text:
\begin{document}
\title[Pfaffian Calabi-Yau Threefolds and Mirror Symmetry]
{Pfaffian Calabi-Yau Threefolds \\ and Mirror Symmetry}
\author{Atsushi Kanazawa}
\date{}

\subjclass[2010]{14J32, 14F33}
\keywords{pfaffian, Calabi-Yau threefold, mirror symmetry, Gromov-Witten, BPS}
\maketitle

\begin{abstract}
The aim of this article is to report on recent progress in understanding mirror symmetry for some non-complete intersection Calabi-Yau threefolds.
We first construct four new smooth Calabi-Yau threefolds with $h^{1,1}=1$ of non-complete intersection type, whose existence was previously conjectured in \cite{van2}.
We then compute the period integral of candidate mirror families for these Calabi-Yau threefolds
and check that the Picard-Fuchs equations coincide with the expected Calabi-Yau equations listed in \cite{van1,van2}.
Some of the mirror families turn out to have two maximally unipotent monodromy points.
\end{abstract}

\section{Introduction}
The aim of this article is to report on recent progress in understanding mirror symmetry for some non-complete intersection Calabi-Yau threefolds.
Throughout this paper, we adopt the following definition.
\begin{Def}
A $d$-dimensional Calabi-Yau variety $X$ is a normal compact variety with at worst Gorenstein canonical singularities
and with trivial dualizing sheaf $\omega_{X} \cong \mathscr{O}_{X}$ such that $H^{i}(X, \mathscr{O}_{X})=0, \ ( i = 1, 2, \dots d-1)$.
\end{Def}
We mainly discuss 3-dimensional Calabi-Yau varieties (Calabi-Yau threefolds) in this paper
but most of the content can be easily generalized to other dimensional Calabi-Yau varieties.
Among Calabi-Yau threefolds, those with $h^{1,1}=1$ dimensional K\"{a}hler moduli space have been attracting much attention
since their expected mirror partners have $h^{2,1}=1$ dimensional complex moduli space and hence one can work on them in detail.
There are around 30 known examples of topologically different smooth Calabi-Yau threefolds with $h^{1,1}=1$,
most of which are complete intersections in toric varieties and homogeneous spaces.
Although non-complete intersection Calabi-Yau threefolds are only partially explored,
they are intriguing on their own and provide important testing grounds for mirror symmetry.
We thus think that the Calabi-Yau threefolds and mirror phenomena we report in this paper are of interest
and will hopefully be the first step toward the future investigations.
This paper is clearly influenced by E. R\o dland's work \cite{rod}, and we owe a lot of arguments to it.
We mention it here, and do not repeat it each time in the sequel.
In the following, we give a brief overview of this paper.\\

Section 2 is mainly devoted to the study of pfaffian threefolds in weighted projective spaces.
pfaffian Calabi-Yau threefolds in $\mathbb{P}^{6}$ was first studied by F. Tonoli in his thesis \cite{to}.

By replacing the ambient space $\mathbb{P}^{6}$ by weighted projective spaces,
we obtain new smooth Calabi-Yau threefolds with $h^{1,1}=1$ of low degree.
We then determine their fundamental topological invariants, namely $\int_{X}H^{3}$, $\int_{X}c_{2}(X)\cdot H$ and $\int_{X}c_{3}(X)$,
which determine the topological type of $X$ when it is simply connected and $h^{1,1}=1$ according to the celebrated Wall's classification theorem \cite{Wall}.
The main result of this section is the following. See Section 2 for the definition of $X_{i}$ and $Y_{i}$.
\begin{Thm}
$X_{5}$, $X_{7}$, $X_{10}$ and $X_{25}$ are smooth Calabi-Yau threefolds with the following topological invariants.
\begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|}  \hline
 $X_{i}$  & $h^{1,1}$ & $h^{1,2}$ & $\int_{X_{i}}H^{3}$ & $\int_{X_{i}}c_{2}(X_{i})\cdot H$ \\ \hline
 $X_{5}$ & $1$ & $51$ & $5$  &  $38$  \\ \hline

 $X_{7}$ &$1$ & $61$ & $7$ &  $46$  \\ \hline
$X_{10}$ &  $1$ & $59$ & $10$ & $52$\\ \hline
$X_{25}$ &  $1$ & $51$ & $25$ & $70$\\ \hline
 \end{tabular}
 \end{center}
\end{Thm}
The existence of Calabi-Yau threefolds with these geometic invariants was previously conjectured
from the view point of classification of Calabi-Yau equations by C. van Enckevort and D. van Straten in \cite{van2}. \\

In Section 3 and 4, we report on mirror symmetry for these Calabi-Yau threefolds.
A pfaffian Calabi-Yau threefold $X_{13}$ of degree $13$ was constructed by F. Tonoli \cite{to}
and a candidate of a mirror family for $X_{13}$ was proposed by J. B\oe hm from the view point of tropical geometry in \cite{boehm}.
We first review the tropical mirror construction and its relation to orbifolding,
and then justify the conjectural mirror family for $X_{13}$ by computing the Picard-Fuchs equation.
After computing the conjectural $g=0,1$ BPS invariants $\{n_{d}^{g}\}_{d\in \N}$,
we heuristically determine the number of the degree 1 rational curves in a general degree 13 pfaffian Calabi-Yau threefold $X_{13}$
and see it coincides with $n_{1}^{0}$ as the mirror symmetry predicts.
Interestingly, the family has a special point where all the indices of the Picard-Fuchs equation are $1/2$, in addition to the usual maximally unipotent monodromy point at $0$. \\

Although the existence of mirror families for given Calabi-Yau threefolds is highly non-trivial,
by following the lead of the degree 13 case, we explicitly exhibit mirror families for Calabi-Yau threefolds $X_{5}$, $X_{7}$ and $X_{10}$ by orbifolding
and check that their Picard-Fuchs equations coincide with the expected Calabi-Yau equations, which are supposed to be the quantum differential equations of the initial threefolds.
However, a generic member of these one-parameter families is quite singular and we could not find any crepant resolution.  \\

Section 5 studies a degree 9 pfaffian Calabi-Yau threefold $X_{9} \subset \mathbb{P}(1^{6},2)$, which is isomorphic to some $\mathbb{P}^{5}_{3^{2}}$.
However, this twofold interpretation allows $X_{9}$ to yield non-isomorphic special one-parameter families, both of which have the same Picard-Fuchs equation. \\

After we posted the preliminary version of this paper to the arXiv,
a physics paper \cite{SS} studied open mirror symmetry of pfaffian Calabi-Yau threefolds constructed in this paper.
It would be also interesting to extend Hori-Tong GLSM description \cite{ht} to our pfaffian threefolds.

\subsection*{Aknowledgement}
The author would like to express his gratitude to Shinobu Hosono for his dedicated support and continuous encouragement.
He greatly appreciates many helpful discussions with Makoto Miura at various stage of this work.
His thanks also go to Duco van Straten for letting him know the new data base \cite{van1} and his comments on the preliminary version of this paper.

\section{Pfaffian Calabi-Yau Threefolds}

\subsection{Pfaffian Threefolds in Projective Spaces}
Suppose that $R$ is a regular local ring and $I\subset R$ is an ideal of height 1 or 2, then J. -P. Serre proved that $R/I$ is Gorenstein if and only if it is complete intersection.
This is no longer true for height 3 ideals, but such Gorenstein ideals are characterized as pfaffian ideals of certain skew-symmetric matrices \cite{be}.
This reasonably motivates us to study pfaffian varieties in the hope that they are not too general non-complete intersection varieties to investigate.
In this subsection, we review the basics of pfaffians in order to make this paper self-contained.
Although a pfaffian variety is, in general, a non-complete intersection, it has an explicit resolution of the structure sheaf, called the pfaffian resolution.
The pfaffian resolution turns out to be indispensable tools to analyze the pfaffian variety. \\

In this paper we work over complex numbers $\C$.
Let SkewSym$(n,\C)$ be the set of  $n\times n$ skew symmetric matrices.
For $N=(n_{i,j}) \in $ SkewSym$(n,\C)$ the pfaffian Pf$(N)$ is defined as
$$
 \mathrm{Pf}(N)= \frac{1}{r!2^{r}}\sum_{\sigma \in \mathfrak{S}_{2r}}\mathrm{sign}(\sigma)\prod_{i=1}^{r} n_{\sigma(i)\sigma(r+i)}
$$
 if $n=2r$ is even, and Pf$(N)=0$ if $n$ is odd.
 It is easy to see that Pf$(N)^{2}=\mathrm{det}(N)$.
 We define $N_{i_{1},\dots,i_{l}}$ as a skew-symmetric matrix obtained by removing all the $i_{j}$-th rows and columns from $N$,
 and $P_{i_{1},\dots,i_{l}}$ as $\mathrm{Pf}(N_{i_{1},\dots,i_{l}})$.
 Set $n=2r+1$ from now on.
 The adjoint matrix of $N$ is the matrix of rank 1, of the form
$$
 \mathrm{adj}(N)=P\cdot P^{t}, \ \ P=(P_{1},P_{2},\dots,P_{2r+1})^{t}.
$$
We thus have $P\cdot N=\det(N)\cdot E_{n}=0$ and if rank$(N)=2r$ then $\{P_{i}\}_{i=1}^{2r+1}$ generate $Ker(N)$.
$\GL(n,\C)$ acts on SkewSym$(n,\C)$ by conjugation with a finite number of orbits $\{O_{2i}\}_{i=0}^{r}$
where the orbit $O_{2i}$ consists of all skew-symmetric matrices of rank $2i$.
The closure $\overline{O_{2i}}$ is singular along its boundary $\overline{O_{2i}}\setminus O_{2i}$ which consists of the union of $\{O_{2j}\}_{j=0}^{i-1}$. \\

Let us first recall the construction of pfaffian varieties in $\mathbb{P}^{n} \ (n>3)$.
We identify $H^{2}(\mathbb{P}^{n},\Z)\cong \Z$ via the hyperplane class.
Given a locally free sheaf $\mathscr{E}$ of odd rank $2r+1$ on $\mathbb{P}^{n}$ and $t \in \mathbb{Z}$,
a global section $N \in H^{0}(\mathbb{P}^{n}, \wedge^{2}\mathscr{E}(t))$ defines an alternating morphism
$\mathscr{E}^{\vee}(-t)\stackrel{N}{\rightarrow} \mathscr{E}$.
Then the pfaffian complex associated to the data $(\mathscr{E},N)$ is the following complex
$$
0 \longrightarrow \mathscr{O}_{\mathbb{P}^{n}}(-t-2s) \stackrel{P^{t}}{\longrightarrow}
\mathscr{E}^{\vee}(-t-s)\stackrel{N}{\longrightarrow}
\mathscr{E}(-s) \stackrel{P}{\longrightarrow} \mathscr{O}_{\mathbb{P}^{n}},
$$
where $s=c_{1}(\mathscr{E})+rt$ and $P$ is defined as;
$$
P = \frac{1}{r!}\wedge^{r} N \in H^{0}(\mathbb{P}^{n},\wedge^{2r}\mathscr{E}(rt)).
$$
The first and third morphisms are given by taking the wedge product with $P$ and $P^{t}$ respectively.
Note that, once we fix a basis of sections $\{e_{i}\}_{i=1}^{2r+1}$ of $\mathscr{E}$, $N$ is just a matrix and we have
$$P=\sum_{i=1}^{2r+1}\mathrm{Pf}(N_{i})\bigwedge_{j\neq i}e_{j}.$$
\begin{Def}
A projective variety $X \subset \mathbb{P}^{n}$ is called a pfaffian variety associated to the data $(\mathscr{E},N)$
if the structure sheaf $\mathscr{O}_{X}$ is given by Coker$(P)$.
The sheaf Im$(P)$ $\subset \mathscr{O}_{\mathbb{P}^{n}}$ is called the pfaffian ideal sheaf and denoted by $\mathscr{I}_{X}$.
\end{Def}

