $g$
are listed in Table 2 below, and in each case we have $g\geq 11$ and
$\mathrm{Cliff}(\Gamma)>2$ by Corollary 2.8, to the effect that Theorem 2.3
applies to $\Gamma$.

As a consequence, we know that any extension of $\mathbf{P}$ of dimension
$1+\alpha(\Gamma,K_{\Gamma})$ is maximal. In particular, in any case for which
$\alpha(\Gamma,K_{\Gamma})=2$, the threefold $\mathbf{P}$ is not extendable
and it is the universal extension of $\Gamma$.

We list in Table 2 examples #9 to #14 coupled with the data of $i_{S}$, the
genera of $\Gamma$ and $C$, where $\Gamma$ is a general member of
$-K_{\mathbf{P}}|_{S}$ and $C$ is a general member of
$-\frac{1}{i_{S}}K_{\mathbf{P}}|_{S}$. The value of
$\alpha(\Gamma,K_{\Gamma})$ and the datum of the singular points of $S$ are
also provided.

$\hypertarget{Table
2}{}\begin{array}[]{|l|l|l|l|l|l|l|}\hline\cr\\#&\mathbf{P}&i_{S}&g=g(\Gamma)&g(C)&\alpha(\Gamma,K_{\Gamma})&\mathrm{Sing}(S)\\\
\hline\cr&&&&&&\\\\[-10.0pt] 9&\mathbf{P}(1,4,5,10)&2&21&6&4&A_{1},2A_{4}\\\
10&\mathbf{P}(1,2,6,9)&3&28&4&3&3A_{1},A_{2}\\\
11&\mathbf{P}(1,2,3,6)&2&25&7&2&2A_{1},2A_{2}\\\
12&\mathbf{P}(1,3,8,12)&2&25&7&3&2A_{2},A_{3}\\\
13&\mathbf{P}(1,6,14,21)&1&22&22&3&A_{1},A_{2},A_{6}\\\
14&\mathbf{P}(2,3,10,15)&1&16&16&4&3A_{1},2A2,A_{4}\\\ \hline\cr\end{array}$

Table 2

The anticanonical divisor $-K_{\mathbf{P}}$ being very ample by Theorem 2.5,
its embeds $\mathbf{P}$ in $\mathbf{P}^{g+1}$ as a variety of degree
$(-K_{\mathbf{P}})^{3}=2g-2$. In that model, recall that $\mathbf{P}$ can only
be extended $\alpha(\Gamma,K_{\Gamma})-2$ times. The only one which is not
extendable in our list is $\mathbf{P}(1,2,3,6)$.

###  2.2 The very ampleness of $-K_{\mathbf{P}}$

Let us prove now that $-K_{\mathbf{P}}$ is very ample in the cases $\\#9$ to
$\\#14$ given above, using the following lemma.

######  Lemma 2.4.

Let $X$ be a projective variety and $D$ an ample Cartier divisor on $X$. Let
$A_{n}=H^{0}(X,\mathcal{O}_{X}(nD))$ for all $n\in\mathbf{N}$ and
$A=\oplus_{n\geq 1}A_{n}$. If $A$ is generated by $A_{1}$ as a
$\mathbf{C}-$algebra, then $D$ is very ample.

###### Proof.

Let $q$ be a positive integer such that $Z=qD$ is very ample. It induces an
embedding of $X$ into a projective space, thus

$X\simeq\mathrm{Proj}(A^{(q)})\simeq\mathrm{Proj}(A),$

where $A^{(q)}$ is the graded ring such that $(A^{(q)})_{d}=A_{dq}$. Let

$\varphi:X\dashrightarrow\mathbf{P}(H^{0}(X,\mathcal{O}_{X}(D)))$

be the map induced by $D$. If $L_{n}$ is the image of the multiplication map

$H^{0}(X,\mathcal{O}_{X}(D))^{\otimes n}\to H^{0}(X,\mathcal{O}_{X}(nD)),$

then it holds that $\varphi(X)=\mathrm{Proj}(L)$ with $L=\oplus_{n\geq
1}L_{n}$. The condition that $A$ is generated by $A_{1}$ is equivalent to
$A=L$, and therefore it yields $\varphi(X)=\mathrm{Proj}(A)\simeq X$. ∎

######  Theorem 2.5.

Let $\mathbf{P}(a_{0},a_{1},a_{2},a_{3})$ be one of the Gorenstein weighted
projective $3-$spaces listed in Table 2. Then $-K_{\mathbf{P}}$ is very ample.

###### Proof.

Let $S$ be a general member of $|-K_{\mathbf{P}}|$ and $\Gamma$ a general
member of $|-K_{\mathbf{P}}|_{S}|$. We first prove that $-K_{\mathbf{P}}|_{S}$
is very ample. Thanks to Lemma 2.4, it is enough to prove that
$H^{0}(S,\mathcal{O}_{S}(-K_{\mathbf{P}}|_{S}))^{\otimes n}\to
H^{n}(S,\mathcal{O}_{S}(-nK_{\mathbf{P}}|_{S}))$ is onto for all $n\geq 1$. It
is naturally the case for $n=1$. Assume now that $n>1$ and that it holds up to
rank $n-1$. Consider the following commutative diagram.

${H^{0}(\Gamma,\mathcal{O}_{\Gamma}(K_{\Gamma}))^{\otimes
n}}$${H^{0}(\Gamma,\mathcal{O}_{\Gamma}(nK_{\Gamma}))}$${H^{0}(S,\mathcal{O}_{S}(-K_{\mathbf{P}}|_{S}))^{\otimes
n}}$${H^{0}(S,\mathcal{O}_{S}(-nK_{\mathbf{P}}|_{S}))}$${H^{0}(S,\mathcal{O}_{S}(-K_{\mathbf{P}}|_{S}))^{\otimes
n-1}}$${H^{0}(S,\mathcal{O}_{S}(-(n-1)K_{\mathbf{P}}|_{S}))}$$\scriptstyle{\otimes
f}$$\scriptstyle{\cdot f}$

Here, $f$ is a global section of $\mathcal{O}_{S}(-K_{\mathbf{P}}|_{S})$ such
that $\Gamma=\left\\{f=0\right\\}$ on $S$. The bottom arrow is onto by the
induction hypothesis.

The right column is the restriction exact sequence. The surjectivity of the
top two vertical arrows follows as a consequence of Lemma 2.1 and Proposition
2.1 in [Sho71]. This applies since $g(\Gamma)\geq 3$ and $\Gamma$ is
nonhyperelliptic, ensuring that $\Gamma$ is projectively normal. The
surjectivity of the top horizontal arrow follows from the fact that
$K_{\Gamma}$ is very ample. We refer to Theorem 2.7 for a case-by-case proof
that $\Gamma$ is nonhyperelliptic and $K_{\Gamma}$ is very ample.

As the vertical sequence on the right is exact, the surjectivity of the middle
arrow follows from the surjectivity of the bottom and the top arrows by
diagram chasing. Hence the induction holds and
$H^{0}(S,\mathcal{O}_{S}(-K_{\mathbf{P}}|_{S}))^{\otimes n}\to
H^{n}(S,\mathcal{O}_{S}(-nK_{\mathbf{P}}|_{S}))$ is onto for all $n\geq 1$.

Now, by the fact that K3 surfaces are projectively normal (see [SD74]), the
restriction map
$H^{0}(\mathbf{P},\mathcal{O}_{\mathbf{P}}(-nK_{\mathbf{P}}))\to
H^{0}(S,\mathcal{O}_{S}(-nK_{\mathbf{P}}|_{S}))$ is onto for all $n\geq 1$.
Consider the following commutative diagram with $n>1$, whose right column is
exact.

${H^{0}(S,\mathcal{O}_{S}(-K_{\mathbf{P}}|_{S}))^{\otimes
n}}$${H^{0}(S,\mathcal{O}_{S}(-nK_{\mathbf{P}}|_{S}))}$${H^{0}(\mathbf{P},\mathcal{O}_{\mathbf{P}}(-K_{\mathbf{P}}))^{\otimes
n}}$${H^{0}(\mathbf{P},\mathcal{O}_{\mathbf{P}}(-nK_{\mathbf{P}}))}$${H^{0}(\mathbf{P},\mathcal{O}_{\mathbf{P}}(-K_{\mathbf{P}}))^{\otimes
n-1}}$${H^{0}(\mathbf{P},\mathcal{O}_{\mathbf{P}}(-(n-1)K_{\mathbf{P}}))}$

We may assume that the bottom arrow is surjective by an induction hypothesis,
as it is the case for $n=1$. By an analogous argument of diagram chasing as
above, the middle arrow
$H^{0}(\mathbf{P},\mathcal{O}_{\mathbf{P}}(-K_{\mathbf{P}}))^{\otimes n}\to
H^{n}(\mathbf{P},\mathcal{O}_{\mathbf{P}}(-nK_{\mathbf{P}}))$ is onto. This is
true for all $n$ and thus the conclusion follows from Lemma 2.4. ∎

The proof of the nonhyperellipticity of $\Gamma$ and the very ampleness of
$K_{\Gamma}$ requires that we state the following lemma.

######  Lemma 2.6.

Let $S$ be a K3 surface and $\Gamma\subset S$ a smooth curve of genus at least
$2$. If $\Gamma$ is hyperelliptic, then there exists a line bundle
$\mathcal{J}$ on $S$ such that the restriction $\mathcal{J}|_{\Gamma}$ is a
$g^{1}_{2}$ (i.e., a pencil of degree $2$ divisors).

A proof of this lemma relies on the first theorem of [GL87], by the fact that
$\Gamma$ is hyperlliptic iff. $\mathrm{Cliff}(\Gamma)=0$, in which case there
is a line bundle on $S$ whose restriction to $\Gamma$ is a pencil of degree
$2$.

We will apply this lemma together with the known fact that, for a given curve
$\Gamma$ of genus $g\geq 2$, $K_{\Gamma}$ is very ample if and only if
$\Gamma$ is nonhyperelliptic (see Proposition IV.5.2. in [Har77]).

######  Theorem 2.7.

Let $\mathbf{P}$ be one of the Gorenstein weighted projective $3-$spaces
listed in Table 2. Then the general linear curve section
$\Gamma\subset\mathbf{P}$ in nonhyperelliptic and $K_{\Gamma}$ is very ample.

###### Proof.

Being a general anticanonical divisor of $\mathbf{P}$, $S$ has isolated
singularities. By Bertini’s Theorem, the general $\Gamma$ in $S$ is smooth.

In case $\\#10$, the index of $-K_{\mathbf{P}}|_{S}$ in $\mathrm{Pic}(S)$ is
equal to $3$. Hence $\Gamma=3C$ where $C$ is a Cartier divisor on $S$. By
Lemma 2.6, if $\Gamma$ is hyperelliptic, there exists a line bundle
$\mathcal{J}$ on $S$ such that $\mathcal{J}|_{\Gamma}$ is a $g_{2}^{1}$. Hence
$\mathcal{J}|_{C}$ has degree $\frac{2}{3}$, which is not possible.

In cases $\\#9,\\#11$ and $\\#12$, the index is equal to $2$, hence
$\Gamma=2C$ for some primitive Cartier divisor $C$ on $S$. Once again, if
$\Gamma$ is hyperelliptic, by Lemma 2.6 there exists a line bundle
$\mathcal{J}$ on $S$ such that $\mathcal{J}|_{\Gamma}$ is a $g_{2}^{1}$.
Therefore, $\mathcal{J}|_{C}$ is a $g_{1}^{1}$, i.e., $C$ is a rational curve.
But we know from the values of $g(C)$ listed in Table 2 that it is not the
case.

Lastly, in cases $\\#13$ and $\\#14$, the index is equal to $1$, so $\Gamma$
is primitive in $\mathrm{Pic}(S)$. We need to use the information which is
given in Table 5 and yields:

  1. $\mathtt{I}$.

for $\\#13$, $\Gamma=-7K_{\Sigma}$ where $\Sigma$ is a general sextic in
$\mathbf{P}(1,1,2,3)$. It holds by the adjunction formula that
$K_{\Gamma}=-6K_{\Sigma}|_{\Gamma}=\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6)|_{\Gamma}$
which is very ample.

  2. $\mathtt{II}$.

for $\\#14$, $\Gamma=-6K_{\Sigma}$ where $\Sigma$ is a general $10-$ic in
$\mathbf{P}(1,2,4,5)$. By the adjunction formula,
$K_{\Gamma}=-5K_{\Sigma}|_{\Gamma}=\mathcal{O}_{\mathbf{P}(1,2,4,5)}(10)|_{\Gamma}$
which is very ample, provided that $\Gamma$ does not meet the base point of
$\mathcal{O}_{\mathbf{P}(1,2,4,5)}(10)$, which holds by the generality
assumption.

In both cases, $K_{\Gamma}$ is very ample and so $\Gamma$ is nonhyperelliptic.
∎

######  Corollary 2.8.

In the setting of Theorem 2.7, the Clifford index of the curve $\Gamma$ is
strictly larger than $2.$

###### Proof.

Since the anticanonical model $\mathbf{P}\subset\mathbf{P}^{g+1}$ satisfies
the $N_{2}$ property, as stated in Proposition 6.1 in [DS21], so does the
curve $\Gamma$. It follows by the appendix of [GL84] that
$\mathrm{Cliff}(\Gamma)>2$. ∎

##  3 Birational models

Before constructing the maximal extensions, we study for each $\mathbf{P}$ in
Table 2 except $\mathbf{P}(1,2,3,6)$ a birational model
$\varphi:\mathbf{P}\dashrightarrow\mathbf{P}^{\prime}$ such that $\varphi$
restricts to an isomorphism on the general $S\in|-K_{\mathbf{P}}|$. This will
allow us to express the general $\Gamma\in|-K_{\mathbf{P}}|_{S}|$ as a
complete intersection of two equations of different degrees in
$\mathbf{P}^{\prime}$ and to construct extensions of $\Gamma$ in Section 4.

The first step consists in the introduction of a suitable Veronese map on each
$\mathbf{P}$, which is an embedding $v_{n}:\mathbf{P}\hookrightarrow X$ where
$X$ is a weighted projective space of dimension $4$ so that the anticanonical
model $\mathbf{P}\to\mathbf{P}^{g+1}$ factors as a composition

${\mathbf{P}}$${X}$${\mathbf{P}^{g+1}}$$\scriptstyle{v_{n}}$

where $X\dashrightarrow\mathbf{P}^{g+1}$ is a rational map that we will
specify.

###  3.1 The Veronese maps

Let $\mathbf{P}=\mathbf{P}(a_{0},a_{1},a_{2},a_{3})$ be a weighted projective
space. Denote $R=\mathbf{C}[x,y,z,w]$ with the suitable grading such that
$\mathbf{P}=\mathrm{Proj}(R)$. Then the following isomorphism holds for all
$n\in\mathbf{N}$,

$\mathbf{P}\simeq\mathrm{Proj}(R^{(n)}),$

where $R^{(n)}$ is the graded ring whose degree $d$ part is
$(R^{(n)})_{d}=R_{nd}$. This gives rise to an embedding $v_{n}$ which we refer
to as the $n-$Veronese map: given a fixed $n$, $R^{(n)}$ is isomorphic to a
quotient of the form $\nicefrac{{\mathbf{C}[y_{0},...,y_{m}]}}{{I}}$ where
$\mathbf{C}[y_{0},...,y_{m}]$ has a given grading $\deg y_{i}=d_{i}$ and $I$
is a homogeneous ideal. This makes $\mathbf{P}$ a subscheme of a larger
weighted projective space, as $\mathbf{P}\simeq
V(I)=\left\\{\mathrm{y}=[y_{0}:\cdots:y_{m}]\>|\>f(\mathrm{y})=0\text{ for all
}f\in I\right\\}$ in $\mathbf{P}(d_{0},...,d_{m})$.

######  Example 3.1.

The $5-$Veronese embedding of $\mathbf{P}=\mathbf{P}(1,4,5,10)$. Taking $n=5$
yields the isomorphism

$\mathbf{P}\simeq\mathrm{Proj}(\mathbf{C}[x^{5},xy,z,w,y^{5}])=\mathrm{Proj}\left(\nicefrac{{\mathbf{C}[u_{0},u_{1},u_{2},v,s]}}{{(u_{0}s-u_{1}^{5})}}\right).$

The grading on the right is the following: the $u_{i}$’s have weight $1$,
while $v$ has weight $2$ and $s$ has weight $4$. This realizes $\mathbf{P}$ as
the degree $5$ hypersurface given by the equation $u_{0}s=u_{1}^{5}$ in
$\mathbf{P}(1,1,1,2,4)$ through the following embedding.

$v_{5}:[x:y:z:w]\in\mathbf{P}\mapsto[u_{0}:u_{1}:u_{2}:v:s]=[x^{5}:xy:z:w:y^{5}].$

The choice $n=5$ is motivated by the fact that it is a divisor of $\sigma=20$,
with $-K_{\mathbf{P}}=\mathcal{O}_{\mathbf{P}}(\sigma)$. We can recover
$-K_{\mathbf{P}}$ from $v_{5}$ using the equality
$-K_{\mathbf{P}}=v_{5}^{*}\mathcal{O}_{\mathbf{P}(1,1,1,2,4)}(4)$.

In general, given $\mathbf{P}$ any Gorenstein space of the list, we can choose
$n$ a divisor of $\sigma$ and embed $\mathbf{P}$ by $v_{n}$ in a larger
weighted projective space $X$. This yields
$-K_{\mathbf{P}}=v_{n}^{*}\mathcal{O}_{X}(\frac{\sigma}{n})$.