When we do not specify the choice of $N$, it is understood that $N$ is generically taken.
In \cite{to}, F. Tonoli took advantage of the following globalized version of the classical theorem of D. A. Buchsbaum and D. Eisenbud

and successfully gave some examples of smooth Calabi-Yau threefolds with $h^{1,1}=1$ in $\mathbb{P}^{6}$.

\begin{Thm}[D. A. Buchsbaum, D. Eisenbud \cite{be}]
Let $X \subset \mathbb{P}^{n}$ be a pfaffian variety associated to the data $(\mathscr{E},N)$.
Then $X$ is the degeneracy locus of the skew-symmetric map $N$
and if $N$ is generically of rank $2r$ it degenerates to rank $2r-2$ in the expected codimension $3$,
in which case, the pfaffian complex gives the self-dual resolution of the ideal sheaf of $X$.
Moreover, $X$ is locally Gorenstein, subcanonical with $\omega_{X} \cong \mathscr{O}_{X}(t+2s-n-1)$.
\end{Thm}

Let $X$ be a pfaffian Calabi-Yau variety of dimension 3 in $\mathbb{P}^{6}$. Here we have $t+2s=7$.
By applying suitable twists, we can assume that $s=3$ and henceforth we consider the pfaffian complex of the following type.
$$
0 \longrightarrow \mathscr{O}_{\mathbb{P}^{6}}(-7) \stackrel{P^{t}}{\longrightarrow}
\mathscr{E}^{\vee}(-4)\stackrel{N}{\longrightarrow}
\mathscr{E}(-3) \stackrel{P}{\longrightarrow} \mathscr{O}_{\mathbb{P}^{n}}
\longrightarrow \mathscr{O}_{X} \longrightarrow 0
$$

It is natural to expect some bounding of topological invariants of pfaffian Calabi-Yau threefolds in $\mathbb{P}^{6}$.
To see the restriction of the possible degree, we reduce the question to the compact complex surface theory by taking a hyperplane section.
Let $S$ be a compact, smooth complex surface.
There are two important numerical invariants of $S$, namely a geometric genus $p_{g}=\dim H^{0}(S,K_{S})$ and the self-intersection of the canonical divisor $K_{S}^{2}$.

\begin{Thm}[Castelnuovo inequality] Let $S$ be a minimal surface of general type.
 If the canonical map $\Phi_{|K_{S}|}:S \rightarrow \mathbb{P}^{n}$ is birational to the image, then $K_{S}^{2} \ge 3p_{g}-7$.
\end{Thm}
We say a variety $X\subset \mathbb{P}^{n}$ is full if $X$ is not contained in any hyperplane.
Let $S$ be a smooth surface obtained by taking a hyperplane section of a full Calabi-Yau threefold $X \subset \mathbb{P}^{6}$.
Then deg$(X)=K_{S}^{2}$ and $K_{S}$ is nef since $S$ is a canonical surface.
As $X \subset \mathbb{P}^{6}$ is full,
the short exact sequence $0 \longrightarrow \mathscr{O}_{X} \longrightarrow \mathscr{O}_{X}(1) \longrightarrow \mathscr{O}_{S}(1) \longrightarrow 0 $
gives the dimension count $p_{g}=6$ and thus the lower bound of the degree of $X$ is $11$.
Since $\mathbb{P}^{5}_{3^{2}}$ is of degree 9, we cannot remove the fullness on $X$.\\

In \cite{to}, F. Tonoli constructed pfaffian Calabi-Yau threefolds of degree $d$ from 11 to 17.
Although Castelnuovo inequality tells us the minimal degree $d$ of a full Calabi-Yau threefold in $\mathbb{P}^{6}$ is 11,
there seems no smooth one with $d=11$, and the case $d=12$ turns out to be just a complete intersection $\mathbb{P}^{6}_{2^{2},3}$.
Therefore the degree 13 is a good starting point to analyze.

\begin{Def}[F. Tonoli \cite{to}]The degree 13 pfaffian Calabi-Yau threefold $X_{13} \subset \mathbb{P}^{6}$ is the degeneracy locus of
a generic alternating morphism $\mathscr{E}^{\vee}(-1)\stackrel{N}{\rightarrow} \mathscr{E}$, where
$\mathscr{E}=\mathscr{O}_{\mathbb{P}^{6}}(1) \oplus \mathscr{O}_{\mathbb{P}^{6}}^{\oplus 4}$.
 \end{Def}
 $X_{13}$ is known to be smooth and the geometric invariants are computed to be
$$
h^{1,1}=1, \ h^{1,2}=61, \ \int_{X_{13}}H^{3}=13, \ \int_{X_{13}}c_{2}(X_{13})\cdot H = 58.
$$

The degree 14 pfaffian Calabi-Yau threefold $X_{14}$ is defined by the locally free sheaf $\mathscr{E}=\mathscr{O}_{\mathbb{P}^{6}}^{\oplus7}$.
This is nothing but the intersection of $\mathbb{P}^{6}$ with Pfaff$(7)\subset \mathbb{P}^{20}$, the rank 4 locus of projectivised general skew-symmetric $7 \times 7 $ matrices
$$
\mathbb{P}  (\bigwedge^{2} \mathbb{C}^{7}) = \mathbb{P}(\mathrm{SkewSym(7,\C)})
\supset \mathrm{Pfaff(7)}=\{[M] \ | \ \mathrm{rank}(M)\le4 \}.
$$
$X_{14}$ and its mirror partner are verified to have beautiful structures in \cite{rod, bc,hos}.
We thus expect that the degree 13 pfaffian Calabi-Yau threefold $X_{13}$ has as rich structures as $X_{14}$.

\subsection{Pfaffian Threefolds in Weighted Projective Spaces}
F. Tonoli's construction is generalized by replacing the ambient space $\mathbb{P}^{6}$ by any Fano variety.
Special care must be taken when the ambient space is singular.
In the following we study the simplest case, when the ambient space is a weighted projective space $\mathbb{P}_{\mathbf{w}}$.
Given a locally free sheaf $\mathscr{E}$ of odd rank $2r+1$ on $\mathbb{P}_{\mathbf{w}}$ and $t \in \mathbb{Z}$,
a global section $N \in H^{0}(\mathbb{P}_{\mathbf{w}}, \wedge^{2}\mathscr{E}(t))$ defines an alternating morphism
$\mathscr{E}^{\vee}(-t)\stackrel{N}{\rightarrow} \mathscr{E}$.
Then the pfaffian complex associated to the data $(\mathscr{E},N)$ is the following complex.
\begin{equation}
0 \longrightarrow \mathscr{O}_{\mathbb{P}_{\mathbf{w}}}(-t-2s) \stackrel{P^{t}}{\longrightarrow}
\mathscr{E}^{\vee}(-t-s)\stackrel{N}{\longrightarrow}
\mathscr{E}(-s) \stackrel{P}{\longrightarrow} \mathscr{O}_{\mathbb{P}_{\mathbf{w}}}\notag,
\end{equation}
where $s=c_{1}(\mathscr{E})+rt$ and $P=\frac{1}{r!}\wedge^{r} N $ are the same as before.
The pfaffian variety $X$ in $\mathbb{P}_{\mathbf{w}}$ associated to the data $(\mathscr{E},N)$ is a variety
whose structure sheaf $\mathscr{O}_{X}$ is given by $Coker(P)$.
We also define $|\mathbf{w}|$ as a sum of weight of $\mathbb{P}_{\mathbf{w}}$.

\begin{Prop}
Let $\mathbb{P}_{\mathbf{w}}$ be a weighted projective space of dimension $6$ and $(\mathscr{E},N)$ data as above.
Moreover, $X$ has trivial dualizing sheaf $\omega_{X} \cong \mathscr{O}_{X}$ if and only if $t+2s=|\mathbf{w}|$.
\end{Prop}
\begin{proof}
Apply the functor $\mathscr{H}om(-, \omega_{\mathbb{P}_{\mathbf{w}}})$ to the pfaffian resolution to compute

the dualizing sheaf $\omega_{X} \cong \mathscr{E}xt^{3}(\mathscr{O}_{X},\omega_{\mathbb{P}_{\mathbf{w}}})$,
which is isomorphic to $\cong \mathscr{O}_{X}$ if and only if $t+2s=|\mathbf{w}|$ by the definition of the pfaffian variety.

 \end{proof}

As we are interested in Calabi-Yau threefolds, we restrict ourselves to the case $t+2s=|\mathbf{w}|$.
Moreover, up to an opportune twist, we may assume $t=1$ for $|\mathbf{w}|$ odd and $t=0$ for $|\mathbf{w}|$ even.

\begin{Def}
Define $X_{5}$, $X_{7}$ and $X_{10}$ as pfaffian varieties associated to the following $\mathbb{P}_{\mathbf{w}_{i}}$, $\mathscr{E}_{i}$,
and generic choices of $N \in H^{0}(\mathbb{P}_{\mathbf{w}_{i}}, \wedge^{2}\mathscr{E}_{i}(t))$.
\begin{center}
 \begin{tabular}{|c|c|c|}  \hline
 $i$ & $\mathbf{w}_{i}$ & $\mathscr{E}_{i}$\\ \hline
 $5$ & $(1^{4},2^{3})$ & $\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{5}}}^{\oplus 5}(1)$  \\ \hline
 $7$ & $(1^{5},2^{2})$ & $\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(1)^{\oplus 2}\oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}^{\oplus 3}$ \\ \hline
 $10$ & $(1^{6},2^{1})$ & $\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(1)^{\oplus 4}\oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}$\\ \hline
 \end{tabular}
 \end{center}
 \end{Def}

There are many other choices of weights $\mathbf{w}$ and locally sheaves $\mathscr{E}$ on $\mathbb{P}_{\mathbf{w}}$ to produce pfaffian Calabi-Yau threefolds
but it seems only these three cases yield smooth Calabi-Yau threefolds in weighted projective spaces of dimension $6$.

\begin{Thm}
For a generic choice of $N$, the pfaffian varieties $X_{5}$, $X_{7}$ and $X_{10}$ are smooth varieties.
\end{Thm}
\begin{proof}
A generic choice of $N$ guarantees quasi-smoothness of $X_{i}$ as follows.
For $X_{5}$ Sing$(\mathbb{P}_{\mathbf{w}_{5}})\cong \mathbb{P}^{2}$ and $X_{5} \cap \mathrm{Sing}(\mathbb{P}_{\mathbf{w}_{5}})$
is identified with the intersection of $\mathbb{P}^{2}$ with the rank 2 locus of projectivised general skew-symmetric 5 $\times$ 5 matrices,
Pfaff$(5)\cap\mathbb{P}^{2}$, which is empty.
For $X_{7}$ on Sing$(\mathbb{P}_{\mathbf{w}_{7}}) \cong \mathbb{P}^{1}$, the defining matrix $N$ has the following form
$$
N  = \begin{pmatrix}
                0           &  0    & g_{1}     & g_{2}    & g_{3} \\
                0            & 0           & g_{4}     & g_{5}   & g_{6}\\
                -g_{1}  & -g_{4} & 0              & 0   & 0 \\
                -g_{2}  & -g_{5} & 0   & 0           &   0\\
                -g_{3}  & -g_{6} & 0   &  0  & 0 \\
                \end{pmatrix} ,
$$
where $\{g_{i}\}_{i=1}^{6}$ are liner polynomials of $x_{5}$ and $x_{6}$, and has rank greater than $2$ for a generic $\{g_{i}\}_{i=1}^{6}$.
Finally for $X_{10}$ generically $P_{5}|_{\mathrm{Sing}(\mathbb{P}_{\mathbf{w}_{10}})}\ne0$
while $P_{i}|_{\mathrm{Sing}(\mathbb{P}_{\mathbf{w}_{10}})}=0$ for $1\le i \le 4$.
So we henceforth assume that $X_{i}$ avoids the singular locus Sing$(\mathbb{P}_{\mathbf{w}_{i}})$.\\

Let $\mathbb{P}_{\mathbf{w}_{i}}^{sm}=\mathbb{P}_{\mathbf{w}_{i}}\setminus  \mathrm{Sing}(\mathbb{P}_{\mathbf{w}_{i}})$.
$H^{0}(\mathbb{P}_{\mathbf{w}_{i}}^{sm}, \wedge^{2}\mathscr{E}_{i}(t))$ is generated by global sections and the surjection
$$
H^{0}(\mathbb{P}_{\mathbf{w}_{i}}^{sm}, \wedge^{2}\mathscr{E}_{i}(t)) \otimes_{\C}\StrO_{\mathbb{P}_{\mathbf{w}_{i}}^{sm}}
\longrightarrow \wedge^{2}\mathscr{E}_{i}(t)
$$
induces a morphism $f$ of $\mathbb{P}_{\mathbf{w}_{i}}^{sm}$-schemes of full rank everywhere
\[\xymatrix{
\mathbb{P}_{\mathbf{w}_{i}}^{sm} \times H^{0}(\mathbb{P}_{\mathbf{w}_{i}}^{sm}, \wedge^{2}\mathscr{E}_{i}(t)) \ar[rr]^{f} \ar[rd]_{\pi_{1}} & &
 E=Spec (Sym(\wedge^{2}\mathscr{E}_{i}(t)))  \ar[dl]^{\pi_{2}} \\
                         & \mathbb{P}_{\mathbf{w}_{i}}^{sm} &   \\
}\]
sending $(x,N)\mapsto N(x)$.
Let $p:\mathbb{P}_{\mathbf{w}_{i}}^{sm} \times H^{0}(\mathbb{P}_{\mathbf{w}_{i}}^{sm}, \wedge^{2}\mathscr{E}_{i}(t))
\rightarrow H^{0}(\mathbb{P}_{\mathbf{w}_{i}}^{sm}, \wedge^{2}\mathscr{E}_{i}(t))$ be the second projection.
Define $E_{2}$ to be the codimension $3$ variety of $E$ whose fiber over a point $x \in \mathbb{P}_{\mathbf{w}_{i}}^{sm}$ is
$$
O_{2} \subset \mathrm{SkewSym}(5,\C) \cong \pi_{2}^{-1}(x)
$$
Note that $O_{2}$ is independent of the identification $\mathrm{SkewSym}(5,\C) \cong \pi_{2}^{-1}(x)$.
Then $Y=f^{-1}(E_{2})$ is of codimension $3$
and singular along $f^{-1}(\mathrm{Sing}(E_{2}))=\mathbb{P}_{\mathbf{w}_{i}}^{sm} \times \{0\}$.
$p|_{Y\setminus (\mathrm{Sing}(Y))}$ is dominant and the generic smoothness of $p|_{Y\setminus (\mathrm{Sing}(Y))}$ proves that
for a generic choice of $N \in H^{0}(\mathbb{P}_{\mathbf{w}_{i}}^{sm}, \wedge^{2}\mathscr{E}_{i}(t))$
$$
p|_{Y\setminus (\mathrm{Sing}(Y))}^{-1}(N)=\{(x,N)\ | \ \rk(N(x))=2\}
$$
is smooth and of dimension $3$.
The quasi-smoothness of $X_{i}$ then shows
$$
\mathbb{P}_{\mathbf{w}_{i}}^{sm}
\supset X_{i}=\pi_{2}(p|_{Y\setminus (\mathrm{Sing}(Y))}^{-1}(N)) \cong p|_{Y\setminus (\mathrm{Sing}(Y))}^{-1}(N).
$$
\end{proof}

For each $X_{i}$, vanishing of $H^{j}(X_{i}, \mathscr{O}_{X_{i}})=0$ for $j=1,2$ readily follows from the pfaffian resolution.
Therefore $X_{5}, X_{7}$ and $X_{10}$ are smooth Calalbi-Yau threefolds.
In the following, we assign to each $X_{i}$ a polarization by the ample generator of Pic$(X_{i})$, $\mathscr{O}_{X_{i}}(H)$
coming from its ambient space $\mathbb{P}_{\mathbf{w}_{i}}$.

\begin{Lem}
The Hilbert series $H_{X_{i}}(t)$ for the pfaffian Calabi-Yau threefold $X_{i}$ $i=5,7,10$ are given by the following;
\begin{align}
H_{X_{5}}(t)=&\frac{1+3t^{2}+t^{4}}{(1-t)^{4}}, \ \ \ \ \
H_{X_{7}}(t)=\frac{1+t+3t^{2}+t^{3}+t^{4}}{(1-t)^{4}}, \notag \\
& H_{X_{10}}(t)=\frac{1+2t+4t^{2}+2t^{3}+t^{4}}{(1-t)^{4}}. \notag
\end{align}
\end{Lem}
\begin{proof}
As we already have a resolution of the structure sheaf of $X_{i}$, the claim easily follows from the additivity of the Hilbert series and
the formula $$H_{\mathbb{P}_{\mathbf{w}_{i}}}(\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{i}}}(k))(t)=\frac{t^{k}}{\prod_{i=0}^{n}(1-t^{w_{i}}_{i})}.$$
\end{proof}
\begin{Prop}
The degree $\int_{X_{i}}H^{3}$ of the pfaffian Calabi-Yau threefold $X_{i}$ is $i$.
\end{Prop}
\begin{proof}
Since pfaffian variety $X_{i}$ is locally a complete intersection, the triple intersection $\int_{X_{i}}H^{3}$ coincides with $d$,
where $d$  is $3!$ times the leading coefficient of the Hilbert polynomial $P_{X_{i}}(t)$, which is readily available thanks to the previous lemma.
\end{proof}

\begin{Prop}$\int_{X_{i}}c_{2}(X_{i})\cdot H$  for each pfaffian Calabi-Yau 3-fold $X_{i}$ is
\begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|}  \hline
 $X_{i}$  &  $X_{5}$    &  $X_{7}$ &  $X_{10}$\\ \hline
$\int_{X_{i}}c_{2}(X_{i})\cdot H$      &  $38$  &  $46$ &  $52$  \\ \hline
 \end{tabular}
 \end{center}
 \end{Prop}
\begin{proof}
Since we know that $X_{i}$ is a smooth Calabi-Yau threefold, Hirzebruch-Riemann-Roch Theorem gives
$$
 \chi(X_{i},\mathscr{O}_{X_{i}}(H))
 =\frac{1}{6}\deg(X_{i})+\frac{1}{12}\int_{X_{i}}c_{2}(X_{i})\cdot H.
$$
 By the Kodaira vanishing theorem, $H^{j}(X_{i},\mathscr{O}_{X_{i}}(H))=0$ except for $j=0$ and hence we have
$$
 \chi(X_{i},\mathscr{O}_{X_{i}}(H))=\dim H^{0}(X_{i},\mathscr{O}_{X_{i}}(H))=\dim H^{0}(\mathbb{P}_{\mathbf{w}_{i}},\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{i}}}(H))
$$
This determine $\int_{X_{i}}c_{2}(X_{i})\cdot H$.
 \end{proof}

We will also need the resolutions of the powers of pfaffian ideals, which are intensively studied in \cite{boffi}.
Let $R$ be a commutative ring.
We consider a free $R$-module $E$ of rank $2r+1$ and a generic alternating map $N : E^{\vee} \rightarrow E$,
then we have the pfaffian resolution.
$$
0 \longrightarrow
R \stackrel{P^{t}}{\longrightarrow} E^{\vee}
\stackrel{N}{\longrightarrow} E
\stackrel{P}{\longrightarrow} R
\longrightarrow R/I \longrightarrow 0.
$$
\begin{Lem}[\cite{boffi}]\label{Lem:0} The resolution  of $I^{2}$ has the following form.
$$
0 \longrightarrow  L_{(2r-1)}E \cong \Lambda^{2r-1}E
\stackrel{\vartheta_{3}}{\longrightarrow} L_{(2r,1)}E
\stackrel{\vartheta_{2}}{\longrightarrow} L_{(2r+1,1^{2})}E \cong S_{2} E
\stackrel{\vartheta_{1}}{\longrightarrow} I^{2}
\longrightarrow 0,
$$
where $\vartheta_{1}$ is the second symmetric power of $P$ and $\vartheta_{3}$ and $\vartheta_{2}$ are induced by the map
\begin{equation}
\Lambda^{a}E \otimes_{S} S_{b}E \rightarrow \Lambda^{a+1}E \otimes_{S} S_{b+1}E, \ \
u\otimes v \mapsto \sum_{i,j}n_{i,j} e_{i}\wedge u \otimes v e_{j}, \notag
\end{equation}
where $N=\sum_{i,j}n_{i,j} e_{i}\otimes e_{j}$ with respect to some fixed basis for $E$.
\end{Lem}

\begin{Lem}\label{Lem:1}
The resolutions of the ideal sheaf
$\mathscr{I}_{X_{5}}^{2}$, $\mathscr{I}_{X_{7}}^{2}$ and $\mathscr{I}_{X_{10}}^{2}$
are of the following form.
 \begin{align}
 0 \longrightarrow
 \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{5}}}(-12) ^{\oplus 10}
 \stackrel{\vartheta_{3}}{\longrightarrow}  \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{5}}}(-10)^{\oplus 24}
\stackrel{\vartheta_{2}}{\longrightarrow} \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{5}}}(-8)^{\oplus15}
\stackrel{\vartheta_{1}}{\longrightarrow} \mathscr{I}_{X_{5}}^{2} \longrightarrow 0 \notag
 \end{align}
 \begin{align}
 0 \longrightarrow
 \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}&(-12) \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-11)^{\oplus 6}
 \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-10)^{\oplus 3}\notag \\
 \stackrel{\vartheta_{3}}{\longrightarrow} &
 \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-10)^{\oplus 6} \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-9)^{\oplus 12}
 \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-8)^{\oplus 6} \notag \notag \\
&\stackrel{\vartheta_{2}}{\longrightarrow}
\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-8)^{\oplus 6} \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-7)^{\oplus 6}
\oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{7}}}(-6)^{\oplus 3}
\stackrel{\vartheta_{1}}{\longrightarrow} \mathscr{I}_{X_{7}}^{2} \longrightarrow 0 \notag
 \end{align}