For the items on our list from #9 to #14 a suitable choice of $n$ is given in
Table 3, realizing each time $\mathbf{P}$ as a hypersurface in a weighted
projective space $X$ of dimension $4$. Similarly as in Table 1, $\sigma$
refers to the degree of $-K_{\mathbf{P}}$ with regard to the grading of
$\mathbf{P}$.

$\hypertarget{Table
3}{}\begin{array}[]{|l|l|l|l|l|l|l|}\hline\cr\\#&\mathbf{P}&\sigma&n&X&v_{n}(\mathbf{P})\subset
X&-K_{\mathbf{P}}\\\ \hline\cr&&&&&&\\\\[-10.0pt]
9&\mathbf{P}(1,4,5,10)&20&5&\mathbf{P}(1,1,1,2,4)_{[u_{0}:u_{1}:u_{2}:v:s]}&\text{quintic
}(u_{0}s=u_{1}^{5})&v_{5}^{*}\mathcal{O}_{X}(4)\\\
10&\mathbf{P}(1,2,6,9)&18&2&\mathbf{P}(1,1,3,5,9)_{[u_{0}:u_{1}:v:s:t]}&10\text{-ic
}(u_{0}t=s^{2})&v_{2}^{*}\mathcal{O}_{X}(9)\\\
11&\mathbf{P}(1,2,3,6)&12&2&\mathbf{P}(1,1,2,3,3)_{[u_{0}:u_{1}:v:s_{0}:s_{1}]}&\text{quartic
}(u_{0}s_{0}=v^{2})&v_{2}^{*}\mathcal{O}_{X}(6)\\\
12&\mathbf{P}(1,3,8,12)&24&3&\mathbf{P}(1,1,3,4,8)_{[u_{0}:u_{1}:v:s:t]}&9\text{-ic
}(u_{0}t=v^{3})&v_{3}^{*}\mathcal{O}_{X}(8)\\\
13&\mathbf{P}(1,6,14,21)&42&7&\mathbf{P}(1,1,2,3,6)_{[u_{0}:u_{1}:v:s:t]}&\text{heptic
}(u_{0}t=u_{1}^{7})&v_{7}^{*}\mathcal{O}_{X}(6)\\\
14&\mathbf{P}(2,3,10,15)&30&3&\mathbf{P}(1,2,4,5,10)_{[u:v:s:t:r]}&12\text{-ic
}(vr=s^{3})&v_{3}^{*}\mathcal{O}_{X}(10)\\\ \hline\cr\end{array}$

Table 3

In each case, the anticanonical embedding
$\mathbf{P}\hookrightarrow\mathbf{P}^{g+1}$ factors as the composite map

${\mathbf{P}}$${X}$${\mathbf{P}^{g+1}.}$$\scriptstyle{v_{n}}$$\scriptstyle{|\mathcal{O}_{X}(\frac{\sigma}{n})|}$

By a divisibility criterion, we can check fairly easily that
$\mathcal{O}_{X}(\frac{\sigma}{n})$ is not always basepoint-free. This
criterion is purely numerical: $\mathcal{O}_{X}(\frac{\sigma}{n})$ is
basepoint-free if and only if $\frac{\sigma}{n}$ is divisible by all the
weights of $X$. Namely, it is not the case for #10, #12 and #14, for which the
induced map $X\dashrightarrow\mathbf{P}^{g+1}$ is nonregular.

###  3.2 The birational models

The goal now is to exhibit a birational map from $\mathbf{P}$ to another
$3-$dimensional weighted projective space $\mathbf{P}^{\prime}$ which realizes
$S$ as a nongeneral anticanonical divisor of $\mathbf{P}^{\prime}$.

In all cases but #11, the image of $X$ in $\mathbf{P}^{g+1}$ is a cone with a
point as vertex. This follows from the fact that $\frac{\sigma}{n}$ equals the
largest weight of $X$; say $X=\mathbf{P}(d_{0},d_{1},d_{2},d_{3},d_{4})$ with
$\frac{\sigma}{n}=d_{4}$. Then the map given by the linear system
$|\mathcal{O}_{X}(\frac{\sigma}{n})|$ is the following.

$[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]\in
X\mapsto[\mathtt{f}_{0}:\mathtt{f}_{1}:\cdots:\mathtt{f}_{s}:x_{4}]$

where the $\mathtt{f}_{i}$’s form a basis of the degree $\frac{\sigma}{n}$
homogeneous polynomials in the variables $x_{0},x_{1},x_{2}$ and $x_{3}$.
Hence the equations for the image of $X$ encode the algebraic relations
between the $\mathtt{f}_{i}$’s and do not involve $x_{4}$.

This suggests projecting from the vertex point of this cone to
$\mathbf{P}^{g}$. This acts trivially on $S$ provided that it is general,
since $S\subset\mathbf{P}^{g}$ is a hyperplane section of $\mathbf{P}$.

This yields a birational map
$\varphi:\mathbf{P}\dashrightarrow\mathbf{P}^{\prime}$, where
$\mathbf{P}^{\prime}$ is the weighted projective $3-$space whose weights are
those of $X$ but the last one. In other words, if
$X=\mathbf{P}(d_{0},d_{1},d_{2},d_{3},d_{4})$ with $d_{4}=\frac{\sigma}{n}$,
then $\mathbf{P}^{\prime}=\mathbf{P}(d_{0},d_{1},d_{2},d_{3})$. The
restriction of this map to $S$ is an isomorphism; the image of $S$ is K3 so it
is a (nongeneral) anticanonical divisor of $\mathbf{P}^{\prime}$. We denote by
$\mathcal{L}$ the (noncomplete) linear system whose members are the
anticanonical divisors of $\mathbf{P}^{\prime}$ which are the direct images of
all $D\in|-K_{\mathbf{P}}|$,

$\mathcal{L}=\varphi_{*}|-K_{\mathbf{P}}|\subset|-K_{\mathbf{P}^{\prime}}|,$

so that he surface $\varphi(S)$ is a general member of $\mathcal{L}$.

Since $S\simeq\varphi(S)$ we will drop the notation $\varphi(S)$ for the sake
of brevity and use $S$ instead. Likewise, we will refer to $\varphi(\Gamma)$
as $\Gamma$. As our computations will show, the restriction of $\mathcal{L}$
to $S$ has $|\mathcal{O}_{\mathbf{P}^{\prime}}(\frac{\sigma}{n})|_{S}|$ as its
mobile part. That way, $\Gamma$ is cut out on $\mathbf{P}^{\prime}$ by two
equations of different degrees.

######  Example 3.2.

We have seen in Example 3.1 that the $5-$Veronese map on
$\mathbf{P}=\mathbf{P}(1,4,5,10)$ embeds it in $X=\mathbf{P}(1,1,1,2,4)$. The
anticanonical model of $\mathbf{P}$ factors through

${X}$${\mathbf{P}^{22}.}$$\scriptstyle{|\mathcal{O}_{X}(4)|}$

Let $\mathbf{P}^{\prime}=\mathbf{P}(1,1,1,2)$. It is embedded in
$\mathbf{P}^{21}$ by $|\mathcal{O}_{\mathbf{P}^{\prime}}(4)|$, making $X$ a
cone over $\mathbf{P}^{\prime}$. As $\mathbf{P}$ passes through the vertex of
$X$, projecting from the vertex point onto $\mathbf{P}^{\prime}$ induces a
rational map $\varphi$ from $\mathbf{P}$ to $\mathbf{P}^{\prime}$. This map
restricts to an isomorphism on $S$, since $S$ is a general hyperplane section
of $\mathbf{P}$ on which the projection map acts as the identity. In
homogeneous coordinates, we can express $\varphi$ as

$\varphi:[x:y:z:w]\in\mathbf{P}\mapsto[u_{0}:u_{1}:u_{2}:v]=[x^{5}:xy:z:w].$

This is the $5-$Veronese map without its last component $y^{5}$.

######  Lemma 3.3.

The map $\varphi$ given in Example 3.2 is birational.

###### Proof.

The map $\varphi$ is toric, so we can consider the transformation it
represents on the respective fans for $\mathbf{P}$ and $\mathbf{P}^{\prime}$.

The map $\varphi$ is regular outside the point for which $x=z=w=0$. We refer
to this point as $p_{y}$ and call it the indeterminacy point of $\varphi$.
Since it is a toric point, we may consider the toric map corresponding to a
weighted blowup of $\mathbf{P}$ at $p_{y}$. This consists in a subdivision of
the cone of the affine chart $\left\\{y\neq 0\right\\}$.

An algorithm for the construction of fans of weighted projective spaces is
given in Proposition 2.8 of [RT13]. This gives rise to the following fan in
$\mathbf{Z}^{3}$ for the toric variety $\mathbf{P}$.

$\Sigma_{\mathbf{P}}=\mathrm{Fan}\left(\left[\begin{array}[]{c}-4\\\ -5\\\
-10\end{array}\right],\left[\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right],\left[\begin{array}[]{c}0\\\ 1\\\
0\end{array}\right],\left[\begin{array}[]{c}0\\\ 0\\\
1\end{array}\right]\right)=:\mathrm{Fan}(\mathtt{e}_{x},\mathtt{e}_{y},\mathtt{e}_{z},\mathtt{e}_{w}),$

in the sense that $\Sigma_{\mathbf{P}}$ is the fan whose cones are all the
cones generated by all the strict subsets of the family
$\left\\{\mathtt{e}_{x},\mathtt{e}_{y},\mathtt{e}_{z},\mathtt{e}_{w}\right\\}$.
Here, $\mathtt{e}_{y},\mathtt{e}_{z}$ and $\mathtt{e}_{w}$ form the canonical
basis of $\mathbf{Z}^{3}$ while $\mathtt{e}_{x}$ is the vector $(-4,-5,-10)$.

There is a one-to-one correspondence between the $1-$cones of
$\Sigma_{\mathbf{P}}$ and the toric coordinates of $\mathbf{P}$, so that for
instance $x$ corresponds to $\mathtt{e}_{x}$ by the fact that the cone
$\mathbf{R}_{+}\mathtt{e}_{x}$ encodes the toric hypersurface
$\left\\{x=0\right\\}$ and likewise, the cone
$\mathbf{R}_{+}\mathtt{e}_{y}+\mathbf{R}_{+}\mathtt{e}_{z}+\mathbf{R}_{+}\mathtt{e}_{w}$
encodes the dense toric orbit $\left\\{x\neq 0\right\\}$.

In particular, the indeterminacy point $p_{y}$ of $\varphi$ is the origin of
the affine chart $\left\\{y\neq 0\right\\}$ and a weighted blowup of
$\mathbf{P}$ at this point results in a subdivision of the cone
$\mathbf{R}_{+}\mathtt{e}_{x}+\mathbf{R}_{+}\mathtt{e}_{z}+\mathbf{R}_{+}\mathtt{e}_{w}$,
i.e., adding a new cone of dimension $1$ which is generated by a linear
combination over $\mathbf{N}$ of $\mathtt{e}_{x},\mathtt{e}_{z}$ and
$\mathtt{e}_{w}$.

We know from the same algorithm given in [RT13] that the following is a fan
for the variety $\mathbf{P}^{\prime}=\mathbf{P}(1,1,1,2)$.

$\Sigma_{\mathbf{P}^{\prime}}=\mathrm{Fan}\left(\left[\begin{array}[]{c}-1\\\
-1\\\ -2\end{array}\right],\left[\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right],\left[\begin{array}[]{c}0\\\ 1\\\
0\end{array}\right],\left[\begin{array}[]{c}0\\\ 0\\\
1\end{array}\right]\right)=:\mathrm{Fan}(\mathtt{e}_{\zeta},\mathtt{e}_{y},\mathtt{e}_{z},\mathtt{e}_{w}).$

Here, $\mathtt{e}_{\zeta}$ refers to the vector $(-1,-1,-2)$.

The goal is then to obtain a multiple of $\mathtt{e}_{\zeta}$ as an element of
$\mathbf{N}\mathtt{e}_{x}+\mathbf{N}\mathtt{e}_{z}+\mathbf{N}\mathtt{e}_{w}$.
This can be done with coefficients $1,1$ and $2$, as the following points out.

$4\left(\begin{array}[]{c}-1\\\ -1\\\
-2\end{array}\right)=\left(\begin{array}[]{c}-4\\\ -5\\\
-10\end{array}\right)+\left(\begin{array}[]{c}0\\\ 1\\\
0\end{array}\right)+2\left(\begin{array}[]{c}0\\\ 0\\\ 1\end{array}\right).$

This gives rise to a variety $\widehat{\mathbf{P}}$ which is the weighted
blowup of $\mathbf{P}$ at $p_{y}$ with weights $1,1$ and $2$ and the fan
associated to $\widehat{\mathbf{P}}$ is
$\mathrm{Fan}(\mathtt{e}_{x},\mathtt{e}_{\zeta},\mathtt{e}_{y},\mathtt{e}_{z},\mathtt{e}_{w})$.

We now refer to the construction of homogeneous coordinates on toric varieties
which is explained in the fifth chapter of [CLS11] and more specifically at
page 205. This allows us to introduce five toric coordinates on
$\widehat{\mathbf{P}}$ which we denote by
$(\mathtt{x},\zeta,\mathtt{y},\mathtt{z},\mathtt{w})$ with the following
grading in $\mathbf{Z}^{2}$, so that
$\widehat{\mathbf{P}}=\mathrm{Proj}(\mathbf{C}[\mathtt{x},\zeta,\mathtt{y},\mathtt{z},\mathtt{w}])$.

$\begin{array}[]{c|ccccc}&\mathtt{x}&\zeta&\mathtt{y}&\mathtt{z}&\mathtt{w}\\\
\hline\cr\text{degree}&1&0&4&5&10\\\ \text{in
}\mathbf{Z}^{2}:&0&1&1&1&2\end{array}$

Besides, $\widehat{\mathbf{P}}$ is also the weighted blowup of
$\mathbf{P}^{\prime}$ along a toric curve, since

$\left(\begin{array}[]{c}-4\\\ -5\\\
-10\end{array}\right)=5\left(\begin{array}[]{c}-1\\\ -1\\\
-2\end{array}\right)+\left(\begin{array}[]{c}1\\\ 0\\\ 0\end{array}\right),$

in other words, $\mathtt{e}_{x}=4\mathtt{e}_{\zeta}+\mathtt{e}_{y}$ where
$\mathtt{e}_{\zeta}$ and $\mathtt{e}_{y}$ are rays of the fan
$\Sigma_{\mathbf{P}^{\prime}}$.

The blowup map $\varepsilon_{1}$ from $\widehat{\mathbf{P}}$ to $\mathbf{P}$
in homogeneous coordinates is the following

$[\mathtt{x}:\zeta:\mathtt{y}:\mathtt{z}:\mathtt{w}]\mapsto[\mathtt{x}\zeta:\mathtt{y}\zeta^{3}:\mathtt{z}\zeta^{4}:\mathtt{w}\zeta^{8}]\in\mathbf{P}$

which is well defined everywhere and contracts the exceptional divisor
$\left\\{\zeta=0\right\\}$ to the point $p_{y}$. Indeed, fix a point with a
representative $(\mathtt{x},\zeta,\mathtt{y},\mathtt{z},\mathtt{w})$ and
$\zeta^{\nicefrac{{1}}{{4}}}$ a fourth root of $\zeta$, then its image in
$\mathbf{P}$ is
$[\mathtt{x}\zeta^{\nicefrac{{1}}{{4}}}:\mathtt{y}:\mathtt{z}\zeta^{\nicefrac{{1}}{{4}}}:\mathtt{w}\zeta^{\nicefrac{{1}}{{2}}}]$.

On the other hand, the blowup map $\varepsilon_{2}$ from
$\widehat{\mathbf{P}}$ to $\mathbf{P}^{\prime}$ is the following

$[\mathtt{x}:\zeta:\mathtt{z}:\mathtt{w}]\mapsto[\mathtt{x}^{5}\zeta:\mathtt{x}\mathtt{y}:\mathtt{z}:\mathtt{w}],$

contracting the exceptional locus $\left\\{\mathtt{x}=0\right\\}$ to a curve.

As a consequence, one checks from the description of $\varphi$ in homogeneous
coordinates that the following diagram commutes.

${\widehat{\mathbf{P}}}$${\mathbf{P}}$${\mathbf{P}^{\prime}}$$\scriptstyle{\varepsilon_{1}}$$\scriptstyle{\varepsilon_{2}}$$\scriptstyle{\varphi}$

and therefore, $\varphi$ is birational, by the fact that $\varepsilon_{1}$ and
$\varepsilon_{2}$ are. ∎

Continuing on Example 3.2, $S$ is general in the basepoint-free linear system
$|-K_{\mathbf{P}}|$, it avoids the indeterminacy point of $\varphi$. Its image
being a K3 surface, it is an anticanonical divisor of $\mathbf{P}^{\prime}$,
i.e., a quintic surface in $\mathbf{P}(1,1,1,2)$. Using the description of
$\varphi$ in homogeneous coordinates, we see that $S$ in $\mathbf{P}(1,1,1,2)$
has equation

$\hypertarget{(1)}{}u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}=0$ (1)

with $f_{4}$ a general homogeneous polynomial of degree $4$ in the variable
$\mathbf{u}=(u_{0},u_{1},u_{2})$ and $v$. Indeed, such an equation pulls back
to an equation on $\mathbf{P}$ of the form $x^{5}f_{20}(x,y,z,w)=0$, where
$f_{20}$ is a general $20$-ic on $\mathbf{P}$. In other words, the pullback to
$\mathbf{P}$ of $S\subset\mathbf{P}^{\prime}$ is $S+(x^{5})$ where the locus
$x=0$ is the exceptional divisor of $\varepsilon_{2}$ and contracted by
$\varphi$.