 \begin{align}
 0 \longrightarrow
 \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-10)^{\oplus6} & \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-9)^{\oplus 4}
 \stackrel{\vartheta_{3}}{\longrightarrow}
 \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-9)^{\oplus 4} \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-8)^{\oplus 16} \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-7)^{\oplus 4} \notag \\
\stackrel{\vartheta_{2}}{\longrightarrow} &
\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-8) \oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-7)^{\oplus 4}
\oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{10}}}(-6)^{\oplus 10}
 \stackrel{\vartheta_{1}}{\longrightarrow} \mathscr{I}_{X_{10}}^{2} \longrightarrow 0 \notag
 \end{align}
Here each term from left to right is regarded as $5 \times 5$ skew-symmetric, general but top left being zero, and symmetric matrices
and the morphisms are given by $\vartheta_{3}(X)=NX-(NX)_{1,1}I$, $\vartheta_{2}(X)=XN+(XN)^{t}$, $\vartheta_{1}(X)=P^{t}XP$.
\end{Lem}
\begin{proof}
By suitably identify $\Lambda^{3}F$, $L_{(4,1)}F$ and $S_{2} F$ with
respectively $5 \times 5$ skew-symmetric, general but top left being zero, and symmetric matrices,
the morphisms $\vartheta_{i}$ are the above forms.
\end{proof}

\begin{Thm}\label{Hodge}
The Hodge numbers $h^{1,1}$ and $h^{1,2}$ of the pfaffian Calabi-Yau threefold $X_{i}$ are given by the following table.
\begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|}  \hline
 $X_{i}$  &  $X_{5}$    &  $X_{7}$  &  $X_{10}$\\ \hline
$h^{1,1}$      &  $1$  &   $1$ &  $1$  \\ \hline
$h^{1,2}$      &  $51$  &   $61$  &  $59$ \\ \hline
 \end{tabular}
 \end{center}
\end{Thm}
\begin{proof}
In the following, we simply write $X=X_{i}$ and $\mathbb{P}_{\mathbf{w}}=\mathbb{P}_{\mathbf{w}_{i}}$ for some $i=5,7,10$.
Twisting the pfaffian resolution of the structure sheaf,
we know that  $H^{i}(X,\mathscr{O}_{X}(-j)) \cong H^{i+3}(\mathbb{P}_{\mathbf{w}},\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}(-|\mathbf{w}|-j))\ (j=1,2)$,
which do not vanish only for $i=3$.
Restricting the weighted projective analogue of the Euler sequence on $X$, we obtain
$
0 \rightarrow \varOmega_{\mathbb{P}_{\mathbf{w}}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{X}
\rightarrow \bigoplus_{i=0}^{6} \mathscr{O}_{X}(-w_{i})
\rightarrow \mathscr{O}_{X} \rightarrow 0.
$
Since we know that $X$ is a smooth Calabi-Yau threefold, the long exact sequence of this gives,
$H^{i}(X,\varOmega_{\mathbb{P}_{\mathbf{w}}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{X})=0 \ (i=0,2)$,
$H^{1}(X,\varOmega_{\mathbb{P}_{\mathbf{w}}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{X}) \cong \mathbb{C}$ and
\begin{align}
0 \longrightarrow H^{2}(X,\mathscr{O}_{X})
& \longrightarrow H^{3}(X,\varOmega_{\mathbb{P}_{\mathbf{w}}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{X}) \notag \\
\longrightarrow & H^{3}(X,\bigoplus_{i=0}^{6} \mathscr{O}_{X}(-w_{i}))
\longrightarrow H^{3}(X, \mathscr{O}_{X}) \longrightarrow 0.\notag
\end{align}
 Hence, we have
 $h^{3}(X,\varOmega_{\mathbb{P}_{\mathbf{w}}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{X})
 =h^{3}(X,\bigoplus_{i=0}^{6} \mathscr{O}_{X}(-w_{i}))-1$.
 From the resolution of $\mathscr{F}_{\bullet}\rightarrow\mathscr{I}_{X}^{2}$ in Lemma \ref{Lem:1} we obtain
$$
h^{4}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X}^{2})- h^{5}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X}^{2})
=\sum_{i=1}^{3}(-1)^{i+1}h^{6}(\mathbb{P}_{\mathbf{w}},\mathscr{F}_{i})-h^{6}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X}^{2}).
$$
The pfaffian resolution gives $H^{4}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X})
\cong H^{6}(\mathbb{P}_{\mathbf{w}},\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}(-|\mathbf{w}|)) \cong \mathbb{C}$ and
$H^{i}(X,\mathscr{I}_{X})=0$ (otherwise). Since we know that $X$ is indeed smooth, we have the following short exact sequence
$$
0 \longrightarrow \mathscr{I}_{X}^{2} \longrightarrow \mathscr{I}_{X} \longrightarrow \mathscr{N}_{X/\mathbb{P}_{\mathbf{w}}}^{\vee} \longrightarrow 0.
$$
The induced long exact sequence gives
$H^{i}(\mathbb{P}_{\mathbf{w}}, \mathscr{N}_{X/\mathbb{P}_{\mathbf{w}}}^{\vee})=0 \ (0 \le i \le 2)$ and
$H^{5}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X}^{2})=H^{6}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X}^{2})=0$.
Moreover, we also have the short exact sequence
$$
0 \longrightarrow H^{3}(\mathbb{P}_{\mathbf{w}}, \mathscr{N}_{X/\mathbb{P}_{\mathbf{w}}}^{\vee})
\longrightarrow   H^{4}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X}^{2})
\longrightarrow H^{4}(\mathbb{P}_{\mathbf{w}},\mathscr{I}_{X}) \longrightarrow 0.
$$
and dimension counting shows
$$
h^{3}(\mathbb{P}_{\mathbf{w}},\mathscr{N}_{X/\mathbb{P}_{\mathbf{w}}}^{\vee})=h^{3}(X,\mathscr{N}_{X/\mathbb{P}_{\mathbf{w}}}^{\vee})
=\sum_{i=1}^{3}(-1)^{i+1}h^{6}(\mathbb{P}_{\mathbf{w}},\mathscr{F}_{i}).
$$
On the other hand, the conormal exact sequence yields
$$
0 \longrightarrow H^{2}(X,\varOmega_{X})
\longrightarrow H^{3}(X,\mathscr{N}_{X/\mathbb{P}_{\mathbf{w}}}^{\vee})
\longrightarrow H^{3}(X, \varOmega_{\mathbb{P}_{\mathbf{w}}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{X}) \longrightarrow 0
$$
and $H^{1}(X,\varOmega_{X}) \cong \mathbb{C}$.
We establish the formula
\begin{align}
h^{2}(X,\varOmega_{X})&=h^{3}(X,\mathscr{N}_{X/\mathbb{P}_{\mathbf{w}}}^{\vee})
-h^{3}(\varOmega_{\mathbb{P}_{\mathbf{w}}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{X_{10}})\notag \\
&=\sum_{i=1}^{3}(-1)^{i+1}h^{6}(\mathbb{P}_{\mathbf{w}},\mathscr{F}_{i})-h^{3}(X,\bigoplus_{i=0}^{6} \mathscr{O}_{X}(-w_{i})).\notag
\end{align}
Therefore $h^{1,2}$ is determined by the explicit description of $\mathscr{F}_{\bullet}\rightarrow\mathscr{I}_{X}^{2}$ derived in Lemma \ref{Lem:1}.
\end{proof}

The existence of smooth Calabi-Yau threefolds $X_{5}$, $X_{7}$ and $X_{10}$ with the computed topological invariants was previously conjectured
from the view point of Calabi-Yau equations in \cite{van2}.
Regrettably it has not been settled yet whether they are simply connected or not.

 \subsection{Complete Intersection Type}
 In this subsection we study complete intersection of pfaffian varieties and hypersufaces in weighted projective spaces.
 The idea is that we use pfaffian varieties as codimension 3 analogue of hypersurfaces in the ambient space.
 \begin{Def}
Let $\mathscr{E}_{25}=\mathscr{O}^{\oplus 5}_{\mathbb{P}^{9}}$ a locally free sheaf on $\mathbb{P}^{9}$.
Two generic global sections $N_{1}, N_{2} \in H^{0}(\mathbb{P}^{9}, \wedge^{2}\mathscr{E}_{25}(1))$ define alternating morphisms
$N_{1},N_{2} : \mathscr{E}_{25}^{\vee}(-t) \rightarrow \mathscr{E}_{25}$.
Define $X_{25}$ as common degeneracy loci of $N_{1}$ and $N_{2}$.
 \end{Def}

 Since the pfaffian sixfold associated to the data $(\mathscr{E}_{25},N_{i})$ is Gr$(2,5) \subset \mathbb{P}^{9}$,
 $X_{25}$ is a complete intersection of two grassmannians embedded in two different ways $i_{j}:\mathrm{Gr}(2,5)\hookrightarrow \mathbb{P}^{9} \ ( j=1,2)$.
 $$
 X_{25}=i_{1}(\mathrm{Gr}(2,5))\cap i_{2}(\mathrm{Gr}(2,5))
$$

 \begin{Lem}
Let $X$ be a pfaffian variety associated to $(\mathscr{E}_{25},N_{i})$.
Then $\mathscr{I}_{X}^{2}$ has the following resolution.
$$
0 \longrightarrow \mathscr{O}_{\mathbb{P}^{9}}(-6)^{\oplus10}\longrightarrow \mathscr{O}_{\mathbb{P}^{9}}(-5)^{\oplus24}
\longrightarrow \mathscr{O}_{\mathbb{P}^{9}}(-4)^{\oplus15}\longrightarrow \mathscr{I}_{X}^{2}\longrightarrow 0
$$
\end{Lem}
\begin{proof}See Lemma \ref{Lem:0}.
\end{proof}

 \begin{Prop}
 $X_{25}$ is a smooth Calabi-Yau threefold with the following topological invariants.
$$
h^{1,1}=1, \ h^{1,2}=51, \ \int_{X_{25}}H^{3}=25, \ \int_{X_{25}}c_{2}(X_{25})\cdot H = 70
$$
 \end{Prop}
\begin{proof}
The basic strategy is to divide the construction of $X_{25}$ into two steps and repeat the similar argumments in the previous subsection.
The grassmannian description guarantees the smoothness of $X_{25}$.
That $X_{25}$ is a Calabi-Yau threefold and the determination of $\int_{X_{25}}H^{3}$ and $\int_{X_{25}}c_{2}(X_{25})\cdot H$ follow immediately in the same manner as before.
The only non-trivial part is the determination of the Hodge numbers and we sketch the proof. \\

Let $X$ be a pfaffian sixfold associated to $(\mathscr{E}_{25},N_{1})$, which is Gr$(2,5)$.
Then $Y=X_{25}$ is a pfaffian threefold associated to $(\mathscr{O}^{\oplus 5}_{X},N_{2})$.
A straightforward computation with the above lemma shows that $h^{3}(Y,\mathscr{N}_{Y/X}^{\vee})=75$ and there is an exact sequence
\begin{align}
0\longrightarrow H^{2}(X,\varOmega_{X}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}}\mathscr{O}_{Y})
&\longrightarrow H^{3}(X,\mathscr{N}_{Y/X}^{\vee}\otimes\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}\mathscr{O}_{Y})\notag \\
\longrightarrow &H^{3}(X,\varOmega_{\mathbb{P}^{9}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}}\mathscr{O}_{Y})
\longrightarrow H^{3}(X,\varOmega_{X}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}}\mathscr{O}_{Y})\longrightarrow 0. \notag
\end{align}
Combining this with the long exact sequence induced from the conormal sequence, we have
\begin{align}
h^{2}(Y,\varOmega_{Y})
=&h^{3}(Y,\mathscr{N}_{Y/X}^{\vee})+h^{2}(X,\varOmega_{X}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{Y})
-h^{3}(X,\varOmega_{X}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{Y}) \notag\\
=&2 h^{3}(Y,\mathscr{N}_{Y/X}^{\vee})-h^{3}(X,\varOmega_{\mathbb{P}^{9}}\otimes_{\mathscr{O}_{\mathbb{P}_{\mathbf{w}}}} \mathscr{O}_{Y})=51.\notag
\end{align}
\end{proof}

The existence of a smooth Calabi-Yau threefold with the computed topological invariants was already predicted in \cite{van2}.

This Calabi-Yau equation has two maximally unipotent monodromy points of the same type,
and this is clearly explained by the self-duality of  Gr$(2,5)$.

\begin{Ex}
A complete intersection of a pfaffian variety associated with $\mathscr{F}_{10}=\StrO_{\mathbb{P}_{\mathbb{P}_{(1^{7},2)}}}^{\oplus5}$
and a quartic hypersurface in $\mathbb{P}_{(1^{7},2)}$ yields a smooth Calabi-Yau threefold $Y_{10}$ with the following topological invariants.
$$
h^{1,1}=1, \ h^{1,2}=101, \ \int_{Y_{10}}H^{3}=10, \ \int_{Y_{10}}c_{2}(Y_{10})\cdot H = 64
$$
We expect this to coincides with the double covering of Fano threefold in the list of C. Borcea \cite{van2}.
\end{Ex}

\begin{Ex}
A complete intersection of a pfaffian variety associated with $\mathscr{F}_{5}=\StrO_{\mathbb{P}_{(1^{6},2,3)}}^{\oplus5}$
and a sextic hypersurface in $\mathbb{P}_{(1^{6},2,3)}$ yields a singular Calabi-Yau threefold $Y_{5}$.
Interestingly, assuming smoothness of this threefold allow us to compute its topological invariants as follows
$$
h^{1,1}=1, \ h^{1,2}=156, \ \int_{Y_{5}}H^{3}=5, \ \int_{Y_{5}}c_{2}(Y_{5})\cdot H = 62.
$$
The existence of a smooth Calabi-Yau threefold with the above invariants was predicted in \cite{van2}.

\end{Ex}

The author is grateful to Makoto Miura for indicating the existence of $Y_{5}$ and $Y_{10}$.
There are many other choices for locally free sheaves $\mathscr{E}$ of odd rank and weights $ \mathbf{w}$ that yield  Calabi-Yau threefolds,
but there do not seem to exist any other smooth examples that are not previously known.
There is, nevertheless, an interesting example $X_{9}$, which we will deal with in Section 5.

\section{Mirror Symmetry for Degree 13 Pfaffian}
\subsection{Mirror Partner}
Our main aim of this subsection is to explicitly construct a mirror family of $X_{13}$.
As $X_{13}$ is not a complete intersection Calabi-Yau threefold, the Batyrev-Borisov mirror construction is not applicable.
Now we briefly review the tropical mirror construction proposed by J. B\oe hm.
His construction reproduces the conventional Batyrev-Borisov mirror construction for complete intersection Calabi-Yau in toric Fano varieties.
For a thorough treatment of the tropical mirror construction, we refer the reader to the original paper \cite{boehm}. \\

Let us start by recalling the Batyrev-Borisov mirror construction, following the usual notations in \cite{cox}.
Let $M$ and $N=\mathrm{Hom}(M,\mathbb{Z})$ dual free abelian groups of rank $d$,
and $M_{\mathbb{R}}$ and $N_{\mathbb{R}}$ be the scalar extension of $M$ and $N$ respectively.
Suppose that $\mathbb{P}_{\Delta}$ is an $n$-dimensional toric varieity associated to the normal fan $\Sigma_{\Delta}$ of an integral polytope $\Delta \subset M$.
The Cox ring $S=\C[x_{r} | r \in \Sigma_{\Delta}(1)]$ of $\mathbb{P}_{\Delta}$ is graded
by Chow group $A_{n-1}(\mathbb{P}_{\Delta})$ via the presentation sequence
$$
0 \longrightarrow M \stackrel{A}{\longrightarrow} \mathbb{Z}^{\Sigma_{\Delta}(1)} \longrightarrow A_{n-1}(\mathbb{P}_{\Delta}) \longrightarrow 0.
$$
Suppose that $\Delta$ is reflexive and given a nef-partition $\Delta=\Delta_{1}+\dots+\Delta_{k}$ or equivalently $\Sigma_{\Delta}(1)=I_{1}\cup \dots \cup I_{k}$,
then a complete intersection Calabi-Yau variety $X=V(I) \subset \mathbb{P}_{\Delta}$ of dimension $d=n-k$ is the zero locus of a generic section
$(f_{i})_{i=1}^{k} \in H^{0}(\mathbb{P}_{\Delta}, \bigoplus_{i=1}^{k} \mathscr{O}_{\mathbb{P}_{\Delta}}(E_{i}))$,
where $\bigotimes_{i=1}^{k} \mathscr{O}_{\mathbb{P}_{\Delta}}(E_{i}) \cong -K_{\mathbb{P}_{\Delta}}$ corresponds to the nef-partition. \\

Define $\nabla_{i}=\mathrm{Conv.}(\{0\}\cup I_{i})$ and the Minkowski sum $\nabla=\nabla_{1}+\dots+\nabla_{k} \subset N$.
Then the following holds.
$$
\Delta^{*}=\mathrm{Conv.}(\nabla_{1}\cup \dots \cup \nabla_{k}), \ \ \ \nabla^{*}=\mathrm{Conv.}(\Delta_{1}\cup \dots \cup \Delta_{k})
$$
$\nabla=\nabla_{1}+\dots+\nabla_{k}$ is again reflexive and this gives a nef-partition of $\nabla$, called the dual nef-partition.
We define a complete intersection Calabi-Yau variety $\check{X}$ by using $\nabla \subset N$.
Choosing a maximal projective subdivision of the normal fan of $\Delta$ and $\nabla$,
we get families $\mathscr{X}$ and $\check{\mathscr{X}}$ of Calabi-Yau varieties, which are conjectured to form a mirror pair.
The important point is that giving a nef-partition is essentially equivalent to determining a union of toric varieties $X_{0}=V(I_{0})$
to which fibers of $\mathscr{X}$ maximally degenerates.\\

Let $I_{0}$ be a reduced monomial ideal in the Cox ring $S$.
The degree $0$ homomorphisms $Hom(I_{0} , S/I_{0})_{0}$ form a finite dimensional vector space.
The torus $T=\C^{\Sigma_{\Delta}(1)}$ acts on $S$ and thus on $Hom(I_{0} , S/I_{0})_{0}$ as well.
So the vector space has a basis of deformations which are characters of $T$.
Being of degree $0$, any such character $\rho$ corresponds to an element $m_{\rho} \in M \cong Im(A)$.
Given a flat family $\mathscr{X}$ of Calabi-Yau varieties in $\mathbb{P}_{\Delta}$ with special fiber $X_{0}$
such that the corresponding ideal $I_{0} \subset S$ is a reduced monomial ideal.
We represent the complex moduli space of a generic fiber $X$ of $\mathscr{X}$ by a one-parameter family $\mathscr{X}'$;
Take a $T$-invariant basis $\rho_{1}, \dots \rho_{l} \in Hom(I_{0} , S/I_{0})_{0}$ of the tangent space
of the component of Hilbert scheme containing $\mathscr{X}$ at $X_{0}$
and assume that the tangent vector $v=\sum_{i}^{l}a_{i}\rho_{i}$ of $\mathscr{X}'$ at $X_{0}$ satisfies $a_{i}\ne 0\ \forall i$.
The elements $\rho_{1}, \dots, \rho_{l}$ correspond to elements $m_{1},\dots, m_{l} \in M$ of the lattice of monomials of $\mathbb{P}_{\Delta}$.
The construction of the first order deformation of a mirror family $\overline{\mathscr{X}'}$ comes with a natural map
via the interpretation of lattice points as deformations and divisors, cf. the monomial divisor map \cite{cox}.
Take the convex hull $\nabla^{*}$ of $m_{1},\dots, m_{l}$
and define $\mathbb{P}_{\nabla}$ the toric variety associated to the normal fan of the (not necessarily integral) polytope $\nabla$.
Then the toric divisors of $\mathbb{P}_{\nabla}$ and the induced divisors on a prospective mirror inside will correspond to deformations of $X_{0}$ in $\mathscr{X}$.
The Bergman complex of $X_{0}$ defines a special fiber $\check{X}_{0} \subset\mathbb{P}_{\nabla}$ and
the first order deformations $\overline{\check{\mathscr{X}}}$ of contributing to the mirror degeneration $\check{X}_{0}$
are constructed by the lattice points of the support of $\mathrm{Strata}(X_{0})^{*} \subset  \Delta^{*}$.
It is sufficient to know given family up to first order deformation in the case of complete intersection or pfaffian varieties as their deformations are not obstructed.\\

To relate $\overline{\check{\mathscr{X}}}$ to the initial family $\mathscr{X}$.
We need to blow-down the ambient toric variety $\mathbb{P}_{\nabla}$
to an orbifold quotient of a weighted projective space $\mathbb{P}_{\mathbf{w}}/G$,
contracting all divisors which do not correspond to Fermat deformation of $\mathscr{X}$.
This blow-down is in general not unique and we choose appropriate one on case-by-case basis.
The next one-parameter family was conjectured to be the mirror family of the degree 13 pfaffian Calabi-Yau threefold $X_{13}$.
Note that this family has a special monomial fiber $\check{X}_{0}$ on $t=0$ and the first order deformation of $\check{X}_{0}$ is given by the Fermat deformation.