Therefore, $\mathcal{L}\subset|-K_{\mathbf{P}^{\prime}}|$ consists of those
quintic surfaces of the form

$u_{0}f_{4}(\mathbf{u},v)+\lambda u_{1}^{5}=0,$

with $\deg(f_{4})=4$ and $\lambda\in\mathbf{C}$. The surface $S$ being general
in $\mathcal{L}$, $\lambda$ is non zero and up to scaling, we may assume
$\lambda=1$ as in (1). The base locus of $\mathcal{L}$ is the curve
$\Delta:=\left\\{u_{0}=u_{1}=0\right\\}$.

######  Lemma 3.4.

In case $\\#9$, given a general $S\in|-K_{\mathbf{P}}|$, the general
$\Gamma\in|-K_{\mathbf{P}}|_{S}|$ is cut out on $S$ in $\mathbf{P}^{\prime}$
by a general quartic.

###### Proof.

Let $S^{\prime}$ be another general member of $\mathcal{L}$, i.e., the image
under $\varphi$ of a general anticanonical divisor of $\mathbf{P}$, which is
the zero locus of $u_{0}f^{\prime}_{4}(\mathbf{u},v)+u_{1}^{5}$ with
$f^{\prime}_{4}$ a homogeneous quartics, then

$S\cap
S^{\prime}=\left\\{u_{0}f_{4}+u_{1}^{5}=u_{0}f^{\prime}_{4}+u_{1}^{5}=0\right\\}=S\cap\left\\{u_{0}(f_{4}-f^{\prime}_{4})=0\right\\},$

and $f_{4}-f^{\prime}_{4}$ is a general quartic of $\mathbf{P}^{\prime}$. This
shows that the restriction $\mathcal{L}|_{S}$ has
$S\cap\left\\{u_{0}=0\right\\}=\Delta$ as its fixed part, and its mobile part
is $|\mathcal{O}_{\mathbf{P}^{\prime}}(4)|_{S}|$. Thus the map from $S$ given
by the restriction of $\mathcal{L}$ is the same map as the one induced by the
quartics of $\mathbf{P}^{\prime}$,

$S\xrightarrow{|\mathcal{O}_{\mathbf{P}^{\prime}}(4)|_{S}|}\mathbf{P}^{21}$

so that the curve $\Gamma$ is the pullback to $S$ of a hyperplane of
$\mathbf{P}^{21}$. Hence, $\Gamma$ is cut out on $S$ by a general quartic of
$\mathbf{P}^{\prime}$. ∎

In conclusion, the curve $\Gamma$ is a complete intersection of degrees $5$
and $4$ given by the two equations

$u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}=g_{4}(\mathbf{u},v)=0$

with $g_{4}$ a general quartic on $\mathbf{P}^{\prime}$. This is summed up in
the following commutative diagram.

${\mathbf{P}}$${\mathbf{P}^{\prime}}$${X}$${\mathbf{P}^{22}}$${\mathbf{P}^{21}}$$\scriptstyle{\varphi}$$\scriptstyle{v_{5}}$$\scriptstyle{|-K_{\mathbf{P}}|}$$\scriptstyle{|\mathcal{O}_{\mathbf{P}^{\prime}}(4)|}$$\scriptstyle{|\mathcal{O}_{X}(4)|}$$\scriptstyle{\mathrm{pr}}$

Here, $\mathrm{pr}$ is the projection map from the vertex point of the cone
$X$ onto $\mathbf{P}^{\prime}$.

In all cases from #9 to #14 but #11, we can apply similar arguments leading to
a description of $\Gamma$ as a complete intersection of two different degrees.
All the needed pieces of information are listed in Table 4 and Table 5 below.
As in the example above, $\Delta$ refers to the base locus of $\mathcal{L}$;
this is also the fixed part of $\mathcal{L}|_{S}$. An expression of $\varphi$
in the coordinates $[x:y:z:w]$ of $\mathbf{P}$ is also provided. It makes it
possible to check the given equation for $S$ in $\mathbf{P}^{\prime}$. It also
makes visible the indeterminacy point of $\varphi$, which we denote by $p$.
The proof that $\varphi$ is birational in each case is along the same lines as
the proof of Lemma 3.3.

In the case of $\mathbf{P}(1,4,5,10)$, the expression of $\varphi$ is
$[x^{5}:xy:z:w]$. The indeterminacy point $p$ is the point for which
$x=z=w=0$, commonly denoted by $p_{y}$. In all cases, the indeterminacy point
is such a coordinate point, as displayed in Table 4 below.

$\hypertarget{Table
4}{}\begin{array}[]{|l|l|l|l|l|}\hline\cr\\#&\mathbf{P}&\mathbf{P}^{\prime}&\text{expression
of }\varphi&p\\\ \hline\cr&&&&\\\\[-10.0pt]
9&\mathbf{P}(1,4,5,10)&\mathbf{P}(1,1,1,2)_{[u_{0}:u_{1}:u_{2}:v]}&[x^{5}:xy:z:w]&p_{y}\\\
10&\mathbf{P}(1,2,6,9)&\mathbf{P}(1,1,3,5)_{[u_{0}:u_{1}:v:s]}&[x^{2}:y:z:xw]&p_{w}\\\
12&\mathbf{P}(1,3,8,12)&\mathbf{P}(1,1,3,4)_{[u_{0}:u_{1}:v:s]}&[x^{3}:y:xz:w]&p_{z}\\\
13&\mathbf{P}(1,6,14,21)&\mathbf{P}(1,1,2,3)_{[u_{0}:u_{1}:v:s]}&[x^{7}:xy:z:w]&p_{y}\\\
14&\mathbf{P}(2,3,10,15)&\mathbf{P}(1,2,4,5)_{[u:v:s:t]}&[y:x^{3}:xz:w]&p_{z}\\\
\hline\cr\end{array}$

Table 4

The anticanonical model $\mathbf{P}\subset\mathbf{P}^{g+1}$ is a hypersurface
of the cone $X$ whose vertex point is $p$. The projection map from the point
$p$ to $\mathbf{P}^{\prime}$ restrict to an isomorphism on the general $S$
such that $p\notin S$, since $S$ is cut out by a hyperplane.

$\hypertarget{Table 5}{}\begin{array}[]{|l|l|l|l|}\hline\cr\\#&\text{equation
for }S\text{ in }\mathbf{P}^{\prime}&\Delta&\Gamma\\\
\hline\cr&&&\\\\[-10.0pt]
9&u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}=0&u_{0}=u_{1}=0&S\cap\mathrm{quartic}\\\
10&u_{0}f_{9}(\mathbf{u},v,s)+s^{2}=0&u_{0}=s=0&S\cap 9\mathrm{-ic}\\\
12&u_{0}f_{8}(\mathbf{u},v,s)+v^{3}=0&u_{0}=v=0&S\cap 8\mathrm{-ic}\\\
13&u_{0}f_{6}(\mathbf{u},v,s)+u_{1}^{7}=0&u_{0}=u_{1}=0&S\cap\mathrm{sextic}\\\
14&vf_{10}(u,v,s,t)+s^{3}=0&v=s=0&S\cap 10\mathrm{-ic}\\\
\hline\cr\end{array}$

Table 5

We always denote by $f_{d}$ a general degree $d$ homogeneous polynomial in
accordance with the grading of $\mathbf{P}^{\prime}$.

##  4 Extensions of $\mathbf{P}$

Recall from Theorem 2.3 and the values of $\alpha(\Gamma,K_{\Gamma})$ given in
Table 2 that $\mathbf{P}(1,2,3,6)$ admits no extension; therefore, we focus
here on items #9 to #14 but #11 and use the description given for $\Gamma$ as
a complete intersection of two different degrees in $\mathbf{P}^{\prime}$ to
construct an extension of $\mathbf{P}$. In all the following cases except #14,
we manage to construct a maximal extension $Y$ as a hypersurface in a weighted
projective space of dimension $2+\alpha(\Gamma,K_{\Gamma})$. The last case
$\\#14$, which is $\mathbf{P}=\mathbf{P}(2,3,10,15)$, will require additional
work.

As a consequence of Theorem 2.3, we are assured that if $Y$ contains all the
surface extensions of $\Gamma$, then it is the universal extension of
$\Gamma$.

###  4.1 $\mathbf{P}=\mathbf{P}(1,4,5,10)$

According to Table 4 and Table 5, the curve $\Gamma$ is cut out on
$\mathbf{P}(1,1,1,2)$ with homogeneous coordinates $[u_{0}:u_{1}:u_{2}:v]$ by
the equations

$u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}=g_{4}(\mathbf{u},v)=0$

where $f_{4}$ and $g_{4}$ are general homogeneous quartic polynomials.
Consider then the equation

$u_{0}s_{0}+u_{1}s_{1}+u_{2}s_{2}=u_{1}^{5}$

where $s_{0},s_{1}$ and $s_{2}$ are coordinates of weight $4$. This defines a
quintic hypersurface $Y$ in $\mathbf{X}=\mathbf{P}(1^{3},2,4^{3})$.

######  Lemma 4.1.

The variety $Y$ has a model in $\mathbf{P}^{24}$ which is a maximal extension
of $\mathbf{P}$, i.e., it has dimension $5=1+\alpha(\Gamma,K_{\Gamma})$
according to Table 2, contains $\mathbf{P}$ as a linear section, and is not a
cone.

###### Proof.

Consider the linear system $|\mathcal{O}_{\mathbf{X}}(4)|$, whose restriction
to $Y$ is very ample and realizes $Y$ in $\mathbf{P}^{24}$ as a variety of
degree
$[\mathcal{O}_{\mathbf{X}}(5)]\cdot[\mathcal{O}_{\mathbf{X}}(4)]^{5}=\frac{4^{5}\times
5}{4^{3}\times 2}=40=2g-2$. This model is an extension of $\Gamma$, as

$Y\cap\left\\{s_{0}=-f_{4}(\mathbf{u},v),s_{1}=s_{2}=g_{4}(\mathbf{u},v)=0\right\\}=\Gamma.$

The fivefold $Y$ is a maximal extension of $\Gamma$ by Theorem 2.3, since it
has dimension $1+\alpha(\Gamma,K_{\Gamma})$ and it contains
$\mathbf{P}=\mathbf{P}(1,4,5,10)$ as a $3-$fold linear section. Indeed, as
indicated in Table 3, $\mathbf{P}$ embeds into $\mathbf{P}(1,1,2,4)$ as the
quintic hypersurface $u_{0}s_{0}=u_{1}^{5}$. It follows that
$\mathbf{P}=Y\cap\left\\{s_{1}=s_{2}=0\right\\}$.

It remains to be proven that $Y$ is not a cone in $\mathbf{P}^{24}$. Its
embedding in $\mathbf{P}^{24}$ is given by the restriction of
$|\mathcal{O}_{\mathbf{X}}(4)|$, whose expression in the weighted coordinates
is

$[u_{0}:u_{1}:u_{2}:v:s_{0}:s_{1}:s_{2}]\mapsto[\mathtt{f}_{0}:\cdots:\mathtt{f}_{r}:s_{0}:s_{1}:s_{2}]$

where the $\mathtt{f}_{i}$’s form a basis of homogeneous quartic polynomials
in $u_{0},u_{1},u_{2}$ and $v$. The variety $\mathbf{P}$ is not a cone and it
is cut out on $Y$ by the two hyperplanes $s_{1}=0$ and $s_{2}=0$. Assume by
contradiction that $Y$ is a cone, then it has a hyperplane section of the form
$Y\cap\left\\{\lambda s_{1}+\mu s_{2}=0\right\\}$ which is a cone over
$\mathbf{P}$ with a point as vertex. One of the two coefficients $\lambda,\mu$
is nonzero, so without loss of generality and up to scaling we may assume that
$\mu=1$, and the variety $Y^{\prime}=Y|_{s_{2}=-\lambda s_{1}}$ which is a
cone over $\mathbf{P}$ is given by the equation

$u_{0}s_{0}+(u_{1}-\lambda u_{2})s_{1}=u_{1}^{5}$

in $\mathbf{P}(1,1,1,2,4,4)$ with coordinates
$[u_{0}:u_{1}:u_{2}:s_{0}:s_{1}]$. Let $F$ be the homogeneous quintic
$u_{0}s_{0}+(u_{1}-\lambda u_{2})s_{1}-u_{1}^{5}$, so that the cone
$Y^{\prime}$ is the vanishing locus of $F$. Besides, one has
$Y^{\prime}\cap\left\\{s_{1}=0\right\\}=\mathbf{P}$. We may consider an
automorphism of $\mathbf{P}^{23}$ fixing the hyperplane
$\left\\{s_{1}=0\right\\}$ and sending the vertex point $p$ of $Y^{\prime}$ to
$p_{s_{1}}=\left\\{u_{0}=u_{1}=u_{2}=v=s_{0}=0\right\\}$. The restriction of
this automorphism to $\mathbf{P}(1,1,1,2,4,4)$ is a polynomial automorphism of
the weighted coordinates $[\mathbf{u}:v:s_{0}:s_{1}]$ which would eliminate
the variable $s_{1}$ from $F$, i.e., which would make the affine chart
$Y^{\prime}|_{s_{1}=1}$ an affine cone.

As the polynomial $F$ doesn’t involve $v$, it is only affected by changes of
the variables $u_{0},u_{1},u_{2},s_{0}$ and $s_{1}$. Such a transformation is
as follows

$[u_{0}:u_{1}:u_{2}:s_{0}:s_{1}]\mapsto[A\mathbf{u}:as_{0}+bs_{1}+h_{1}(\mathbf{u},v):cs_{0}+ds_{1}+h_{2}(\mathbf{u},v)]$

with
$A=(A_{ij})_{i,j\in\left\\{0,1,2\right\\}}\in\mathrm{GL}_{3}(\mathbf{C})$,
$ad\neq bc$ and $h_{1},h_{2}$ are homogeneous quartics.

Such a change of variables applied to the equation $F=0$ yields

$\begin{array}[]{lccl}(A_{21}u_{0}+A_{22}u_{1}+A_{23}u_{2})^{5}&=&&(A_{00}u_{0}+A_{01}u_{1}+A_{02}u_{2})(as_{0}+bs_{1}+h_{1})\\\
&&+&(A_{10}u_{0}+A_{11}u_{1}+A_{12}u_{2})(cs_{0}+ds_{1}+h_{2})\\\ &&-&(\lambda
A_{20}u_{0}+\lambda A_{21}u_{1}+\lambda
A_{22}u_{2})(cs_{0}+ds_{1}+h_{2}).\end{array}$

The condition that this does not involve $s_{1}$ implies the following:

$A_{00}b+A_{10}d-\lambda A_{20}d=A_{01}b+A_{11}d-\lambda
A_{21}d=A_{02}b+A_{12}d-\lambda A_{22}d=0.$

As $b$ and $d$ cannot be both zero, the columns of $A$ are linearly dependent
and thus $\det(A)=0$, a contradiction. ∎

Letting $(\lambda_{0},\lambda_{1},\lambda_{2})$ move in $\mathbf{C}^{3}$, we
get a family of K3 surfaces

$Y\cap\left\\{s_{0}=\lambda_{0}g_{4}(\mathbf{u},v)-f_{4}(\mathbf{u},v),s_{1}=\lambda_{1}g_{4}(\mathbf{u},v),s_{2}=\lambda_{2}g_{4}(\mathbf{u},v)\right\\}$

which are all linear sections of $Y$ and contain $\Gamma$ as a hyperplane
section. Indeed, the curve $\Gamma$ is cut out on all of them by
$\left\\{g_{4}(\mathbf{u},v)=0\right\\}$. Among them, those that are members
of the linear system $\mathcal{L}$ are those parameterized by
$\lambda_{1}=\lambda_{2}=0$.

######  Lemma 4.2.

The variety $Y$ in $\mathbf{P}^{24}$ is the universal extension of $\Gamma$.

###### Proof.

By Theorem 2.3 we need to show that $Y$ contains all the surface extensions of
$\Gamma$ as linear sections, and that these surfaces in $Y$ are unique up to
projective automorphisms of $\mathbf{P}^{24}$ fixing $\Gamma$.

Given $\lambda=(\lambda_{0},\lambda_{1},\lambda_{2})\in\mathbf{C}^{3}$ as
above, consider the following surface in
$\mathbf{P}^{\prime}=\mathbf{P}(1,1,1,2)$:

$S_{\lambda}=Y\cap\left\\{s_{0}=\lambda_{0}g_{4}(\mathbf{u},v)-f_{4}(\mathbf{u},v),s_{1}=\lambda_{1}g_{4}(\mathbf{u},v),s_{2}=\lambda_{2}g_{4}(\mathbf{u},v)\right\\}.$

It is a linear section of $Y$ and contains $\Gamma$ as a hyperplane section in
$\mathbf{P}^{21}$. Letting $\lambda$ move, we obtain a family indexed by the
affine space $\mathbf{C}^{3}$.

Assume by contradiction that there exists $\lambda$ such that $S_{\lambda}$ is
a cone. Then in particular, it contains a line of $\mathbf{P}^{21}$, i.e., a
curve $L\subset S_{\lambda}$ such that
$L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(4)]=1$. But $\mathbf{P}^{\prime}$ is
embedded in $\mathbf{P}^{6}$ by the linear system
$|\mathcal{O}_{\mathbf{P}^{\prime}}(2)|$, and we have
$L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(2)]=\frac{1}{2}$, which is a
contradiction.

Assume now by contradiction that there exist $\lambda\neq\lambda^{\prime}$ in
$\mathbf{C}^{3}$ and an automorphism $\rho$ of $\mathbf{P}^{22}$ which acts
trivially on $\Gamma$ and such that $\rho(S_{\lambda})=S_{\lambda^{\prime}}$.
Let us denote by $\langle S_{\lambda},S_{\lambda^{\prime}}\rangle$ the linear
space spanned by $S_{\lambda}\cup S_{\lambda^{\prime}}$ in $\mathbf{P}^{24}$,
and consider the threefold $T=Y\cap\langle
S_{\lambda},S_{\lambda^{\prime}}\rangle$ which contains $S_{\lambda}$ and
$S_{\lambda^{\prime}}$ as hyperplane sections. This threefold is a cone, or
else it would be spanned by a pencil of surface linear sections of the
universal extension of $\Gamma$ and by Theorem 2.3 we would have
$\rho(S_{\lambda})\neq S_{\lambda^{\prime}}$.

Since $Y$ is a hypersurface of $\mathbf{X}=\mathbf{P}(1^{3},2,4^{3})$ and $T$
is cut out on $Y$ by two quartic hypersurfaces, the threefold $T$ is naturally
realized as a hypersurface of $X=\mathbf{P}(1,1,1,2,4)$. Since $T$ is a cone
in $\mathbf{P}^{22}$, it is covered by lines, i.e., curves $L\subset T$ such
that $L\cdot[\mathcal{O}_{X}(2)]=\frac{1}{2}$.

Let $p$ be the unique basepoint of $\mathcal{O}_{X}(2)$. With respect to the
weighted coordinates $[u_{0}:u_{1}:u_{2}:v:s]$ on $X$, the point $p$ is given
by the equations $u_{0}=u_{1}=u_{2}=v=0$. If $p$ is not the vertex of $T$,
then there is a line $L\subset T$ such that $p\notin L$ and thus the
restriction of $\mathcal{O}_{X}(2)$ to $L$ is Cartier, which is not compatible
with $L\cdot[\mathcal{O}_{X}(2)]=\frac{1}{2}$. If $p$ is the vertex point of
$T$, then the affine chart $T|_{s\neq 0}$ is an affine cone, i.e., the
equation for $T$ in $X$ does not involve the coordinate $s$. But we know from
the defining equation of $Y$ in $\mathbf{X}=\mathbf{P}(1^{3},2,4^{3})$ that it
is not possible.

The conclusion is that $Y$ contains a family of pairwise unique surface
extensions of $\Gamma$ which is parameterized by the affine space
$\mathbf{C}^{3}$. Let $\mathcal{H}$ be the family which parameterizes
$21-$planes $\Lambda\subset\mathbf{P}^{24}$ such that $\Gamma\subset\Lambda$,
and $\mathcal{S}=\mathbf{P}(\mathrm{ker}({}^{t}\Phi_{K_{\Gamma}}))$ the family
of surface extensions of $\Gamma$. Then we have $\mathcal{H}\simeq
S\simeq\mathbf{P}^{3}$, and the map $\mathcal{H}\to\mathcal{S}$ which maps
$\Lambda$ to $Y\cap\Lambda$ is linear (see [CDS20]) and its image contains a
dense open subset of $\mathcal{S}$ by the above. It follows that it is an
isomorphism and therefore, $Y$ is universal by Theorem 2.3. ∎

As a sanity check, let us see what happens if we try to build a larger
extension of $\mathbf{P}$. One might consider adding a coordinate $s_{3}$ of
weight $4$ and the hypersurface

$\ell_{0}s_{0}+\ell_{1}s_{1}+\ell_{2}s_{2}+\ell_{3}s_{3}=u_{1}^{5}$

where $\ell_{0},\ell_{1},\ell_{2},\ell_{3}$ are degree $1$ homogeneous
polynomials in the variables $u_{0},u_{1}$ and $u_{2}$. This is
$6-$dimensional and indeed contains $\Gamma$ as a linear section; however, the
$\ell_{i}$’s are linearly dependent and thus the variety given by the equation
above is a cone over $Y$.

###  4.2 $\mathbf{P}=\mathbf{P}(1,2,6,9)$

By Table 4 and Table 5, the curve $\Gamma$ is given in $\mathbf{P}(1,1,3,5)$
by the following two equations

$u_{0}f_{9}(\mathbf{u},v,s)+s^{2}=g_{9}(\mathbf{u},v,s)=0$

where $f_{9}$ and $g_{9}$ are general homogeneous polynomials of degree $9$.
Adding two coordinates $t_{0}$ and $t_{1}$ of weight $9$, we consider the
$10-$ic hypersurface $Y$ in $\mathbf{X}=\mathbf{P}(1^{2},3,5,9^{2})$ given by
the equation

$u_{0}t_{0}+u_{1}t_{1}=s^{2}.$

######  Lemma 4.3.

The variety $Y$ has a model in $\mathbf{P}^{30}$ which is a maximal extension
of $\mathbf{P}$.

###### Proof.

The linear system $|\mathcal{O}_{\mathbf{X}}(9)|$ has one base point in
$\mathbf{X}$ but its restriction to $Y$ defines an embedding which realizes
$Y$ as a projective variety in $\mathbf{P}^{30}$ of degree $\frac{9^{4}\times
10}{9^{2}\times 5\times 3}=54=2g-2$. It has dimension
$4=1+\alpha(\Gamma,K_{\Gamma})$ by Table 2 and contains $\Gamma$ as a linear
section in $\mathbf{P}^{27}$:

$Y\cap\left\\{t_{0}=-f_{9}(\mathbf{u},v,s),t_{1}=g_{9}(\mathbf{u},v,s)=0\right\\}=\Gamma.$

Besides, $Y$ is not a cone, by the same arguments as those mentioned in the
proof of Lemma 4.1. The fourfold $Y$ is a maximal extension of $\Gamma$.
Recall from Table 3 that $\mathbf{P}$ is the hypersurface of
$\mathbf{P}(1,1,3,5,9)$ given by the equation $u_{0}t_{0}=s^{2}$. This shows
that $\mathbf{P}=Y\cap\left\\{t_{1}=0\right\\}$. ∎

In particular,

$Y\cap\left\\{t_{0}=\lambda_{0}g_{9}(\mathbf{u},v,s)-f_{9}(\mathbf{u},v,s),t_{1}=\lambda_{1}g_{9}(\mathbf{u},v,s)\right\\}$

describes a family of K3 surfaces in $Y$ indexed by
$(\lambda_{0},\lambda_{1})\in\mathbf{C}^{2}$ which all contain $\Gamma$ as a
hyperplane section. Among them, the members of $\mathcal{L}$ are the ones for
which $\lambda_{1}=0$.

######  Lemma 4.4.

The variety $Y$ in $\mathbf{P}^{30}$ is the universal extension of $\Gamma$.

###### Proof.

The proof is along the same lines as the proof of Lemma 4.2. The surface
extensions of $\Gamma\subset\mathbf{P}^{27}$ are parameterized by
$\mathbf{P}^{2}$ and we have a dense family indexed by
$\lambda=(\lambda_{0},\lambda_{1})\in\mathbf{C}^{2}$ of surfaces in
$\mathbf{P}^{\prime}=\mathbf{P}(1,1,3,5)$:

$S_{\lambda}=Y\cap\left\\{t_{0}=\lambda_{0}g_{9}(\mathbf{u},v,s)-f_{9}(\mathbf{u},v,s),t_{1}=\lambda_{1}g_{9}(\mathbf{u},v,s)\right\\},$

all of which are linear sections of $Y$ and contain $\Gamma$ as a hyperplane
section in $\mathbf{P}^{28}$. None of them is a cone over $\Gamma$, since such
a cone would contain a line, i.e., a curve $L\subset S_{\lambda}$ such that
$L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(9)]=1$ and thus
$L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(3)]=\frac{1}{3}$, which is not
possible since $\mathcal{O}_{\mathbf{P}^{\prime}}(3)$ induces by restriction
on $S_{\lambda}$ a double cover of $\mathbf{P}(1,1,3)$. Besides, the surfaces
$S_{\lambda}$ are unique up to automorphisms of $\mathbf{P}^{28}$ fixing
$\Gamma$ by a similar argument as in Lemma 4.2 and thus $Y$ is the universal
extension of $\Gamma$. ∎

###  4.3 $\mathbf{P}=\mathbf{P}(1,3,8,12)$

By Table 4 and Table 5, the curve $\Gamma$ is given in $\mathbf{P}(1,1,3,4)$
by the following equations

$u_{0}f_{8}(\mathbf{u},v,s)+v^{3}=g_{8}(\mathbf{u},v,s)=0$

where $f_{8}$ and $g_{8}$ are general homogeneous polynomials of degree $8$.
After adding two coordinates $t_{0}$ and $t_{1}$ of weight $8$, we consider
the $9-$ic hypersurface $Y$ in $\mathbf{X}=\mathbf{P}(1^{2},3,4,8^{2})$ of
equation

$u_{0}t_{0}+u_{1}t_{1}=v^{3}.$

######  Lemma 4.5.

The variety $Y$ has a model in $\mathbf{P}^{27}$ which is a maximal extension
of $\mathbf{P}$.

###### Proof.

It is embedded in $\mathbf{P}^{27}$ by the restriction of the linear system
$|\mathcal{O}_{\mathbf{X}}(8)|$. This model has degree $\frac{8^{4}\times
9}{8^{2}\times 4\times 3}=46=2g-2$ and dimension
$4=1+\alpha(\Gamma,K_{\Gamma})$ by Table 2 and contains $\Gamma$ as a linear
section:

$Y\cap\left\\{t_{0}=-f_{8}(\mathbf{u},v,s),t_{1}=g_{8}(\mathbf{u},v,s)=0\right\\}=\Gamma.$

Hence it is a maximal extension of $\Gamma$. It is also an extension of
$\mathbf{P}$; indeed, we know from Table 3 that $\mathbf{P}$ is the
hypersurface $u_{0}t_{0}=v^{3}$ is $\mathbf{P}(1,1,3,4,8)$. This exhibits
$\mathbf{P}$ as a hyperplane section of $Y$, which is
$\mathbf{P}=Y\cap\left\\{t_{1}=0\right\\}$. The fact that $Y$ is not a cone
can be proven in the same way as in Lemma 4.1. ∎

Letting $(\lambda_{0},\lambda_{1})$ move in $\mathbf{C}^{2}$, we get a family

$Y\cap\left\\{t_{0}=\lambda_{0}g_{8}(\mathbf{u},v,s)-f_{8}(\mathbf{u},v,s),t_{1}=\lambda_{1}g_{8}(\mathbf{u},v,s)\right\\}$

of K3 surfaces in $Y$ which contain $\Gamma$ as a hyperplane section. The
surfaces in this family that are members of $\mathcal{L}$ are the ones for
which $\lambda_{1}=0$.

######  Lemma 4.6.

The variety $Y$ in $\mathbf{P}^{27}$ is the universal extension of $\Gamma$.

###### Proof.

The proof is similar as the proof of Lemma 4.2. The surface extensions of
$\Gamma$ are parameterized by $\mathbf{P}^{2}$ and we have a dense family
indexed by $\lambda=(\lambda_{0},\lambda_{1})\in\mathbf{C}^{2}$ of surfaces in
$\mathbf{P}^{\prime}=\mathbf{P}(1,1,3,4)$:

$S_{\lambda}=Y\cap\left\\{t_{0}=\lambda_{0}g_{8}(\mathbf{u},v,s)-f_{8}(\mathbf{u},v,s),t_{1}=\lambda_{1}g_{8}(\mathbf{u},v,s)\right\\},$

which are linear sections of $Y$ and contain $\Gamma$ as a hyperplane section.
There exists no $\lambda$ such that $S_{\lambda}$ is a cone, since otherwise
it would contain a line of $\mathbf{P}^{25}$, i.e., a curve $L\subset
S_{\lambda}$ such that $L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(8)]=1$ and
thus $L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(4)]=\frac{1}{2}$, which is not
possible since the restriction of $\mathcal{O}_{\mathbf{P}^{\prime}}(4)$ to
$S_{\lambda}$ is basepoint-free. Besides, the surfaces $S_{\lambda}$ are
unique up to automorphisms of $\mathbf{P}^{25}$ fixing $\Gamma$ by a similar
argument as in Lemma 4.2 and thus $Y$ is the universal extension of $\Gamma$.
∎

###  4.4 $\mathbf{P}=\mathbf{P}(1,6,14,21)$

By Table 4 and Table 5, the curve $\Gamma$ is given in $\mathbf{P}(1,1,2,3)$
by the following equations

$u_{0}f_{6}(\mathbf{u},v,s)+u_{1}^{7}=g_{6}(\mathbf{u},v,s)=0$

with $f_{6}$ and $g_{6}$ general homogeneous polynomials of degree $6$. Adding
two coordinates $t_{0}$ and $t_{1}$ of weight $6$, we consider the heptic
hypersurface $Y$ in $\mathbf{X}=\mathbf{P}(1^{2},2,3,6^{2})$ given by

$u_{0}t_{0}+u_{1}t_{1}=u_{1}^{7}.$

######  Lemma 4.7.

The variety $Y$ has a model in $\mathbf{P}^{24}$ which is a maximal extension
of $\mathbf{P}$.

###### Proof.

It is embedded in $\mathbf{P}^{24}$ by restriction of the linear system
$|\mathcal{O}_{\mathbf{X}}(6)|$. This model has degree $\frac{6^{4}\times
7}{6^{2}\times 3\times 2}=42=2g-2$ and contains $\Gamma$ as a linear section:

$Y\cap\left\\{t_{0}=-f_{6}(\mathbf{u},v,s),t_{1}=g_{6}(\mathbf{u},v,s)=0\right\\}=\Gamma.$

Besides, $Y$ is not a cone, by the same arguments as those mentioned in the
proof of Lemma 4.1. By Table 2, it has dimension
$4=1+\alpha(\Gamma,K_{\Gamma})$. Hence, it a maximal extension of $\Gamma$. It
is also an extension of $\mathbf{P}$: recall from Table 3 that $\mathbf{P}$ is
the hypersurface $u_{0}t_{0}=u_{1}^{7}$ in $\mathbf{P}(1,1,2,3,6)$. As a
consequence, we have the equality $Y\cap\left\\{t_{1}=0\right\\}=\mathbf{P}$.
∎

Letting $(\lambda_{0},\lambda_{1})$ move in $\mathbf{C}^{2}$, we have a family

$Y\cap\left\\{t_{0}=\lambda_{0}g_{6}(\mathbf{u},v,s)-f_{6}(\mathbf{u},v,s),t_{1}=\lambda_{1}g_{6}(\mathbf{u},v,s)\right\\}$

of K3 surfaces in $Y$ which are extensions of $\Gamma$. Those surfaces which
are members of $\mathcal{L}$ are the ones for which $\lambda_{1}=0$. Let
$\lambda=(\lambda_{0},\lambda_{1})$, then the surface given by the
intersection above is the following hypersurface in
$\mathbf{P}^{\prime}=\mathbf{P}(1,1,2,3)$:

$S_{\lambda}=\left\\{u_{1}^{7}+u_{0}f_{6}(\mathbf{u},v,s)=(\lambda_{0}u_{0}+\lambda_{1}u_{1})g_{6}(\mathbf{u},v,s)\right\\},$

so that $\Gamma=S_{\lambda}\cap\left\\{g_{6}(\mathbf{u},v,s)=0\right\\}$. The
question arises whether there exists $\lambda$ such that $S_{\lambda}$ is a
cone over $\Gamma$ in $\mathbf{P}^{22}$. If the answer is no, then $Y$ is the
universal extension of $\Gamma$ since it contains all its surface extensions.

However, the argument used in Lemma 4.2 doesn’t apply here, since a curve
$L\subset S_{\lambda}$ such that
$L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(6)]=1$ could pass through the base
points of $\mathcal{O}_{\mathbf{P}^{\prime}}(2)$ and
$\mathcal{O}_{\mathbf{P}^{\prime}}(3)$, allowing
$L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(2)]=\frac{1}{3}$ and
$L\cdot[\mathcal{O}_{\mathbf{P}^{\prime}}(3)]=\frac{1}{2}$.

###  4.5 $\mathbf{P}=\mathbf{P}(2,3,10,15)$

By Table 4 and Table 5, the curve $\Gamma$ is given in $\mathbf{P}(1,2,4,5)$
by the equations

$vf_{10}(u,v,s,t)+s^{3}=g_{10}(u,v,s,t)=0,$

with $f_{10}$ and $g_{10}$ general homogeneous polynomials of degree $10$.
After adding two coordinates $r_{0}$ and $r_{1}$ of weight $10$ we construct
an extension of $\Gamma$ as the hypersurface

$u^{2}r_{0}+vr_{1}=s^{3}$

which we denote by $Y_{1}$, in $\mathbf{X}=\mathbf{P}(1,2,4,5,10^{2})$.

######  Lemma 4.8.

The variety $Y_{1}$ has a model in $\mathbf{P}^{18}$ which is a nonmaximal
extension of $\mathbf{P}$.