\begin{Def}[J. B\oe hm \cite{boehm}]
Define $\check{\mathscr{X}}=\{\check{X}_{t}\}_{t\in \mathbb{P}^{1}}$ as the one-parameter flat family of the pfaffian Calabi-Yau threefolds
associated to the following special skew-symmetric $5\times 5$ matrix $N_{t}$ parametrized by $t \in \mathbb{P}^{1}$.
$$
N_{t} = \begin{pmatrix}
                0                            & tx_{0}^{2}                          & x_{5}x_{6}      & x_{3}x_{4} & tx_{2}^{2} \\
                -tx_{0}^{2}       & 0                                             & t(x_{3}+x_{4})& x_{2}                    & x_{1} \\
                -x_{5}x_{6}  & -t(x_{3}+x_{4})             & 0                             & tx_{1}                  & x_{0} \\
                -x_{3}x_{4}  & -x_{2}                                     & -tx_{1}                    & 0                            & t(x_{5}+x_{6}) \\
                -tx_{2}^{2}       & -x_{1}                                     & -x_{0}                      & -t(x_{5}+x_{6}) & 0 \\
                \end{pmatrix}
$$
 \end{Def}

 Of course, $\check{\mathscr{X}}$ is nothing but a special one-parameter family of degree 13 pfaffian Calabi-Yau threefolds.
More explicitly, the pfaffian ideal sheaf of this family $\mathscr{I}_{\check{X}}=\langle P_{1}, P_{2}, P_{3}, P_{4}, P_{5} \rangle$ is generated by
\begin{align}
 & P_{1} = x_{0}x_{2} - tx_{1}^{2} - t^{2}(x_{3} + x_{4})(x_{5} + x_{6}) \notag \\
 & P_{2} = x_{0}x_{3}x_{4} - tx_{5}x_{6}(x_{5} + x_{6}) -t^{2}x_{1}x_{2}^{2} \notag \\
 & P_{3} = x_{1}x_{3}x_{4} - tx_{2}^{3} - t^{2}x_{0}^{2}(x_{5} + x_{6}) \notag \\
 & P_{4} = x_{1}x_{5}x_{6} - tx_{0}^{3} - t^{2}x_{2}^{2}(x_{3} + x_{4}) \notag \\
 & P_{5} = x_{2}x_{5}x_{6} - tx_{3}x_{4} (x_{3} + x_{4}) -t^{2}x_{0}^{2}x_{1}.\notag
 \end{align}
Since  $\check{X}_{t}$ is originally contained in the toric variety $\mathbb{P}^{6}/\mathbb{Z}_{13}$,
$\mathbb{Z}_{13}$ acts on $\check{X}_{t}$ as
$$
\zeta_{13} \cdot [x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] =
[x_{0}:\zeta_{13}^{4}x_{1}:\zeta_{13}^{8}x_{2}:\zeta_{13}^{10}x_{3}:\zeta_{13}^{10}x_{4}:\zeta_{13}^{11}x_{5}:\zeta_{13}^{11}x_{6}],
$$
where $\zeta_{13} = e^{\frac{2\pi i}{13}}$.
The fixed points of $\check{X}_{t}$ under this action are 6 points, which do not depend on the value of parameter $t$.
 Among them, there are 4 singular points $p_{i} = \{x_{i} \ne 0, x_{j}=0\ (j \ne i) \} \ (i=3,4,5,6)$,
 and 2 smooth ones $p_{i,i+1} = \{x_{i} + x_{i+1} = 0, \ x_{i} \ne 0 \ x_{j}=0\ (j \ne i,i+1)\} \ (i=3,5)$.

\begin{Prop}
For generic $t \in \mathbb{P}^{1}$, singular points of $\check{X}_{t}$ appear only on $\{p_{i}\}_{i=3}^{6}$ and each has multiplicity 12.
\end{Prop}
\begin{proof}
We work on the singular point $p_{3}$, for example.
In a neighborhood of $p_{3}$, since $P_{1,4,5}\neq 0$, $\check{X}_{t}$ is defined by the complete intersection of $P_{1}$, $P_{4}$ and $P_{5}$.
Then it is easily seen that the germ of this singularity is isomorphic to the following cDV singularity
$$
f = x^{2} + y^{3} + z^{5} + zw^{2}, \ (x,y,z,w) \in \mathbb{C}^{4}.
$$
Here the action of $\mathbb{Z}_{13}$ is given by $\zeta_{13} \cdot (x,y,z,w) = (\zeta_{13}^{11}x, \zeta_{13}^{3}y, \zeta_{13}^{7}z, \zeta_{13} w)$.
The Milnor number of this singularity turns out to be 12.
On the other hand, the Jacobian ideal of $\mathscr{I}_{\check{X}}$ has dimension 0 and degree 48 \footnote{This is done by Macaulay2.}.
Due to the symmetry, other singular points are of multiplicity 12 as well and hence we conclude the singular points are only $\{p_{i}\}_{i=3}^{6}$.
\end{proof}

Now we have a family of Calabi-Yau threefolds $\check{\mathscr{X}}=\{\check{X}_{t}\}_{t\in \mathbb{P}^{1}}$ parametrized by $t \in \mathbb{P}^{1}$.
However, this is not an effective family, as $\check{X}_{t} \cong \check{X}_{\zeta_{7} t}$ for $\zeta_{7} = e^{\frac{2\pi i}{7}} $
via the map
$$
[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] \mapsto [x_{0}:\zeta_{7}^{3}x_{1}:x_{2},\zeta_{7}^{6}x_{3}:\zeta_{7}^{6}x_{4}:\zeta_{7}^{6}x_{5}:\zeta_{7}^{6}x_{6}].
$$

\begin{Rem}
It is proved in \cite{sk} that the above $cD_{6}$ singularity admits no crepant resolution.
However, the quotient $\{f=0\}/\mathbb{Z}_{13}$ may have a crepant resolution.
The definition of this family will be partly justified by calculating the Picard-Fuchs equation of it in the next subsection.
\end{Rem}

\subsection{Period Map and Picard-Fuchs Equation}
Since $X_{13}$ is a smooth Calabi-Yau threefold, it has a nowhere vanishing holomorphic 3-form up to multiplication with a non-zero constant.
Although a pfaffian variety is not defined as a complete intersection and there is no way of explicitly getting one in general,
there is an analogous way of obtaining a global section of $\omega_{X_{13}}\cong\varOmega_{X_{13}}^{3}$.
For the sake of simplicity, we restrict ourselves to the degree 13 case, but the generalization to the case of general pfaffian Calabi-Yau threefolds is straightforward.

\begin{Prop}[E. R\o dland \cite{rod}]
Let $\sigma \in \mathfrak{S}_{5}$.
We have a nowhere vanishing global section of
$\varOmega_{X_{13}}^{3} \cong \mathscr{O}_{\mathbb{P}^{6}}(-7)\otimes \bigwedge^{3} \mathscr{N}_{X_{13}/\mathbb{P}^{6}}$,
up to multiplication with a non-zero constant, explicitly given by
$$
\alpha = C_{\sigma} \mathrm{Res}_{X}\frac{P_{\sigma(1),\sigma(2),\sigma(3)}\Omega_{0}}{P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}},
$$
where
$$
\Omega_{0}=\frac{1}{(2\pi i)^{6}}\sum_{i=0}^{6}(-1)^{i}x_{i}dx_{0}\wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{6}.
$$
This expression is independent of the choice of $\sigma$ so long as the constant $C_{\sigma}$ is chosen appropriately.
\end{Prop}

\begin{proof}
First of all, the invariance of the integrand under scaling of the coordinates can be checked for each $\sigma$
and thus it is well-defined as a rational 6-form on $\mathbb{P}^{6}$.
On the affine open $U_{\sigma(4),\sigma(5)}=\{P_{\sigma(1),\sigma(2),\sigma(3)}\ne0\}$, $\{P_{\sigma(i)}\}_{i=1}^{3}$ form a complete intersection
and thus $\{P_{\sigma(i)}\}_{i=1}^{3}$ can be seen as a part of the local coordinate.
We may assume that $\{P_{\sigma(1)}, P_{\sigma(2)}, P_{\sigma(3)},x_{4}, x_{5}, x_{6}, x_{7}\}$ form the coordinate of $\mathbb{A}^{7}$,
i.e. $\frac{\partial(P_{\sigma(1)},P_{\sigma(2)},P_{\sigma(3)})}{\partial(x_{1},x_{2},x_{3})}\ne0$.
Since we have
$$
dP_{\sigma(1)}\wedge dP_{\sigma(2)}\wedge dP_{\sigma(3)}
=\sum_{i<j<k}\frac{\partial(P_{\sigma(1)},P_{\sigma(2)},P_{\sigma(3)})}{\partial(x_{i},x_{j},x_{k})}dx_{i}\wedge dx_{j}\wedge dx_{k},
$$
the residue theorem provides the following holomorphic 3-form on $X_{13} |_{U_{\sigma(4),\sigma(5)}}$,
$$
\alpha=C_{\sigma} \frac{P_{\sigma(1),\sigma(2),\sigma(3)}}
{(2\pi i)^{3} \frac{\partial(P_{\sigma(1)},P_{\sigma(2)},P_{\sigma(3)})}{\partial(x_{0},x_{1},x_{2})}}
\sum_{i=4}^{7}(-1)^{i}x_{i}dx_{4}\wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{7}.
$$
On the other hand, $P_{\sigma(1),\sigma(2),\sigma(3)}$ vanishes if and only if $\{P_{\sigma(i)}\}_{i=1}^{3}$ do not form a complete intersection.
Therefore $\frac{\partial(P_{\sigma(1)},P_{\sigma(2)},P_{\sigma(3)})}{\partial(x_{1},x_{2},x_{3})}$ divides $P_{\sigma(1),\sigma(2),\sigma(3)}$, and $\alpha$ is globally defined.
Furthermore, the local expression of $\alpha$ shows it is nowhere vanishing on $X_{13}$.
Since $X_{13}$ is Calabi-Yau threefold, $\varOmega_{X_{13}}^{3}$ is trivial and hence the expression of $\alpha$ for each $\nu$ is merely different by a constant.
\end{proof}

Although $\check{X}_{t}$ has some singularities, the nowhere vanishing holomorphic 3-form $\alpha$ is defined
on the non-singular locus $\check{X}_{t} \setminus \mathrm{Sing}(\check{X}_{t})$
and the period map of the family $\check{\mathscr{X}}$ is defined as usual since integration can be performed on $3$-cycle away from the singular locus.
For the sake of convenience, we work with the variety $\check{X}_{t}$ in $\mathbb{P}^{6}$ instead of $\check{X}_{t}/\mathbb{Z}_{13}$.
Note that Picard-Fuchs equations is preserved under taking a quotient of the variety by a finite group under whose action $\alpha$ is invariant.
More precisely, we can perform the integration on $\check{X}_{t}$ and obtain the genuine period by dividing by $13$. \\