###### Proof.

It is embedded in $\mathbf{P}^{18}$ by the restriction of the linear system
$|\mathcal{O}_{\mathbf{X}}(10)|$. This model has degree $\frac{10^{4}\times
12}{10^{2}\times 5\times 4\times 2}=30=2g-2$ and contains $\Gamma$ as a linear
section:

$Y_{1}\cap\left\\{r_{1}=-f_{10}(u,v,s,t),r_{0}=g_{10}(u,v,s,t)=0\right\\}=\Gamma.$

In accordance with the equation $vr_{1}=s^{3}$ which is given for $\mathbf{P}$
in Table 3 as a hypersurface in $\mathbf{P}(1,2,4,5,10)$, one checks that
$Y_{1}\cap\left\\{r_{0}=0\right\\}=\mathbf{P}$. However, $Y_{1}$ has dimension
$4$, while $1+\alpha(\Gamma,K_{\Gamma})=5$, so the extension isn’t maximal. ∎

The next two subsections are devoted to the construction of a maximal
extension. The strategy is to introduce another birational model for
$\mathbf{P}$ to construct another nonmaximal extension $Y_{2}$. The data of
$Y_{1}$ and $Y_{2}$ will allow us to construct in §4.5.2 a maximal extension
of $\mathbf{P}$.

####  4.5.1 An alternative model for $\mathbf{P}=\mathbf{P}(2,3,10,15)$

Here, we construct $Y_{2}$. Introducing homogeneous coordinates
$[u^{\prime}:v^{\prime}:s^{\prime}:t^{\prime}]$ on the weighted projective
space $\mathbf{P}(1,3,5,9)$, consider the following rational map $\psi$ from
$\mathbf{P}$ to $\mathbf{P}(1,3,5,9)$

$[x:y:z:w]\in\mathbf{P}\mapsto[u^{\prime}:v^{\prime}:s^{\prime}:t^{\prime}]=[x:y^{2}:z:yw].$

The expression of $\psi$ in homogeneous coordinates is obtained from the
$2-$Veronese map $v_{2}$ on $\mathbf{P}$,

$[x:y:z:w]\mapsto[x:y^{2}:z:yw:w^{2}],$

by removing the last component $w^{2}$. This is a similar construction as the
one for $\varphi$ displayed in Table 4, which was obtained from the
$3-$Veronese map $v_{3}$.

######  Lemma 4.9.

The map $\psi$ is birational and it restricts to an isomorphism on the general
anticanonical divisor of $\mathbf{P}$.

The proof that $\psi$ is birational, which we shall not detail here, consists
in a resolution of the indeterminacy point of $\psi$, as was done in the proof
of Lemma 3.3. A similar argument applies to $\psi$ as the one which was given
for $\varphi:\mathbf{P}\dashrightarrow\mathbf{P}(1,2,4,5)$ which ensured that
$\varphi(S)\simeq S$ for a general $S\in|-K_{\mathbf{P}}|$. It revolves around
the following commutative diagram.

${\mathbf{P}}$${\mathbf{P}(1,3,5,9,15)}$${\mathbf{P}(1,3,5,9)}$${\mathbf{W}=\mathrm{cone}(\mathbf{V})}$${\mathbf{V}}$$\scriptstyle{v_{2}}$$\scriptstyle{\psi}$$\scriptstyle{|\mathcal{O}(15)|}$$\scriptstyle{|\mathcal{O}(15)|}$$\scriptstyle{\mathrm{pr}}$

Here, $v_{2}$ is the $2-$Veronese map from $\mathbf{P}$ to
$\mathbf{P}(1,3,5,9,15)$,
$-K_{\mathbf{P}}=v_{2}^{*}\mathcal{O}_{\mathbf{P}(1,3,5,9,15)}(15)$,
$\mathbf{W}$ is a cone over $\mathbf{V}$ with a point as vertex and
$\mathrm{pr}$ is the projection map from the vertex point of $\mathbf{W}$ onto
$\mathbf{V}$.

A consequence of this is that $S$ can be realized as a nongeneral
anticanonical divisor of $\mathbf{P}(1,3,5,9)$, namely, one of equation

$v^{\prime}f_{15}(u^{\prime},v^{\prime},s^{\prime},t^{\prime})+t^{\prime 2}=0$

where $f_{15}$ is a homogeneous polynomial of degree $15$. One checks that the
pullback to $\mathbf{P}$ of such a hypersurface is $S+(y^{2})$, and the locus
$y=0$ is contracted by $\psi$.

In $\mathbf{P}(1,3,5,9)$, the curve $\Gamma$ is cut out on $S$ by a general
hypersurface of degree $15$, as the diagram above shows. Hence $\Gamma$ is
given in $\mathbf{P}(1,3,5,9)$ by the following equations

$v^{\prime}f_{15}(u^{\prime},v^{\prime},s^{\prime},t^{\prime})+t^{\prime
2}=g_{15}(u^{\prime},v^{\prime},s^{\prime},t^{\prime})=0$

where $g_{15}$ is a general homogeneous polynomial of degree $15$. Let’s add
two coordinates $r_{0}^{\prime}$ and $r_{1}^{\prime}$ of weight $15$ and
examine the hypersurface $Y_{2}$ in
$\mathbf{X}^{\prime}=\mathbf{P}(1,3,5,9,15^{2})$ given by the equation

$u^{\prime 3}r_{0}^{\prime}+v^{\prime}r_{1}^{\prime}=t^{\prime 2}.$

######  Lemma 4.10.

The variety $Y_{2}$ has a model in $\mathbf{P}^{18}$ which is also a
nonmaximal extension of $\mathbf{P}$.

###### Proof.

It has dimension $4$ and is embedded in $\mathbf{P}^{18}$ by restriction of
$|\mathcal{O}_{\mathbf{X}^{\prime}}(15)|$. This model contains $\Gamma$ as a
linear section; indeed, given two constants $\lambda_{0}$ and $\lambda_{1}$ :

$Y_{2}\cap\left\\{\begin{array}[]{l}r_{0}^{\prime}=\lambda_{0}g_{15}(u^{\prime},v^{\prime},s^{\prime},t^{\prime})\\\
r_{1}^{\prime}=\lambda_{1}g_{15}(u^{\prime},v^{\prime},s^{\prime},t^{\prime})-f_{15}(u^{\prime},v^{\prime},s^{\prime},t^{\prime})\\\
g_{15}(u^{\prime},v^{\prime},s^{\prime},t^{\prime})=0\end{array}\right\\}=\Gamma.$

This is an extension of $\mathbf{P}$ as well. Indeed, we have
$Y_{2}\cap\left\\{r_{0}^{\prime}=0\right\\}=\left\\{v^{\prime}r_{1}^{\prime}=t^{\prime
2}\right\\}=\mathbf{P}$ in $\mathbf{P}(1,3,5,9,15)$. ∎

####  4.5.2 The maximal extension of $\mathbf{P}(2,3,10,15)$

In the preceding subsections we constructed $Y_{1}$ and $Y_{2}$ two fourfold
extensions of $\Gamma$. Now we construct a maximal extension $Y$ of
$\mathbf{P}$, such that $Y$ contains both $Y_{1}$ and $Y_{2}$ as hyperplane
sections in $\mathbf{P}^{19}$. This construction involves a weighted
projective bundle over $\mathbf{P}^{1}$, i.e., a quotient of a vector bundle
such that the fiber is a weighted projective space.

Let $\Lambda=\mathbf{P}^{17}$ be the linear subspace spanned by $\mathbf{P}$
in $\mathbf{P}^{19}$; for $i=1,2$ we have $Y_{i}=Y\cap H_{i}$ where $H_{i}$ is
a hyperplane in $\mathbf{P}^{19}$ such that $\Lambda\subset H_{i}$. The
fourfolds $Y_{1}$ and $Y_{2}$ generate a pencil of hyperplane sections of $Y$
which all contain $\mathbf{P}$.

The construction of $Y$ will require a realization of $Y_{1}$ and $Y_{2}$ as
complete intersections in $\mathbf{P}(1^{2},2,3,5^{3})$. Note that the image
of the $6-$Veronese map $v_{6}$ on $\mathbf{P}(2,3,10,15)$ lies in
$\mathbf{P}(1^{2},2,3,5^{2})$, so one might think that it could be possible to
recover $Y_{1}$ and $Y_{2}$ from $v_{6}$. However, all my attempts in trying
so were unsuccessful.

On the one hand, $Y_{1}$ is given as the $12-$ic hypersurface in $\mathbf{X}$
of equation $u^{2}r_{0}+vr_{1}=s^{3}$. On the other hand, $Y_{2}$ is the
$18-$ic hypersurface in $\mathbf{X}^{\prime}$ given by the equation $u^{\prime
3}r_{0}^{\prime}+v^{\prime}r_{1}^{\prime}=t^{\prime 2}$. Both $\mathbf{X}$ and
$\mathbf{X}^{\prime}$ can be embedded in $\mathbf{P}(1^{2},2,3,5^{3})$ by the
following Veronese maps.

$(v_{2})_{\mathbf{X}}:\begin{cases}\mathbf{X}=\mathbf{P}(1,2,4,5,10^{2})&\longrightarrow\mathbf{P}(1^{2},2,3,5^{3})\\\
[u:v:s:t:r_{0}:r_{1}]&\longmapsto[u^{2}:v:s:ut:t^{2}:r_{1}:r_{0}].\end{cases}$
$(v_{3})_{\mathbf{X}^{\prime}}:\begin{cases}\mathbf{X}^{\prime}=\mathbf{P}(1,3,5,9,15^{2})&\longrightarrow\mathbf{P}(1^{2},2,3,5^{3})\\\
[u^{\prime}:v^{\prime}:s^{\prime}:t^{\prime}:r_{0}^{\prime}:r_{1}^{\prime}]&\longmapsto[v^{\prime}:u^{\prime
3}:u^{\prime}s^{\prime}:t^{\prime}:r_{1}^{\prime}:s^{\prime
3}:r_{0}^{\prime}].\end{cases}$

We may choose $[U_{0}:U_{1}:V:W:X_{0}:X_{1}:X_{2}]$ as homogeneous coordinates
on $\mathbf{P}(1^{2},2,3,5^{3})$, whose pullbacks by the Veronese maps are

$\begin{array}[]{c|ccccccc}&U_{0}&U_{1}&V&W&X_{0}&X_{1}&X_{2}\\\
\hline\cr\text{pullback to }\mathbf{X}&u^{2}&v&s&ut&t^{2}&r_{1}&r_{0}\\\
\text{pullback to }\mathbf{X}^{\prime}&v^{\prime}&u^{\prime
3}&u^{\prime}s^{\prime}&t^{\prime}&r_{1}^{\prime}&s^{\prime
3}&r_{0}^{\prime}\end{array}$

Hence the above realizes $\mathbf{X}$ (respectively $\mathbf{X}^{\prime}$) as
the hypersurface of equation $U_{0}X_{0}=W^{2}$ (respectively
$U_{1}X_{1}=V^{3}$). The descriptions we know for $Y_{1}$ and $Y_{2}$ in
$\mathbf{X}$ and $\mathbf{X}^{\prime}$ yield

$Y_{1}=\left\\{\begin{array}[]{l}U_{0}X_{0}=W^{2}\\\
U_{1}X_{1}+U_{0}X_{2}=V^{3}\end{array}\right\\}$

and

$Y_{2}=\left\\{\begin{array}[]{l}U_{0}X_{0}+U_{1}X_{2}=W^{2}\\\
U_{1}X_{1}=V^{3}\end{array}\right\\}.$

Besides, we know from Lemma 4.8 and Lemma 4.10 that $\mathbf{P}$ is cut out on
$Y_{1}$ and $Y_{2}$ by the same hyperplane in $\mathbf{P}^{18}$, namely:
$Y_{1}\cap\left\\{X_{2}=0\right\\}=Y_{2}\cap\left\\{X_{2}=0\right\\}=\mathbf{P}$.
In particular, $\mathbf{P}$ in $\left\\{X_{2}=0\right\\}$ is given by the
equations

$\hypertarget{(2)}{}U_{0}X_{0}=W^{2},\hskip 5.69046ptU_{1}X_{1}=V^{3}.$ (2)

We introduce now two coordinates $\lambda,\mu$ and consider
$F=\mathrm{Proj}(R)$ with

$R=\mathbf{C}[\lambda,\mu,U_{0},U_{1},V,W,X_{0},X_{1},X_{2}]$

endowed with the following grading in $\mathbf{Z}^{2}$:

$\begin{array}[]{c|ccccccccc}&\lambda&\mu&U_{0}&U_{1}&V&W&X_{0}&X_{1}&X_{2}\\\
\hline\cr\text{degree}&1&1&0&0&0&0&0&0&-1\\\ \text{in
}\mathbf{Z}^{2}:&0&0&1&1&2&3&5&5&5\end{array}$

It is a bundle over $\mathbf{P}^{1}$ with fiber $\mathbf{P}(1^{2},2,3,5^{3})$,
whose bundle map to $\mathbf{P}^{1}$ is $[\lambda:\mu]$ and the locus
$X_{2}=0$ is the trivial subbundle
$\mathbf{P}^{1}\times\mathbf{P}(1^{2},2,3,5^{2})$.

There is a morphism $\phi:F\to\mathbf{P}(1^{2},2,3,5^{4})$ which is given in
coordinates by the expression $[U_{0}:U_{1}:V:W:X_{0}:X_{1}:\lambda X_{2}:\mu
X_{2}]$ and the projective model in $\mathbf{P}^{19}$ induced by the linear
system $|\mathcal{O}_{F}(0,5)|$ decomposes as the composite map

${F}$${\mathbf{P}(1^{2},2,3,5^{4})}$${\mathbf{P}^{19}.}$$\scriptstyle{\phi}$$\scriptstyle{|\mathcal{O}(5)|}$

Notice that $\phi$ contracts the trivial bundle
$\mathbf{P}^{1}\times\mathbf{P}(1^{2},2,3,5^{2})$ given by the equation
$X_{2}=0$ onto $\mathbf{P}(1^{2},2,3,5^{2})$. Hence the image of
$\left\\{X_{2}=0\right\\}$ by $|\mathcal{O}_{F}(0,5)|$ is the image of
$\mathbf{P}(1^{2},2,3,5^{2})$ in $\mathbf{P}^{17}$.

Consider the complete intersection $Z$ in $F$ given by the two homogeneous
equations

$\displaystyle U_{0}X_{0}+\lambda U_{1}X_{2}$ $\displaystyle=W^{2},$
$\displaystyle U_{1}X_{1}+\mu U_{0}X_{2}$ $\displaystyle=V^{3}.$

######  Lemma 4.11.

The image of $Z$ in $\mathbf{P}^{19}$ is not a cone and contains $Y_{1}$ and
$Y_{2}$ as hyperplane sections. By Table 2, it has dimension
$1+\alpha(\Gamma,K_{\Gamma})$, and thus it is a maximal extension of
$\mathbf{P}$.

###### Proof.

The restriction of $Z$ to $\left\\{X_{2}=0\right\\}$ is the complete
intersection in $\mathbf{P}^{1}\times\mathbf{P}(1^{2},2,3,5^{2})$ yielded by
the equations $U_{0}X_{0}=W^{2}$ and $U_{1}X_{1}=V^{3}$. These are the
defining equations for $\mathbf{P}$ in $\mathbf{P}(1^{2},2,3,5^{2})$ as
mentioned in (2), hence

$Z\cap\left\\{X_{2}=0\right\\}=\mathbf{P}^{1}\times\mathbf{P}$

and it is contracted by $\phi$ to $\mathbf{P}$.

Let $Y$ be the image of $Z$ in $\mathbf{P}^{19}$. Let us show that it contains
$Y_{1}$ and $Y_{2}$ as hyperplane sections. On the one hand, $\left\\{\lambda
X_{2}=0\right\\}$ is the pullback to $F$ of a hyperplane in $\mathbf{P}^{19}$,
such that

$Z\cap\left\\{\lambda X_{2}=0\right\\}=Z|_{\lambda=0}+Z|_{X_{2}=0}.$

In the above, $Z|_{X_{2}=0}$ is contracted onto $\mathbf{P}$, and
$Z|_{\lambda=0}$ has image $Y_{1}$.

On the other hand,

$Z\cap\left\\{\mu X_{2}=0\right\\}=Z|_{\mu=0}+Z|_{X_{2}=0}$

where once again, $Z|_{X_{2}=0}$ is contracted onto $\mathbf{P}$, and
$Z|_{\mu=0}$ has image $Y_{2}$.

It remains to be proven that $Y$ is not a cone. The pencil of fourfold
extensions of $\mathbf{P}$ contained in $Y$ consists of all the $Y\cap H$,
where $H\subset\mathbf{P}^{19}$ is a hyperplane such that $\mathbf{P}\subset
H$. These fourfolds are each cut out on $Y$ by $\ell(\lambda,\mu)X_{2}=0$,
with $\ell$ a linear form. Hence, they are complete intersections in
$\mathbf{P}(1^{2},2,3,5^{3})$ of the form

$\displaystyle U_{0}X_{0}+\lambda U_{1}X_{2}$ $\displaystyle=W^{2},$
$\displaystyle U_{1}X_{1}+\mu U_{0}X_{2}$ $\displaystyle=V^{3}$

where $\lambda$ and $\mu$ are fixed constant coefficients (to be precise,
solutions to $\ell(\lambda,\mu)=0$). Let $Y_{(\lambda,\mu)}$ be the fourfold
section of $Y$ given by the equations above, so that $Y_{1}=Y_{(0,1)}$ and
$Y_{2}=Y_{(1,0)}$. We first notice that $Y_{(\lambda,\mu)}\simeq
Y_{(\alpha\lambda,\beta\mu)}$ for all $\alpha,\beta\in\mathbf{C}^{*}$; indeed,
the automorphism which consists in the change of variables $U_{1}\mapsto\alpha
U_{1},U_{0}\mapsto\beta U_{0},X_{1}\mapsto\frac{1}{\alpha}X_{1}$ and
$X_{0}\mapsto\frac{1}{\beta}X_{0}$ identifies $Y_{(\alpha\lambda,\beta\mu)}$
with $Y_{(\lambda,\mu)}$. Therefore, among the $Y_{(\lambda,\mu)}$ there are
at most three isomorphism classes: $Y_{(1,0)},Y_{(0,1)}$ and $Y_{(1,1)}$. In
particular, the class represented by $Y_{(1,1)}$ is dense in the pencil
$\left\\{Y\cap H\>|\>\mathbf{P}\subset H\right\\}$.

Assume now by contradiction that $Y$ is a cone. It contains $\mathbf{P}$ as a
linear section of codimension $2$, and $\mathbf{P}$ is not a cone, so there
are only two possible cases: either the vertex of $Y$ is a point, or it is a
line. In the latter case, all the $Y_{(\lambda,\mu)}$’s are cones over
$\mathbf{P}$ with each time a point as vertex; in the former case, there is a
unique member $Y_{(\lambda,\mu)}$ which is a cone over $\mathbf{P}$. This
unique member is either $Y_{(1,0)}$ or $Y_{(0,1)}$ since the class of
$Y_{(1,1)}$ is dense in the pencil, so without loss of generality we may
assume that $Y_{(1,0)}$ is a cone over $\mathbf{P}$ with a point as vertex
(the rest of the proof is analogous if the cone is $Y_{(0,1)}$).

Let us recall the equations for $Y_{(1,0)}$ in $\mathbf{P}(1^{2},2,3,5^{3})$
with respect to the coordinates $[U_{0}:U_{1}:V:W:X_{0}:X_{1}:X_{2}]$.

$\displaystyle U_{0}X_{0}+U_{1}X_{2}$ $\displaystyle=W^{2},$ $\displaystyle
U_{1}X_{1}$ $\displaystyle=V^{3}.$

We recall as well the fact that $\mathbf{P}$ is the hyperplane section
$Y_{(1,0)}\cap\left\\{X_{2}=0\right\\}$ in $\mathbf{P}^{18}$. There is a
change of variable which fixes the hyperplane $\left\\{X_{2}=0\right\\}$ and
moves the vertex point to $p_{X_{2}}$. This change of variables makes the
affine chart $Y_{(1,0)}|_{X_{2}=1}$ an affine cone, i.e., it eliminates the
variable $X_{2}$ from the equations above.

Indeed, let $F=G=0$ be the defining equations of a cone whose vertex point is
$p_{X_{2}}$, such that $F$ and $G$ are two homogeneous sextics on
$\mathbf{P}(1^{2},2,3,5^{3})$, not divisible by $X_{2}$, and set
$f=F|_{X_{2}=1},g=G|_{X_{2}=1}$. If one of them is not homogeneous, say $f$,
then

$f=f_{6}+\tilde{f}$

where $f_{6}=f_{6}(U_{0},U_{1},V,W,X_{0},X_{1})$ is homogeneous of degree $6$
and $\tilde{f}$ has degree $5$ or less. By the fact that $\deg(\tilde{f})<6$
and $g$ and $f$ are sextics, we have $\tilde{f}\notin(f,g)$ and thus there
exists a point $q=(U_{0},U_{1},V,W,X_{0},X_{1})$ in the affine cone
$\left\\{f=g=0\right\\}$ such that $\tilde{f}(q)\neq 0$. From the condition
$f(q)=0$, we have the equality $f_{6}(q)=-\tilde{f}(q)$, and by the fact that
$\left\\{f=g=0\right\\}$ is an affine cone, then for all
$\lambda\in\mathbf{C}^{*}$ the point

$\lambda\cdot q=(\lambda U_{0},\lambda
U_{1},\lambda^{2}V,\lambda^{3}W,\lambda^{5}X_{0},\lambda^{5}X_{1})$

also belongs to $\left\\{f=g=0\right\\}$. If $\lambda$ is general, we have

$f_{6}(\lambda\cdot
q)=\lambda^{6}f_{6}(q)=-\lambda^{6}\tilde{f}(q)\neq-\tilde{f}(\lambda\cdot q)$

since the equality $\lambda^{6}\tilde{f}(q)=\tilde{f}(\lambda\cdot q)$ is a
polynomial condition of degree $6$ on $\lambda$. This leads to the
contradiction that $\lambda\cdot q\notin\left\\{f=g=0\right\\}$ and the
conclusion that $f$ and $g$ are homogeneous.

As a consequence, there exists a transformation of the form

$[U_{0}:U_{1}:V:W:X_{0}:X_{1}:X_{2}]\mapsto[A\mathbf{U}:V:W:M\mathbf{X}]$

where $A\in GL_{2}(\mathbf{C})$ and $M\in GL_{3}(\mathbf{C})$, which
eliminates $X_{2}$ from the equations of $Y_{(1,0)}$. Let us denote

$A=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\text{ and
}M=\left(\begin{array}[]{ccc}\alpha_{0}&\beta_{0}&\gamma_{0}\\\
\alpha_{1}&\beta_{1}&\gamma_{1}\\\
\alpha_{2}&\beta_{2}&\gamma_{2}\end{array}\right).$

This change of variables applied to the equations of $Y_{(1,0)}$ yields

$\displaystyle(aU_{0}+bU_{1})(\alpha_{0}X_{0}+\beta_{0}X_{1}+\gamma_{0}X_{2})+(cU_{0}+dU_{1})(\alpha_{2}X_{0}+\beta_{2}X_{1}+\gamma_{2}X_{2})$
$\displaystyle=W^{2},$
$\displaystyle(cU_{0}+dU_{1})(\alpha_{1}X_{0}+\beta_{1}X_{1}+\gamma_{1}X_{2})$
$\displaystyle=V^{3}.$

By the fact that this does not involve the variable $X_{2}$, we have

$a\gamma_{0}+c\gamma_{2}=c\gamma_{1}=b\gamma_{0}+d\gamma_{2}=d\gamma_{1}=0$

in other words,

$\left(\begin{array}[]{cc}a&c\\\
b&d\end{array}\right)\left(\begin{array}[]{c}\gamma_{0}\\\
\gamma_{2}\end{array}\right)=0\text{ and }\left(\begin{array}[]{cc}a&c\\\
b&d\end{array}\right)\left(\begin{array}[]{c}0\\\
\gamma_{1}\end{array}\right)=0.$

This contradicts either $\det(A)\neq 0$, or $\det(M)\neq 0$. The conclusion is
that $Y_{(1,0)}$ is not a cone over $\mathbf{P}$ and therefore, $Y$ is not a
cone ∎

##  5 The primitive polarizations of the K3 surfaces

We recall that the index of the polarized K3 surface
$(S,-K_{\mathbf{P}}|_{S})$, which is denoted by $i_{s}$ in Table 2, is the
divisibility of $-K_{\mathbf{P}}|_{S}$ in the Picard group of $S$, i.e., the
largest integer $r$ such that $-\frac{1}{r}K_{\mathbf{P}}|_{S}$ is a Cartier
divisor on $S$. Here, $\Gamma$ is a general member of
$|-K_{\mathbf{P}}|_{S}|$, and we introduce $C$ a general member of
$|-\frac{1}{i_{S}}K_{\mathbf{P}}|_{S}|$, so that $\Gamma=i_{S}C$ in
$\mathrm{Pic}(S)$.

In what follows, we go through all the cases $\\#9$ to $\\#14$ that are listed
in Table 2 and give a geometric description of the curve $C$.

###  5.1 $\mathbf{P}=\mathbf{P}(1,4,5,10)$

According to Table 2, the index $i_{S}$ is equal to $2$. The genus of $C$ is
$6$. In Table 5, $S$ is explicitly given as the quintic hypersurface
$u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}=0$ in $\mathbf{P}(1,1,1,2)$, with $\deg
f_{4}=4$, and $\Gamma$ is cut out on $S$ by a quartic. Therefore
$C=\frac{1}{2}\Gamma$ is cut out by a quadric, so its defining equations are

$u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}=g_{2}(\mathbf{u},v)=0$

with $g_{2}$ a general homogeneous quadric polynomial.

######  Lemma 5.1.

The curve $C$ is isomorphic to a plane quintic with a total inflection point,
i.e., there is a line $\Delta$ which is tangent to $C$ in $\mathbf{P}^{2}$ and
$C|_{\Delta}$ is a quintuple point.

Conversely, any such plane quintic can be realized as a member of
$|-\frac{1}{2}K_{\mathbf{P}}|_{S}|$ for a general $S\in|-K_{\mathbf{P}}|$.

###### Proof.

Up to scaling, we may choose $g_{2}(\mathbf{u},v)=v-\alpha(u_{0},u_{1},u_{2})$
where $\alpha$ is a conic. Hence $C$ is cut out by

$u_{0}f_{4}(u_{0},u_{1},u_{2},v)+u_{1}^{5}=0,v=\alpha(u_{0},u_{1},u_{2}).$

Substituting $\alpha(u_{0},u_{1},u_{2})$ for $v$ in the first equation
naturally realizes $C$ as a quintic in $\mathbf{P}^{2}$ with coordinates
$[u_{0}:u_{1}:u_{2}]$. Moreover, the restriction of $C$ to the line $u_{0}=0$
is a quintuple point. Let $\Delta=\left\\{u_{0}=0\right\\}$,

$C|_{\Delta}=5p$

where $p=\left\\{u_{0}=u_{1}=0\right\\}$ in $\mathbf{P}^{2}$. This is an
inflection point of order $5$ of the curve $C$. The tangent cone of $C$ at
this point is the reduced line $\Delta$ by generality of $f_{4}$ (the curve
$C$ is indeed smooth, since it is a general hyperplane section of $S$ in
$\mathbf{P}^{6}$ and $S$ has isolated singularities).

Conversely, let $C^{\prime}$ be such a plane quintic. Up to a choice of
coordinates, $C^{\prime}$ is given by an equation of the form
$u_{0}g_{4}(u_{0},u_{1},u_{2})+u_{1}^{5}=0$ with $\deg g_{4}=4$. It holds that
$C^{\prime}$ in its canonical model can be extended by a quintic surface in
$\mathbf{P}(1,1,1,2)$, as the following points out. Following the construction
that was done in A.2 in the appendix of [LD21], there exists a quintic
polynomial $f_{5}(\mathbf{u},v)$ and a quadric
$\alpha(\mathbf{u})=\alpha(u_{0},u_{1},u_{2})$ such that

$C^{\prime}=\left\\{f_{5}(\mathbf{u},v)=0,v=\alpha(\mathbf{u})\right\\}$

in $\mathbf{P}(1,1,1,2)$. Hence the quintic surface
$\left\\{f_{5}(\mathbf{u},v)=0\right\\}$ in $\mathbf{P}(1,1,1,2)$ is an
extension of $C^{\prime}$.

Here $f_{5}$ and $\alpha$ are so that
$f_{5}|_{v=\alpha(\mathbf{u})}=u_{0}g_{4}(u_{0},u_{1},u_{2})+u_{1}^{5}$. Thus
$f_{5}=u_{1}^{5}+\lambda\beta(\mathbf{u},v)(v-\alpha(\mathbf{u}))\text{ (mod
}u_{0})$ in $\mathbf{C}[u_{0},u_{1},u_{2},v]$ for some constant $\lambda$ and
$\deg\beta=3$. Picking $\lambda=0$ doesn’t change $C^{\prime}$, and yields
$f_{5}=u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}$ for some homogeneous quartic
$f_{4}$ on $\mathbf{P}(1,1,1,2)$.

Hence $C^{\prime}$ is cut out on $S^{\prime}$ by a quadric, where $S^{\prime}$
is the quintic $u_{0}f_{4}(\mathbf{u},v)+u_{1}^{5}=0$. Recall from Table 4
that $\varphi:\mathbf{P}\dashrightarrow\mathbf{P}(1,1,1,2)$ restricts to an
isomorphism on the general member of $|-K_{\mathbf{P}}|$; here $S^{\prime}$ is
a member of $\mathcal{L}=\varphi_{*}|-K_{\mathbf{P}}|$ so it is isomorphic to
a general anticanonical divisor of $\mathbf{P}$. Moreover,
$C^{\prime}=-\frac{1}{2}K_{\mathbf{P}}|_{S^{\prime}}$ in
$\mathrm{Pic}(S^{\prime})$. ∎

###  5.2 $\mathbf{P}=\mathbf{P}(1,2,6,9)$

According to Table 2, the index $i_{S}$ of the polarization
$(S,-K_{\mathbf{P}}|_{S})$ is equal to $3$. The curve $C$ has genus $4$. We
know by Table 5 that $S$ is a degree $10$ hypersurface in
$\mathbf{P}(1,1,3,5)$, of equation $u_{0}f_{9}(\mathbf{u},v,s)+s^{2}=0$ and
$\Gamma$ is the intersection of $S$ with a general $9-$ic. Hence $C$ is cut
out on $S$ by a general cubic of $\mathbf{P}(1,1,3,5)$, i.e., its equations
are

$u_{0}f_{9}(\mathbf{u},v,s)+s^{2}=0,\hskip 5.69046ptv=\alpha(u_{0},u_{1})$

where $\alpha$ is a homogeneous cubic polynomial on $\mathbf{P}^{1}$.

######  Lemma 5.2.

The curve $C$ is isomorphic to a $10-$ic curve in $\mathbf{P}(1,1,5)$ (i.e., a
quadric section of the cone over a rational quintic curve). In other words,
$C$ is a smooth hyperelliptic curve of genus $4$.

Conversely, any such curve can be realized as a member of
$|-\frac{1}{3}K_{\mathbf{P}}|_{S}|$ for a general $S\in|-K_{\mathbf{P}}|$.

###### Proof.

By the above equations, $C$ is naturally realized as a degree $10$ curve on
$\mathbf{P}(1,1,5)$ with coordinates $[u_{0}:u_{1}:s]$ of the form
$u_{0}h_{9}(\mathbf{u},s)+s^{2}=0$. Hence the linear system
$|\mathcal{O}_{\mathbf{P}(1,1,5)}(1)|$, whose base locus
$\left\\{u_{0}=u_{1}=0\right\\}$ does not meet $C$, restricts to a $g_{2}^{1}$
on $C$.

Conversely, let $C^{\prime}$ be a curve in $\mathbf{P}(1,1,5)$ of degree $10$.
In a suitable choice of coordinates, the point $[u_{0}:u_{1}]=[0:1]$ belongs
to the branch locus of the double cover $C\to\mathbf{P}^{1}$. Hence the line
$\Delta=\left\\{u_{0}=0\right\\}$ in $\mathbf{P}(1,1,5)$ is tangent to
$C^{\prime}$. As a result, the restriction $C^{\prime}|_{u_{0}=1}$ is a double
point, which yields the following equation for $C^{\prime}$:

$u_{0}h^{\prime}_{9}(\mathbf{u},s)+s^{2}=0,$

with $\deg h^{\prime}_{9}=9$. Introducing $v$ a coordinate of weight $3$ and
$f^{\prime}_{9}(\mathbf{u},v,s)$ a degree $9$ homogeneous polynomial on
$\mathbf{P}(1,1,3,5)$ such that
$f^{\prime}_{9}(\mathbf{u},v,s)|_{v=\alpha(u_{0},u_{1})}=h^{\prime}_{9}(\mathbf{u},v)$,
we realize $C^{\prime}$ as a complete intersection in $\mathbf{P}(1,1,3,5)$ of
equations

$u_{0}f^{\prime}_{9}(\mathbf{u},v,s)+s^{2}=0,v=\alpha(u_{0},u_{1}).$

This makes $C^{\prime}$ a curve in the surface
$S^{\prime}=\left\\{u_{0}f^{\prime}_{9}(\mathbf{u},v,s)+s^{2}=0\right\\}$,
which is a member of $\mathcal{L}=\varphi_{*}|-K_{\mathbf{P}}|$, meaning that
$S^{\prime}$ is isomorphic to a general anticanonical divisor of $\mathbf{P}$.
Moreover, the moving part of $\mathcal{L}|_{S^{\prime}}$ is the restriction to
$S^{\prime}$ of the $9-$ics, and therefore $3C$ is a member of the moving part
of $\mathcal{L}|_{S}$. This makes $3C$ the class of the hyperplane sections of
$S$ in $\mathbf{P}^{28}$, in other words $3C=-K_{\mathbf{P}}|_{S}$ in
$\mathrm{Pic}(S)$. This yields $C=-\frac{1}{3}K_{\mathbf{P}}|_{S^{\prime}}$. ∎

###  5.3 $\mathbf{P}=\mathbf{P}(1,2,3,6)$

This is the only example of our list for which
$-\frac{1}{i_{S}}K_{\mathbf{P}}$ is Cartier. The index $i_{S}$ is equal to $2$
and $-\frac{1}{2}K_{\mathbf{P}}$ is the class of sextic surfaces. Hence, $S$
is a general surface of degree $12$ and $C$ is cut out on $S$ by a sextic. The
projective model associated to $-\frac{1}{2}K_{\mathbf{P}}$ in which $C$ is a
hyperplane section of $S$ is a realization of $\mathbf{P}$ as a variety of
degree $(-\frac{1}{2}K_{\mathbf{P}})^{3}=6$ in $\mathbf{P}^{7}$. It factors as
the composite map

${\mathbf{P}}$${\mathbf{P}(1,1,2,3,3)}$${\mathbf{P}^{7}}$$\scriptstyle{v_{2}}$$\scriptstyle{|\mathcal{O}(3)|}$

where $v_{2}$ is the $2-$Veronese embedding mentioned in Table 3.

Let $[u_{0}:u_{1}:v:s_{0}:s_{1}]$ be coordinates on $\mathbf{P}(1,1,2,3,3)$,
then $v_{2}$ realizes $\mathbf{P}$ as the hypersurface $u_{0}s_{0}=v^{2}$, $S$
as the intersection of $\mathbf{P}$ with a general sextic, and $C$ as the
intersection of $S$ with a general cubic.