At $t=0$, the variety $\check{X}_{t}$ decomposes into $13$ $3$-dimensional planes and
thus the origin is a good candidate for a maximally unipotent monodromy point of this one-parameter family $\check{\mathscr{X}}$.
The fundamental period can be obtained by integrating the holomorphic 3-form on a torus cycle that vanishes the at $t=0$.
Fix a 3-dimensional plane $H$ defined by $H=\{ x_{1}=x_{2}=x_{3}=0 \}$.
On the domain $H\setminus (\{x_{4}=0\}\cup\{x_{5}=0\}\cup\{x_{6}=0\})$,
there is a cycle given by $|\frac{x_{4}}{x_{0}}|=|\frac{x_{5}}{x_{0}}|=|\frac{x_{6}}{x_{0}}|=\epsilon$,
which extends to a $3$-dimensional torus cycle $\gamma \in H_{3}(\check{X}_{t}, \mathbb{C})$ $(|t|\ll 1)$. \\

\begin{Thm}
Let $\Phi_{0}(t) = \int_{\gamma}\alpha $ be a period map of the one-parameter family $\{\check{X}_{t}\}_{t\in \mathbb{P}^{1}}$.
Then $\Phi_{0}(t)$ has the following expansion near $t=0$.
$$
\Phi_{0}(t)=\sum_{n=0}^{\infty} \binom{2n}{n}^{2} \sum_{k=0}^{n} \binom{2n+k}{n} \binom{n}{k}^{2} t^{7n}
$$
It can be observed that $\phi=t^{7} \in \mathbb{P}^{1}$ is the genuine moduli parameter of $\{\check{X}_{t}\}_{t\in \mathbb{P}^{1}}$.
Thus we shall write $\Phi_{0}(\phi)$ and $\{\check{X}_{\phi}\}_{\phi\in \mathbb{P}^{1}}$.
Moreover, the Picard-Fuchs operator $\mathscr{D}$ of this family is
\begin{align}
 \mathscr{D} = & 13^{2}\Theta^{4}-\phi(59397\Theta^{4}+117546\Theta^{3}+86827\Theta^{2}+28054\Theta+3380) \notag \\
&+2^{4}\phi^{2}(6386\Theta^{4}-1774\Theta^{3}-17898\Theta^{2}-11596\Theta-2119) \notag \\
&+2^{8}\phi^{3}(67\Theta^{4}+1248\Theta^{3}+1091\Theta^{2}+312\Theta+26) -2^{12}\phi^{4}(2\Theta+1)^{4}, \notag
 \end{align}
where $\Theta$ is the Euler operator $\phi \frac{\partial}{\partial \phi}$.
\end{Thm}

\begin{proof}
Let us work on an afine open affine subset $U_{0}=\{x_{0}=1\}$ and fix the permutation $\nu = (2,4,5,1,3)$.
For the sake of convenience, we define $a_{i,j}$ to be
$$
(a_{i,j})= \begin{pmatrix}
                \frac{x_{5}^{2}x_{6}}{x_{3}x_{4}}t   &  \frac{x_{5}x_{6}^{2}}{x_{3}x_{4}}t  &  \frac{x_{1}x_{2}^{2}}{x_{3}x_{4}}t^{2} \\
                \frac{1}{x_{1}x_{5}x_{6}}t  &  \frac{x_{2}^{2}x_{3}}{x_{1}x_{5}x_{6}}t^{2}  &  \frac{x_{2}^{2}x_{4}}{x_{1}x_{5}x_{6}}t^{2}\\
                \frac{x_{3}^{2}x_{4}}{x_{2}x_{5}x_{6}}t  &  \frac{x_{3}x_{4}^{2}}{x_{2}x_{5}x_{6}}t  &  \frac{x_{1}}{x_{2}x_{5}x_{6}}t^{2}\\
                \end{pmatrix}.
$$
Then, near the origin, the period integral is described as
$$
\Phi_{0}(t)=\int_{\gamma}\mathrm{Res}_{X}\frac{P_{2,4,5}}{P_{2}P_{4}P_{5}} \Omega_{0}
=\int_{\Gamma}\frac{1}{\prod_{i=1,3,4}(1-\sum_{j=1}^{3}a_{i,j})} \cdot \bigwedge_{k=1}^{6} \frac{dx_{k}}{2 \pi i x_{k}},
$$
where $\Gamma=\{|x_{i}|=\epsilon \ (i=1,\dots 6)\}$.
We then expand the denominator of the integrand as a power series in terms of $a_{i,j}$
and see that the only terms that contribute the period integral is the products $\prod_{}a_{i,j}^{n_{i,j}}$ that is independent of $x_{i}$.
Suppose $\prod_{}a_{i,j}^{n_{i,j}} \ (n_{i,j}\in \mathbb{Z}_{\ge0})$ does not contain any $x_{i}$,
then $\prod_{}a_{i,j}^{n_{i,j}}$ is a product of
\begin{align}
& t_{1}=a_{1,1}a_{1,2}a_{2,3}a_{3,1}a_{3,3}=t^{7}=\phi \notag \\
& t_{2}=a_{1,1}a_{1,2}a_{2,2}a_{3,2}a_{3,3}=t^{7}=\phi \notag\\
& t_{3}=a_{1,1}a_{1,2}a_{1,3}a_{2,1}a_{3,1}a_{3,2}=t^{7}=\phi \notag
\end{align}
and it is easily checked that this expression is unique.
Therefore the period integral $\Phi_{0}(t)$ is essentailly a function of $\phi=t^{7}$, and henceforth we write $\Phi_{0}(\phi)$.
Note that this is compatible with the previous observation that $\check{X}_{t} \cong \check{X}_{\zeta_{7}t}$.
Since
$$
t_{1}^{a}t_{2}^{b}t_{3}^{c}=a_{1,1}^{a+b+c}a_{1,2}^{a+b+c}a_{1,3}^{c}a_{3,1}^{c}a_{3,2}^{b}a_{3,3}^{a}a_{4,1}^{a+c}a_{4,2}^{b+c}a_{4,3}^{a+c}
$$ and
the coefficient of $\prod a_{i,j}^{n_{i,j}}$ appearing as an integrand $\Phi_{0}(\phi)$
is $\prod_{i=1,3,4}\binom{n_{i,1}+n_{i,2}+n_{i,3}}{n_{i,1},n_{i,2},n_{i,3}}$, the period integral $\Phi_{0}(\phi)$ can be summarized as follows.
\begin{align}
\Phi_{0}(\phi) = & \sum_{n=0}^{\infty}\sum_{a+b+c=n} \binom{2a+2b+3c}{a+b+c,a+b+c,c}\binom{a+b+c}{c,b,a}\binom{2a+2b+2c}{a+c,b+c,a+b}\phi^{n} \notag\\
= & \sum_{n=0}^{\infty} \sum_{k=0}^{n} \sum_{l=0}^{n-k} \binom{2n+k}{n} \binom{n+k}{n} \binom{n}{k} \binom{n-k}{l} \binom{2n}{n-l} \binom{n+l}{k+l} \phi^{n} \notag\\
= & \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{2n+k}{n} \binom{n+k}{n} \binom{n}{k} \sum_{l=0}^{n-k} \binom{n-k}{l} \binom{2n}{n+k} \binom{n+k}{n-l} \phi^{n} \notag\\
= & \sum_{n=0}^{\infty} \binom{2n}{n} \sum_{k=0}^{n} \binom{n+k}{n} \binom{2n}{n}^{2} \sum_{l=0}^{n-k} \binom{n-k}{l} \binom{n+k}{n-l} \phi^{n} \notag\\
= & \sum_{n=0}^{\infty} \binom{2n}{n}^{2} \sum_{k=0}^{n} \binom{2n+k}{n} \binom{n}{k}^{2} \phi^{n},\notag
 \end{align}
where we used relations
\begin{gather}
\binom{2n}{n-l}\binom{n+l}{k+l}=\binom{2n}{n+k}\binom{n+k}{n-l}, \ \  \binom{n+k}{n}\binom{2n}{n+k}=\binom{2n}{n}\binom{n}{k}, \notag \\
 \sum_{l=0}^{n-k}\binom{n-k}{l} \binom{n+k}{n-l} =\binom{2n}{k}.\notag
\end{gather}
$\Phi_{0}(\phi)$ coinsides with the power series solution of the Calabi-Yau equation of No.99 classified in \cite{van1},
whose Picard-Fuchs equation is exactly what we are looking for.
\end{proof}

\begin{Cor}Let $\alpha_{1},\alpha_{2}$ be the roots of $256\phi^{2}+349\phi-1$.
Then the Riemann's P-Scheme of $\mathscr{D}$ is given by the following.
 The conifold points are $\alpha_{1},\alpha_{2}$.
$$
 \begin{Bmatrix}
 \begin{tabular}{c|ccccc}
$\phi$  &  $0$  &  $\alpha_{1}$  &  $\alpha_{2}$   &  $13/16$  &  $\infty$ \\ \hline
 $\rho_{1}$               &  $0$  &  $0$  &  $0$  &  $0$  &  $1/2$  \\ \hline
 $\rho_{2}$               &  $0$  &  $1$  &  $1$  &  $1$  &  $1/2$  \\ \hline
 $\rho_{3}$               &  $0$  &  $1$  &  $1$  &  $3$  &  $1/2$  \\ \hline
 $\rho_{4}$              &  $0$  &  $2$  &  $2$  &  $4$  &  $1/2$ \notag
 \end{tabular}
 \end{Bmatrix}
$$
\end{Cor}

As we have expected, $\mathscr{D}$ has a maximally unipotent monodromy point at $\phi=0$
and other singular points of $\mathscr{D}$ are on $\mathbb{R} \cup \infty$.
Observe that $\infty$ is not a maximally unipotent monodromy point in the usual sense but very similar to that.
This will be discussed later in this paper.

 \subsection{Picard-Fuchs Equation around $0$ and Curve Counting}
We now briefly review Gromov-Witten and BPS invariants.
Let $X$ be a Calabi-Yau threefold and let $N_{\beta}^{g}(X)=\int_{[\overline{M}_{g,0}(X,\beta)]^{vir}}1$
be the $0$-point genus $g$ Gromov-Witten invariant of $X$ in the curve class $\beta \in H_{2}(X,\mathbb{Z})$.
Here $[\overline{M}_{g,0}(X,\beta)]^{vir}$ is the virtual fundamental class of the coarse moduli space of stable maps $\overline{M}_{g,0}(X,\beta)$ of
expected complex dimension $(1-g)(\dim X-3) + \int_{\beta}c_{1}(X)=0$.

\begin{Def}
Define BPS (Gopakumar-Vafa) invariants $n_{\beta}^{g}(X)$ by the formula
$$
\sum_{\beta \ne0}\sum_{g\ge0}N_{\beta}^{g}(X)\lambda^{2g-2}q^{\beta}
=\sum_{\beta \ne0}\sum_{g\ge0}n_{\beta}^{g}(X)\sum_{k>0}\frac{1}{k}(2\sin(\frac{kt}{2}))^{2g-2}q^{k\beta}.
$$
LHS is the generating function of Gromov-Witten invariants of $N_{\beta}^{g}(X)$ of $X$ in all genera and all nonzero curve classes.
Matching the coefficients of the two series yields equations determining $n_{\beta}^{g}(X)$ recursively in terms $N_{\beta}^{g}(X)$.
\end{Def}