Consider now $\mathbf{P}^{\prime}:=\mathbf{P}(1,1,1,3)$ with coordinates
$[a_{0}:a_{1}:a_{2}:b]$ and the rational map
$\psi:\mathbf{P}^{\prime}\dashrightarrow\mathbf{P}(1,1,2,3,3)$ given by the
expression

$[u_{0}:u_{1}:v:s_{0}:s_{1}]=[a_{0}:a_{1}:a_{0}a_{2}:a_{0}a_{2}^{2}:b].$

Its image satisfies the same equation as $\mathbf{P}$, hence it is equal to
$\mathbf{P}$. There is a birational map $\varphi$ which makes the following
diagram commute.

${\mathbf{P}}$${\mathbf{P}^{\prime}}$${\mathbf{P}(1,1,2,3,3)}$$\scriptstyle{\varphi}$$\scriptstyle{v_{2}}$$\scriptstyle{\psi}$

One checks from the expression of $v_{2}$ in Table 3 and that of $\psi$ that
$\varphi$ has the following expression with regard to the weighted
coordinates.

$\varphi:[x:y:z:w]\mapsto[a_{0}:a_{1}:a_{2}:b]=[x^{3}:xy:z:x^{3}w].$

It contracts the vanishing locus of $x$ to a point $p$. The divisor
$D=\left\\{x=0\right\\}$ has degree $2$ on $C$; indeed, $C$ is cut out on
$\mathbf{P}$ by two general equations of respective degree $12$ and $6$ with
regard to the grading of $\mathbf{P}$, and
$D\in|\mathcal{O}_{\mathbf{P}}(1)|$, therefore:

$\deg D|_{C}=D\cdot
C=[\mathcal{O}_{\mathbf{P}}(1)]\cdot[\mathcal{O}_{\mathbf{P}}(12)]\cdot[\mathcal{O}_{\mathbf{P}}(6)]=2.$

This ensures that the restriction $\varphi|_{C}$ maps $2$ distinct points to
$p$. The indeterminacy locus $x=z=0$ does not meet $C$ by the generality
assumption, so the map $\varphi$ induces by restriction to $C$ a morphism
$C\to\mathbf{P}(1,1,1,3)$ which has degree $1$ and makes $C$ the normalization
of its image.

######  Lemma 5.3.

Let $C_{0}$ be the image of $C$ by $\varphi$. Then $C_{0}$ is isomorphic to a
plane sextic which has an oscnode at a point $p$ and there is a line
$\Delta\subset\mathbf{P}^{2}$ through $p$ such that $C_{0}|_{\Delta}=6p$.
Besides, $C$ is the normalization of $C_{0}$.

###### Proof.

We denote by $\varphi^{-1}$ the rational inverse of $\varphi$. One checks from
the expressions of $v_{2}$ and $\psi$ that $\varphi^{-1}$ is the map

$[a_{0}:a_{1}:a_{2}:b]\in\mathbf{P}^{\prime}\mapsto[x:y:z:w]=[a_{0}:a_{0}a_{1}:a_{0}^{2}a_{2}:a_{0}^{3}b].$

Say $q$ is a fixed point in $\mathbf{P}^{\prime}$ with a chosen representative
$(a_{0},a_{1},a_{2},b)$, and $\sqrt{a_{0}}$ is a square root of $a_{0}$, then
the image of $q$ by $\varphi^{-1}$ is

$[\sqrt{a_{0}}:a_{1}:\sqrt{a_{0}}a_{2}:b]\in\mathbf{P}$

and one checks that the composition $v_{2}\circ\varphi^{-1}$ indeed maps $q$
to $\psi(q)$. Let $\Sigma$ be the direct image of $S$ under $\varphi$; it is
the proper transform of $S$ by $\varphi^{-1}$. From the above, we identify the
exceptional locus of $\varphi^{-1}$ as $\left\\{a_{0}=0\right\\}$ and the
pullback to $\mathbf{P}^{\prime}$ of the general $12-$ic surface $S$ is

$(\varphi^{-1})^{*}S=\left\\{a_{0}^{6}(a_{0}f_{5}(\mathbf{a},b)+\lambda
a_{1}^{6}+\mu a_{1}^{3}b+\gamma b^{2})=0\right\\}$

where $f_{5}(\mathbf{a},b)$ is a quintic on $\mathbf{P}^{\prime}$ and
$\lambda,\mu,\gamma$ are constant. Hence the proper transform $\Sigma$ of $S$
is a nongeneral sextic of $\mathbf{P}^{\prime}=\mathbf{P}(1,1,1,3)$ of
equation

$a_{0}f_{5}(\mathbf{a},b)+\lambda a_{1}^{6}+\mu a_{1}^{3}b+\gamma b^{2}=0.$

On the one hand, we have $h^{0}\mathcal{O}_{\mathbf{P}}(-K_{\mathbf{P}})=27$,
while $h^{0}\mathcal{O}_{\mathbf{P}^{\prime}}(5)+3=30$, so even the quintic
$f_{5}$ can’t be general, and it must belong to a subspace $V\subset
H^{0}(\mathbf{P}^{\prime},\mathcal{O}_{\mathbf{P}^{\prime}}(5))$ with $\dim
V=\dim H^{0}(\mathbf{P}^{\prime},\mathcal{O}_{\mathbf{P}^{\prime}}(5))-3=24$.
Namely, it follows from the expression of $\varphi^{-1}$ that $f_{5}$ does not
involve the monomials $a_{2}^{5},a_{2}^{4}a_{1}$ and $a_{2}^{3}a_{1}^{2}$.

As $C$ is a hyperplane section of $S$ in $\mathbf{P}^{7}$, it is cut out on
$S$ by a general cubic $\alpha(u_{0},u_{1},v,s_{0},s_{1})=0$ in
$\mathbf{P}(1,1,2,3,3)$, with

$[u_{0}:u_{1}:v:s_{0}:s_{1}]=[a_{0}:a_{1}:a_{0}a_{2}:a_{0}a_{2}^{2}:b],$

and thus its image $C_{0}$ in $\mathbf{P}^{\prime}$ is cut out on $\Sigma$ by
a nongeneral cubic of the form

$b=\tau a_{1}^{3}+a_{0}q(\mathbf{a})$

with $\tau$ a constant and $q$ a general quadric.

$C_{0}=\Sigma\cap\left\\{b=\tau
a_{1}^{3}+a_{0}q(\mathbf{a})\right\\}=\left\\{\begin{array}[]{l}a_{0}f_{5}(\mathbf{a},b)+\lambda
a_{1}^{6}+\mu a_{1}^{3}b+\gamma b^{2}=0\\\ b=\tau
a_{1}^{3}+a_{0}q(\mathbf{a})\end{array}\right\\}.$

This makes $C_{0}$ a sextic curve in $\mathbf{P}^{2}$ with coordinates
$[a_{0}:a_{1}:a_{2}]$ such that $C_{0}|_{a_{0}=0}=(a_{1}^{6})$. Hence the
restriction of $C_{0}$ to the line $\Delta=\left\\{a_{0}=0\right\\}$ is a
sextic point. This point is the intersection point of $C_{0}$ with $\Delta$,
i.e., the point $p=\left\\{a_{0}=a_{1}=0\right\\}$.

The morphism from $C$ to $C_{0}$ has degree $1$ and maps two distinct points
to $p$. Hence $C_{0}$ has two local branches at the point $p$, say $B_{1}$ and
$B_{2}$, such that

$6p=C_{0}|_{\Delta}=B_{1}|_{\Delta}+B_{2}|_{\Delta}.$

This implies that $B_{i}|_{\Delta}=a_{1}p$ for $a_{i}\in\mathbf{N}$ for
$i=1,2$ and $a_{1}+a_{2}=6$. There are three cases to distinguish.

  1. $(i)$

If $(a_{1},a_{2})=(1,5)$, then $C_{0}$ is a sextic plane curve with a node at
$p$, and $g(C)=\frac{(6-1)(6-2)}{2}-1=9$. But $g(C)=7$ by Table 2.

  2. $(ii)$

If $(a_{1},a_{2})=(2,4)$, then $C_{0}$ has a tacnode at $p$, and
$g(C)=\frac{(6-1)(6-2)}{2}-2=8$, which is also a contradiction.

  3. $(iii)$

If $(a_{1},a_{2})=(3,3)$, then $C_{0}$ has an oscnode at $p$, and we indeed
have $g(C)=\frac{(6-1)(6-2)}{2}-3=7$.

Hence, around $p$, $C_{0}$ has two local branches which meet at an oscnode. By
the generality assumption, $C$ is smooth, as a consequence of Bertini’s
Theorem. It follows that $p$ is the only singular point of $C_{0}$ and $C$ is
the normalization of $C_{0}$. ∎

###  5.4 $\mathbf{P}=\mathbf{P}(1,3,8,12)$

As stated in Table 2, the curve $C$ has genus $7$. It follows from Table 5
that $S$ is isomorphic to the $9-$ic hypersurface
$u_{0}f_{8}(\mathbf{u},v,s)+v^{3}=0$ in $\mathbf{P}(1,1,3,4)$ with coordinates
$[u_{0}:u_{1}:v:s]$ and $\Gamma$ is cut out on $S$ by a degree $8$
hypersurface of $\mathbf{P}(1,1,3,4)$. The index $i_{S}$ is equal to 2,
therefore $C=\frac{1}{2}\Gamma$ in $\mathrm{Pic}(S)$ and $C$ is the
intersection of $S$ with a general quartic. Such a quartic has equation
$s=\alpha(\mathbf{u},v)$, where $\deg\alpha=4$. Hence $C$ is cut out by the
equations

$u_{0}f_{8}(\mathbf{u},v,s)+v^{3}=0,s=\alpha(\mathbf{u},v).$

######  Lemma 5.4.

The curve $C$ is isomorphic to a degree $9$ curve in $\mathbf{P}(1,1,3)$
(i.e., a cubic section of the cone over a rational cubic curve) with an
inflection point of order $3$ along a line of the ruling, i.e., there is a
line $\Delta\in|\mathcal{O}_{\mathbf{P}(1,1,3)}(1)|$ which is tangent to $C$
at a point $p$, and $C|_{\Delta}=3p$. In other words, $C$ is a trigonal curve
of genus $7$ with a total ramification point.

Conversely, any such curve in $\mathbf{P}(1,1,3)$ is isomorphic to a member of
$|-\frac{1}{2}K_{\mathbf{P}}|_{S}|$ for a general $S\in|-K_{\mathbf{P}}|$.

###### Proof.

By the above equations, $C$ is naturally realized as the curve of degree $9$
in $\mathbf{P}(1,1,3)$ given by the following

$u_{0}h_{8}(\mathbf{u},v)+v^{3}=0$

where $\deg h_{8}=8$. Let $\Delta$ be the line $\left\\{u_{0}=0\right\\}$,
then $C|_{\Delta}=3p$ where $p$ is the point $\left\\{u_{0}=v=0\right\\}$. The
tangent cone of $C$ at $p$ is the reduced line $\Delta$, hence $C$ is smooth
and has an inflection point of order $3$ at $p$.

Conversely, let $C^{\prime}\subset\mathbf{P}(1,1,3)$ be such a curve. Then for
a fitting choice of coordinates, it is cut out by an equation of the form

$u_{0}h^{\prime}_{8}(\mathbf{u},v)+v^{3}=0$

with $\deg h^{\prime}_{8}=8$. We introduce a coordinate $s$ of weight $4$ and
a homogeneous degree $8$ polynomial $f^{\prime}_{8}(\mathbf{u},v,s)$ on
$\mathbf{P}(1,1,3,4)$ such that
$f^{\prime}_{8}(\mathbf{u},v,s)|_{s=\alpha(\mathbf{u},v)}=h^{\prime}_{8}(\mathbf{u},v)$.
In this setting, $C^{\prime}$ is the complete intersection in
$\mathbf{P}(1,1,3,4)$ given by

$u_{0}f^{\prime}_{8}(\mathbf{u},v,s)+v^{3}=0,s=\alpha(\mathbf{u},v)$

meaning it is cut out by a general quartic on the surface
$S^{\prime}=\left\\{u_{0}f^{\prime}_{8}(\mathbf{u},v,s)+v^{3}=0\right\\}$.
Recall from Table 4 and Table 5 that the birational map $\varphi$ from
$\mathbf{P}$ to $\mathbf{P}(1,1,3,4)$ restricts to an isomorphism on the
general anticanonical divisors of $\mathbf{P}$. Here $S^{\prime}$ is a member
of $\mathcal{L}=\varphi_{*}|-K_{\mathbf{P}}|$, hence it is isomorphic to a
general member of $|-K_{\mathbf{P}}|$, and furthermore the moving part of
$\mathcal{L}|_{S^{\prime}}$ is the restriction to $S^{\prime}$ of the $8-$ics
of $\mathbf{P}(1,1,3,4)$. As a result, $2C$ is a member of the moving part of
$\mathcal{L}|_{S}$, i.e., it is a hyperplane section of $S$ in
$\mathbf{P}^{25}$. Hence $2C=-K_{\mathbf{P}}|_{S}$ in $\mathrm{Pic}(S)$, and
the conclusion follows that $C=-\frac{1}{2}-K_{\mathbf{P}}|_{S}$. ∎

###  5.5 $\mathbf{P}=\mathbf{P}(1,6,14,21)$

We know from Table 5 that $S$ is a heptic hypersurface in
$\mathbf{P}(1,1,2,3)$ with coordinates $[u_{0}:u_{1}:v:s]$ of equation
$u_{0}f_{6}(\mathbf{u},v,s)+u_{1}^{7}=0$. In this case, the index $i_{S}$ is
equal to $1$, hence $C$ and $\Gamma$ are two curves of genus $22$ which
represent the same Cartier divisor on $S$, which is cut out by a general
sextic of $\mathbf{P}(1,1,2,3)$. Such a sextic is smooth by generality, since
$\mathbf{P}(1,1,2,3)$ has only two isolated singularities and the linear
system of its sextics doesn’t have base points. Moreover, the general sextic
is a double cover of $\mathbf{P}(1,1,2)$ ramified over a general curve of
degree $6$. It is indeed given by an equation of the form
$s^{2}=h_{6}(\mathbf{u},v)$ with $\deg(h_{6})=6$, and the ramification locus
in $\mathbf{P}(1,1,2)$ is the curve $h_{6}(\mathbf{u},v)=0$. This sextic of
$\mathbf{P}(1,1,2,3)$ is then a Del Pezzo surface of degree $1$ and we shall
denote it by $DP_{1}$. In particular, it can be obtained from $\mathbf{P}^{2}$
by blowing up $8$ general points.

######  Lemma 5.5.

The curve $C$ is the blowup of a plane $21-$ic curve $C_{0}$ at $8$ heptuple
points $p_{1},...,p_{8}$. Moreover, if $p$ is the ninth base points of the
pencil $\mathcal{P}$ whose members are the plane cubics through the
$p_{i}^{\prime}$s, then there exists $\gamma$ a member of $\mathcal{P}$ such
that

$C_{0}|_{\gamma}=7p+7p_{1}+\cdots+7p_{8}.$

Conversely, the proper transform in $DP_{1}$ by the blowup map
$DP_{1}\to\mathbf{P}^{2}$ of any such plane curve $C_{0}$ of degree $21$ is
isomorphic to a member of $|-K_{\mathbf{P}}|_{S}|$ for a general
$S\in|-K_{\mathbf{P}}|$.

###### Proof.

Let $\varepsilon:DP_{1}\to\mathbf{P}^{2}$ be the blowup map,
$H=\varepsilon^{*}[\mathcal{O}_{\mathbf{P}^{2}}(1)]$ the pullback of the lines
and $E_{i}$ the exceptional curve over $p_{i}$, $i\in\\{1,...,8\\}$. On the
one hand, by the discrepancy of $\varepsilon$, we have
$-K_{DP_{1}}=3H-\sum_{i=1}^{8}E_{i}$. On the other hand, the adjunction
formula yields $-K_{DP_{1}}=[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(1)|_{DP_{1}}]$.
Since $C=DP_{1}\cap S$, where $S$ is a heptic in $\mathbf{P}(1,1,2,3)$, it
holds that

$C=-7K_{DP_{1}}=21H-\sum_{i=1}^{8}7E_{i}$

and thus it is the proper transform of a degree $21$ curve $C_{0}$ in
$\mathbf{P}^{2}$ which passes through the points $p_{i}$, each with
multiplicity $7$.

The curve $C$ is given by the two following equations.

$u_{0}f_{6}(\mathbf{u},v,s)+u_{1}^{7}=g_{6}(\mathbf{u},v,s)=0$

for $g_{6}$ a general homogeneous sextic polynomial, so that $g_{6}=0$ is the
defining equation of $DP_{1}$. The base point of $-K_{DP_{1}}$ is the
intersection point of $DP_{1}$ with the locus
$\left\\{u_{0}=u_{1}=0\right\\}$, which we denote by $p$. Let $B$ be the curve
$DP_{1}\cap\left\\{u_{0}=0\right\\}$. It is an anticanonical curve of $DP_{1}$
and by the equations above we have

$C|_{B}=(DP_{1}\cap\left\\{u_{0}f_{6}(\mathbf{u},v,s)+_{1}^{7}=0\right\\})|_{u_{0}=0}=DP_{1}|_{u_{0}=0}\cap(u_{1}^{7})|_{u_{0}=0}=7p.$

If $\gamma=\varepsilon(B)$, which is a plane cubic through
$p,p_{1},...,p_{8}$, then the above implies that
$C_{0}|_{\gamma}=7p+7p_{1}+\cdots+7p_{8}$.

Conversely, if $C_{0}^{\prime}$ is a plane $21-$ic curve, then its blowup
$C^{\prime}\subset DP_{1}$ at the points $p_{i}$ is in the Cartier class
$-7K_{DP_{1}}$, and by the surjectivity of

$H^{0}(\mathbf{P}(1,1,2,3),\mathcal{O}_{\mathbf{P}(1,1,2,3)}(7))\twoheadrightarrow
H^{0}(DP_{1},\mathcal{O}_{DP_{1}}(-7K_{DP_{1}}))$

which follows from the restriction short exact sequence

$0\to\mathcal{O}_{\mathbf{P}(1,1,2,3)}\to\mathcal{O}_{\mathbf{P}(1,1,2,3)}(7)\to\mathcal{O}_{DP_{1}}(-7K_{DP_{1}})\to
0$

and the vanishing $h^{1}(\mathcal{O}_{\mathbf{P}(1,1,2,3)})=0$, we have
$C^{\prime}=S^{\prime}\cap DP_{1}$ where $S^{\prime}$ is a heptic surface in
$\mathbf{P}(1,1,2,3)$. It follows that $C^{\prime}$ in $\mathbf{P}(1,1,2,3)$
has equations

$f^{\prime}_{7}(\mathbf{u},v,s)=g_{6}(\mathbf{u},v,s)=0.$

Besides, there exists $B^{\prime}$ an anticanonical curve of $DP_{1}$ such
that $C^{\prime}|_{B^{\prime}}=7p$. We may choose the coordinates
$[u_{0}:u_{1}:v:s]$ on $\mathbf{P}(1,1,2,3)$ such that
$B^{\prime}=DP_{1}\cap\left\\{u_{0}=0\right\\}$. This yields

$f^{\prime}_{7}(\mathbf{u},v,s)|_{u_{0}=g_{6}(\mathbf{u},v,s)=0}=u_{1}^{7}.$

In other words,
$f^{\prime}_{7}=u_{1}^{7}+\lambda\alpha(u_{0},u_{1})g_{6}(\mathbf{u},v,s)(\text{mod
}u_{0})$ for some constant $\lambda$ and $\deg\alpha=1$. We may choose
$\lambda=0$, which does not change $C^{\prime}$ and realizes it as a complete
intersection in $\mathbf{P}(1,1,2,3)$ of the form

$u_{0}f^{\prime}_{6}(\mathbf{u},v,s)+u_{1}^{7}=g_{6}(\mathbf{u},v,s)=0$

with $\deg f^{\prime}_{6}=0$. Thus $C^{\prime}$ lies on the surface
$S^{\prime}=\left\\{u_{0}f^{\prime}_{6}(\mathbf{u},v,s)+u_{1}^{7}=0\right\\}$.
It is a member of $\mathcal{L}=\varphi_{*}|-K_{\mathbf{P}}|$, for $\varphi$
the birational map displayed in Table 4. Therefore $S^{\prime}$ is isomorphic
to a general member of $|-K_{\mathbf{P}}|$ and
$C^{\prime}=-K_{\mathbf{P}}|_{S^{\prime}}$. ∎

###  5.6 $\mathbf{P}=\mathbf{P}(2,3,10,15)$

The index $i_{S}$ is equal to $1$, meaning that both $C$ and $\Gamma$
represent the same Cartier divisor on $S$. The curve $C$ is then the
intersection of $\mathbf{P}$ in $\mathbf{P}^{17}$ with two general
hyperplanes. Recall from (2) that $\mathbf{P}$ is realized as a complete
intersection in $\mathbf{P}(1^{2},2,3,5^{2})$ with coordinates
$[U_{0}:U_{1}:V:W:X_{0}:X_{1}]$ of equations

$U_{0}X_{0}=W^{2},U_{1}X_{1}=V^{3}$

and that its hyperplane sections in $\mathbf{P}^{17}$ are its sections by the
quintics of $\mathbf{P}(1^{2},2,3,5^{2})$. For the sake of notation, let us
use lower case letters instead of upper case ones to designate the
coordinates, as there is no risk of confusion.

The equations that cut out the curve $C$ in $\mathbf{P}$ are thus general
quintics of $\mathbf{P}(1^{2},2,3,5^{2})$, and by the generality assumption we
may choose them to be $x_{0}=f_{5}(\mathbf{u},v,w)$ and
$x_{1}=h_{5}(\mathbf{u},v,w)$ where $f_{5}$ and $h_{5}$ are homogeneous of
degree $5$.

This makes $C$ the curve in $\mathbf{P}(1,1,2,3)$ given by the two following
sextic equations

$v^{3}=u_{1}h_{5}(\mathbf{u},v,w),w^{2}=u_{0}f_{5}(\mathbf{u},v,w).$

In other words, $C$ is the intersection of the two sextic surfaces
$\Sigma=\left\\{v^{3}=u_{1}h_{5}(\mathbf{u},v,w)\right\\}$ and
$\Theta=\left\\{w^{2}=u_{0}f_{5}(\mathbf{u},v,w)\right\\}$. The former is
birational to the surface $\mathbf{P}(1,1,2)$, while the latter is a double
cover of it.

######  Lemma 5.6.

The curve $C$ is the normalization of a $1-$nodal curve $C_{0}$ in
$\mathbf{P}(1,1,2)$ of degree $12$ (i.e., a sextic section of the cone over a
conic). Let $p$ be the node of $C_{0}$ such that $C=Bl_{p}C_{0}$, then it is a
smooth point of $\mathbf{P}(1,1,2)$ and the line
$\Delta\in|\mathcal{O}_{\mathbf{P}(1,1,2)}(1)|$ through $p$ is such that
$C_{0}|_{\Delta}=6p$. Furthermore, there is another line
$\Delta^{\prime}\in|\mathcal{O}_{\mathbf{P}(1,1,2)}(1)|$ such that $C_{0}$ is
tri-tangent to $\Delta^{\prime}$, meaning

$C_{0}|_{\Delta^{\prime}}=2p_{1}+2p_{2}+2p_{3}$

where $p_{1},p_{2}$ and $p_{3}$ are general points of $\Delta^{\prime}$. In
other words, $C_{0}$ is a $6-$gonal curve of genus $16$ such that one member
of the $g_{6}^{1}$ is a sextuple point, and another member consists of three
double points.

Conversely, the normalization of any such $12-$ic curve
$C_{0}\subset\mathbf{P}(1,1,2)$ can be realized as a member of
$|-K_{\mathbf{P}}|_{S}|$ for a general $S\in|-K_{\mathbf{P}}|$.

###### Proof.

Note first that $C$ is smooth. Indeed, $S$ has only isolated singularities and
$C$ is a general hyperplane section of $S\subset\mathbf{P}^{16}$ which avoids
the singular points. The smoothness is then a consequence of Bertini’s
Theorem.

Consider the map
$\varepsilon:\mathbf{P}(1,1,2,3)\dashrightarrow\mathbf{P}(1,1,2)$ which maps
$[u_{0}:u_{1}:v:w]$ to $[u_{0}:u_{1}:v]$, well defined at all the points for
which $(u_{0},u_{1},v)\neq(0,0,0)$. Its indeterminacy point
$p_{w}=\left\\{u_{0}=u_{1}=v=0\right\\}$ belongs to $\Sigma$. The fiber
$\varepsilon$ over a general smooth point of $\mathbf{P}(1,1,2)$, say
$[u_{0}:u_{1}:v]$ fixed with $(u_{0},u_{1})\neq(0,0)$, is a $\mathbf{P}(1,3)$
in $\mathbf{P}(1,1,2,3)$ parameterized by

$[\mathtt{u}:\mathtt{w}]\in\mathbf{P}(1,3)\to[\mathtt{u}u_{0}:\mathtt{u}u_{1}:\mathtt{u}^{2}v:\mathtt{w}]\in\varepsilon^{-1}([u_{0}:u_{1}:v]).$

It follows from the defining equation of $\Sigma$ in $\mathbf{P}(1,1,2,3)$
that its restriction to such a fiber is the locus in $\mathbf{P}(1,3)$ given
by an equation of the form $\mathtt{u}^{3}(\mu\mathtt{u}^{3}+\mathtt{w})=0$,
with $\mu$ a constant. Hence the restriction of $\Sigma$ to this fiber
consists of two points, a general one and the indeterminacy point of
$\varepsilon$ (for which $\mathtt{u}=0$). The only exception is the particular
fiber $\mathfrak{f}:=\left\\{u_{1}=v=0\right\\}$, which is a $\mathbf{P}(1,3)$
with coordinates $[u_{0}:w]$, contained in $\Sigma$ and contracted to a smooth
point of $\mathbf{P}(1,1,2)$.

Meanwhile, the fiber over the singular point $p_{v}\in\mathbf{P}(1,1,2)$,
i.e., all the points $[u_{0}:u_{1}:v:w]$ for which $(u_{0},u_{1})=(0,0)$, is a
$\mathbf{P}(2,3)$ with coordinates $[v:w]$ and the restriction of $\Sigma$ to
this particular fiber is given by the equation $v^{3}=0$. Hence $\Sigma$ meets
this particular fiber only at the indeterminacy point of $\varepsilon$.

Hence, $\varepsilon|_{\Sigma}$ is a birational map from $\Sigma$ to
$\mathbf{P}(1,1,2)$ with indeterminacy point
$p_{w}=\left\\{u_{0}=u_{1}=v\right\\}$ and it induces a regular map
$\Sigma-\left\\{p_{w}\right\\}\to\mathbf{P}(1,1,2)-\left\\{p_{v}\right\\}$
which contracts the curve $\mathfrak{f}$ to a point.

Using the parameterization of $\mathfrak{f}$ as a $\mathbf{P}(1,3)$ with
coordinates $[u_{0}:w]$, we know from the equations for $C$ in
$\mathbf{P}(1,1,2,3)$ that the restriction $C|_{\mathfrak{f}}$ is cut out by
the equation $\lambda u_{0}^{3}w+w^{2}=0$, where $\lambda$ is a constant.
Hence, $C$ has degree $2$ on $\mathfrak{f}$ and does not contain the
indeterminacy point $p_{w}$ of $\varepsilon$. This implies in particular that
the restriction $\varepsilon|_{C}$ is a regular map with image a curve
$C_{0}\subset\mathbf{P}(1,1,2)$. Since $\varepsilon|_{\Sigma}$ is birational,
then the morphism from $C$ to $C_{0}$ is birational; besides, it maps the two
points $C\cap\mathfrak{f}$ to a single point $p$, so the curve $C_{0}$ has a
node at $p$ and $C=Bl_{p}C_{0}$. Since $C$ is smooth, it makes it the
normalization of $C_{0}$.

Furthermore, $C_{0}$ is a member of $|\mathcal{O}_{\mathbf{P}(1,1,2)}(d)|$
such that

$\frac{d}{2}=C_{0}\cdot[\mathcal{O}_{\mathbf{P}(1,1,2)}(1)]=C\cdot[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(1)]=[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6)]^{2}\cdot[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(1)]=6.$

The curve $C_{0}$ is then given by a degree $12$ equation on
$\mathbf{P}(1,1,2)$; in other words, it is a sextic section of the cone over a
conic.

In particular, let $\Delta=\left\\{u_{1}=0\right\\}$ be the line through $p$
in $\mathbf{P}(1,1,2)$. By the above, $C_{0}|_{\Delta}$ has degree $6$. But
the intersection of $C_{0}$ with $\Delta$ is the image by $\varepsilon$ of
$C\cap\left\\{u_{1}=0\right\\}$, and since $C\subset\Sigma$ and
$\Sigma\cap\left\\{u_{1}=0\right\\}=\left\\{u_{1}=v=0\right\\}=\mathfrak{f}$
we have

$C_{0}\cap\Delta=\varepsilon(C|_{u_{1}=0})\subset\varepsilon(\mathfrak{f})=\left\\{p\right\\}.$

Hence $C_{0}|_{\Delta}=6p$.

Now let $\Delta^{\prime}$ be the line $\left\\{u_{0}=0\right\\}$ in
$\mathbf{P}(1,1,2)$. The restriction of $C_{0}$ to $\Delta^{\prime}$ is the
image by $\varepsilon$ of $C|_{u_{0}=0}$. By the fact that
$C=\Sigma\cap\Theta$ and

$\Theta|_{u_{0}=0}=(w^{2})=2\ell$

where $\ell$ is the curve $u_{0}=w=0$, which is a $\mathbf{P}(1,2)$ with
coordinates $[u_{1}:v]$, and $\Sigma|_{\ell}$ has degree $3$,

$C|_{u_{0}=0}=\Theta|_{u_{0}=0}\cap\Sigma|_{u_{0}=0}=2\ell\cap\Sigma|_{u_{0}=0}$

which consists of three double points. Therefore,
$C_{0}|_{\Delta^{\prime}}=2p_{1}+2p_{2}+2p_{3}$ where $p_{1},p_{1},p_{3}$ are
general points of $\Delta^{\prime}$. By the fact that $C_{0}$ is smooth
outside its node, it means that the three contact points of $C_{0}$ with
$\Delta^{\prime}$ are tangency points.

Conversely, let $C_{0}^{\prime}$ be such a curve in $\mathbf{P}(1,1,2)$ and
$C^{\prime}$ its proper transform in $\Sigma$. As $C_{0}^{\prime}$ is given by
an equation of degree $12$,

$6=C_{0}^{\prime}\cdot[\mathcal{O}_{\mathbf{P}(1,1,2)}(1)]=C^{\prime}\cdot[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(1)]=\Sigma\cdot[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6)]\cdot[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(1)].$

The surjectivity of
$H^{0}(\mathbf{P}(1,1,2,3),\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6))\to
H^{0}(\Sigma,\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6)|_{\Sigma})$ which follows
from the restriction exact sequence

$0\to\mathcal{O}_{\mathbf{P}(1,1,2,3)}\to\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6)\to\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6)|_{\Sigma}\to
0$

and the vanishing of $h^{1}(\mathcal{O}_{\mathbf{P}(1,1,2,3)})$ implies that
there exists $\Theta^{\prime}$ a sextic of $\mathbf{P}(1,1,2,3)$ such that
$C^{\prime}=\Sigma\cap\Theta^{\prime}$. Let $f_{6}(\mathbf{u},v,w)=0$ be an
equation for $\Theta^{\prime}$. As $C_{0}^{\prime}$ is tri-tangent to the line
$\Delta^{\prime}$, the set $C_{0}^{\prime}\cap\left\\{u_{0}=0\right\\}$ has
cardinality $3$, and moreover

$\deg
C^{\prime}|_{u_{0}=0}=\Theta^{\prime}|_{u_{0}=0}\cdot\Sigma|_{u_{0}=0}=[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(6)]^{2}\cdot[\mathcal{O}_{\mathbf{P}(1,1,2,3)}(1)]=6,$

so necessarily $\Theta^{\prime}|_{u_{0}=0}$ is a nonreduced curve
$2\ell^{\prime}$ of $\mathbf{P}(1,2,3)_{[u_{1}:v:w]}$ such that
$\ell^{\prime}\cdot\Sigma|_{u_{0}=0}=3$. Hence
$f_{6}(\mathbf{u},v,w)|_{u_{0}=0}=h_{3}(u_{1},v,w)^{2}$ with $h_{3}$ a
homogeneous cubic. Up to scaling, we have $h_{3}=w+\alpha(u_{1},v)$ with
$\deg\alpha=3$, and the change of variables $w\mapsto w+\alpha(u_{1},v)$,
which is an automorphism of $\mathbf{P}(1,1,2,3)$, yields
$f_{6}(\mathbf{u},v,w)|_{u_{0}=0}=w^{2}$ and thus
$f_{6}(\mathbf{u},v,w)=u_{0}f^{\prime}_{5}(\mathbf{u},v,w)+w^{2}$ for some
homogeneous quintic $f^{\prime}_{5}$. Hence $C^{\prime}$ lies on the surface
$\Theta^{\prime}$ of equation

$u_{0}f^{\prime}_{5}(\mathbf{u},v,w)+w^{2}=0.$

As a consequence, $C^{\prime}$ is the complete intersection in
$\mathbf{P}(1,1,2,3)$ given by the following.

$u_{0}f^{\prime}_{5}(\mathbf{u},v,w)+w^{2}=0,v^{3}=u_{1}h_{5}(\mathbf{u},v,w).$

As $\mathbf{P}$ is cut out in $\mathbf{P}(1^{2},2,3,5^{2})$ by the equations
$u_{0}x_{0}=w^{2}$ and $u_{1}x_{1}=v^{3}$, and the pullbacks of the
hyperplanes of $\mathbf{P}^{17}$ are the quintics hypersurfaces of
$\mathbf{P}(1^{2},2,3,5^{2})$, it makes it visible that $C^{\prime}$ is a
linear section of $\mathbf{P}$ in $\mathbf{P}^{17}$:

$C^{\prime}=\mathbf{P}\cap\left\\{x_{0}=-f^{\prime}_{5}(\mathbf{u},v,w),x_{1}=h_{5}(\mathbf{u},v,w)\right\\}.$

In particular, there exists a general hyperplane section
$S^{\prime}\in|-K_{\mathbf{P}}|$ of $\mathbf{P}$ in $\mathbf{P}^{17}$ such
that $C^{\prime}$ is a hyperplane section of $S^{\prime}$, which yields
$C^{\prime}=-K_{\mathbf{P}}|_{S^{\prime}}$. ∎

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INSTITUT DE MATHÉMATIQUES DE TOULOUSE (CNRS UMR 5219), UNIVERSITÉ PAUL
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