As the Picard-Fuchs operator $\mathscr{D}$ of $\check{\mathscr{X}}=\{\check{X}_{\phi}\}_{\phi\in \mathbb{P}^{1}}$ has a maximally unipotent monodromy point at $\phi=0$,
we can define the mirror map $q(\phi)$ there and calculate the conjectural genus $g$ BPS invariants $\{n_{d}^{g}\}_{d\in \N}$ of $X_{13}$.
In what follows, we work on the case $g=0,1$ for simplicity.
Since it is a routine work to calculate the mirror map and the Yukawa couplings, we omit the detail of those computations below.
For a complete description, see for example \cite{COGP, cox}. \\

A good integral basis of $H_{3}(\check{X}_{\phi}, \mathbb{Z})$,
which corresponds to the normalized solutions of $\mathscr{D}$ below, determines a canonical coordinate $q$ of complexified K\"{a}hler moduli of $X_{13}$.
At $\phi=0$, we have two normalized solutions of $\mathscr{D}$, $\Phi_{0}(\phi)$ and $\Phi_{1}(\phi)$.
$\Phi_{0}(\phi)$ is of the form of power series such that $\Phi_{0}(0)$=1
and $\Phi_{1}(\phi)$ satisfies
$$
\Phi_{1}(\phi) =(\log(\phi)) \Phi_{0}(\phi)+\Psi(\phi),
$$
where $\Psi(\phi)$ is regular at $\phi=0$ and $\Psi(0)=0$.
These give the mirror map $q(\phi)=\exp(\frac{\Phi_{1}(\phi)}{\Phi_{0}(\phi)})$.
$$
q(\phi)=\phi+86\phi^{2}+12901\phi^{3}+2460318\phi^{4}+536898026\phi^{5}+ \dots
$$
The quantum corrected Yukawa coupling
$$
K_{ttt} (q) = \int_{X}H^{3}+(q\frac{d}{dq})^{3}\sum_{d\ge1}N_{d}(X_{13})q^{d} \in \mathbb{Q}[[q]]
$$
is then determined by the mirror map $q(\phi)$ to be
$$
K_{ttt} (q)= 13+647q+129975q^{2}+25451198q^{3}+5134100919q^{4}+ \dots
$$
We will apply the following BCOV formula for $g=1$ Gromov-Witten potential $F_{1}(\phi)$ to $X_{13}$.
$$
F_{1}(\phi)=\frac{1}{2}\log \left\{\frac{\Phi_{0}(\phi)^{\frac{\chi(X)}{12}-3-h^{1,1}}(q\frac{d\phi}{dq})}
{disc(\phi)^{\frac{1}{6}}\phi^{\frac{\int_{X}c_{2}(X)\cdot H}{12}+1}}\right\}
$$
Here we assumed that the exponent of the discriminant is $1/6$ as usual.
Then the $g=0,1$ BPS invariants are computed to be
\begin{center}
 \begin{tabular}{| c | l | l |} \hline
 $d$  &  $n_{d}^{0}$  &  $n_{d}^{1}$\\ \hline
 1      &  647 &  0  \\
 2      &  16166   &  0  \\
 3      &  942613   &  176\\
 4      &  80218296   &  164696\\
 5      &   8418215008   &  78309518\\
 \hline
 \end{tabular}
 \end{center}

 Since we have the defining equations of $X_{13}$, we can in principle count the number of degree $d$ rational curves on general $X_{13}$
 and check the coincidence of it with $n_{d}^{0}$ as follows.
Write a map $\mathbb{P}^{1} \longrightarrow \mathbb{P}^{6}$ as
$$
 [u:v] \mapsto [\sum_{i=0}^{d}a_{i}u^{i}v^{d-i}:\sum_{i=0}^{d}b_{i}u^{i}v^{d-i}:\dots:\sum_{i=0}^{d}g_{i}u^{i}v^{d-i}].
$$
Then the image of this map is contained in $X_{13}$ if and only if $P_{i}(\mathbf{x}(u,v))=0 \ (i=1,\dots, 5)$ for all $[u:v] \in \mathbb{P}^{1}$.
This containment condition yields dependent equations in $\{a_{0},a_{1},\dots,g_{d-1},g_{d}\}$.
Since what we want to count is not maps from $\mathbb{P}^{1}$ to $X_{13}$ but rational curves in $X_{13}$,
we must kill Aut$(\mathbb{P}^{1})$ by suitably normalizing the map.
As the proportional polynomials also define the same map, the number of the independent parameters turns out to be $7(d+1)-3-1$.
We predict that the ideal generated by the dependent equations has dimension 0 and the degree $n_{d}^{0}$. \\

When $d=1$, we have 19 dependent equations and 10 parameters.
For a generic choice of $N$ and with suitable normalization, we actually compute the degree of the ideal and get the answer $647$,
as the mirror symmetry predicts\footnote{This is done by Macaulay2.}.
Explicit calculation is available upon request.

\subsection{Picard-Fuchs equation around $\infty$}

In \cite{rod}, E. R\o dland constructed a mirror family for the degree 14 pfaffian Calabi-Yau threefolds
$X_{14}=\mathrm{Pfaff}(7) \cap \mathbb{P}^{6}$ by orbifolding the initial threefolds.
His work is notable from two aspects.
Firstly, this is the first example of mirror symmetry for a non-complete intersection Calabi-Yau threefold with $h^{1,1}=1$.
Secondly, the Picard-Fuchs equation of the mirror family $\check{X}_{14}$ has two maxiamlly degenerating points;
$\infty$ corresponds to the initial $X_{14}$ and  $0$ corresponds to $\mathrm{Gr}(2,7)\cap \mathbb{\check{P}}^{13} \subset \mathbb{\check{P}}^{20}$,
which is the projective dual of $\mathrm{Pfaff}(7) \cap \mathbb{P}^{6} \subset \mathbb{P}^{20}$.
In fact, this pair is the first example of a derived equivalence between non-birational Calabi-Yau threefolds \cite{bc}.
K. Hori and D. Tong presented how to describe these Calabi-Yau threefolds
with GLSM using a non-abelian gauge group in two dimensions \cite{ht}.
The link between the pfaffian $X_{14}$ and the grassmannian sections $\mathrm{Gr}(2,7)_{1^{7}}$ was further studied in \cite{hos},
in which a thought-provoking phenomenon in the higher genus Gromov-Witten invariants is discovered.\\

In our case, $\infty$ is apparently an interesting point of $\mathscr{D}$ and it seems worthwhile to analyze it in detail
\footnote{This type of special point also appears when we consider a Calabi-Yau threefold $\mathbb{P}^{7}_{2^{4}}$.
The quantum differential equation of  $\mathbb{P}^{7}_{2^{4}}$ is $\theta^{4}-2^{4}q(2\theta+1)^{4}$.}.
We will first see that it makes sense to call $\infty$ a maximally unipotent monodromy point and calculate the virtual invariants there.
Changing the coordinate from $\phi$ to $1/\phi$ and transforming the gauge by $\sqrt{\phi}$ amount to the change, $\Theta \rightarrow -\Theta \rightarrow -\Theta-1/2,$
in the Euler operator.
Let us also change the variable from $\phi$ to $-\phi$ for later use.
Then the Picard-Fuchs operator becomes
\begin{align}
\mathscr{D}^{'}=&2^{20}\Theta^{4}-2^{8}\phi(1072\Theta^4-17824\Theta^3-10888\Theta^2-1976\Theta-145)\notag \\
&+2^{5}\phi^{2}(51088\Theta^4+116368\Theta^3-45264\Theta^2-14228\Theta-1397) \notag \\
&+13\phi^{3}(73104\Theta^4+1536\Theta^3-488\Theta^2+384\Theta+97) +13^{2}\phi^{4}(2\Theta+1)^4. \notag
\end{align}
 Although $\mathscr{D}^{'}$ has a maximally unipotent monodromy point at $\phi=0$, the integrality of mirror symmetry breaks.
 It is observed that there is a preferable choice of variable $\tilde{\phi}=\phi/2^{16}$, and then everything miraculously goes well.
 The coefficients $1/2^{16}$ is chosen to be the minimum, for which the integrality of the normalized period, the mirror map and the BPS invariants holds, as we will see
 \footnote{S. Hosono pointed out that the change of the sign and the coefficient $1/2^{16}$ can be justified
 by the analytic continuation of the local solutions about $0$ to $\infty$.}.\\ \\
 The Picard Fuchs operator $\tilde{\mathscr{D}}$ with respct to this new variable is
 \begin{align}
 \tilde{\mathscr{D}}=&
 \tilde{\Theta}^{4} -2^{4}\tilde{\phi}(1072\tilde{\Theta}^4-17824\tilde{\Theta}^3-10888\tilde{\Theta}^2-1976\tilde{\Theta}-145) \notag \\
 &+2^{17}\tilde{\phi}^{2}(51088\tilde{\Theta}^4+116368\tilde{\Theta}^3-45264\tilde{\Theta}^2-14228\tilde{\Theta}-1397)  \notag \\
 &+13\cdot2^{28}\tilde{\phi}^{3}(73104\tilde{\Theta}^4+1536\tilde{\Theta}^3-488\tilde{\Theta}^2+384\tilde{\Theta}+97) \notag \\
 &+ 13^{2}2^{44}\tilde{\phi}^{4}(2\tilde{\Theta}+1)^4. \notag
 \end{align}
 This is the Calabi-Yau equation of No.225 in \cite{van2}.
 However, the expected positive Euler number corresponding to this point \cite{van1} excludes a geometric interpretation by a Calabi-Yau threefold with $h^{1,1}=1$.
 Since $\tilde{\mathscr{D}}$ has a maximally unipotent monodromy point at $0$,
 it still makes sense to speak of the virtual BPS invarinats $\{\tilde{n}_{d}^{0}\}_{d\in \N}$ corresponding to $0$,
\begin{center}
 \begin{tabular}{|c | l|} \hline
 $d$  &  $\tilde{n}_{d}^{0}$ \\ \hline
 1      &  70944$a$ \\
 2      &  107300032$a$ \\
 3      &  3707752060576$a$ \\
 4      &  66327758316665792$a$ \\
 5      &   1970671594871618215520$a$ \\
  \hline
 \end{tabular}
 \end{center}
where $a$ is supposed to be the degree of the virtual geometry\footnote{$a$ is expected to be $1$ in \cite{van2}.}.
We hope to understand the meaning of these virtual invariants, which may not come from Calabi-Yau geometry.

\subsection{Conclusion}
It is classically known that the monodromy matrix of the quantum differential equation with respect to an appropreate basis
is expressed in terms of the geometric invariants of the underlying Calabi-Yau threefold with one dimensional moduli.
In what follows, we assume that the origin is a maximally unipotent monodromy point.
Then  $\int_{X}H^{3}$ and $\int_{X}c_{2}(X)\cdot H$ can be read off from the monodromy around the origin and the conifold point.
After it is analytically continued to the origin, the conifold-period $z_{2}(t)$ has the form
$$
z_{2}(t)=\frac{\int_{X}H^{3}}{6}t^{3}+\frac{\int_{X}c_{2}(X)\cdot H}{24}t+\frac{\int_{X}c_{3}(X)}{(2\pi i)^{3}}\zeta(3)
+\sum_{d=1}^{\infty}N_{d}^{0}(X)q^{d},
$$
where $q=e^{2\pi i t}$.
So we obtain $\int_{X}c_{3}(X)$ as well and have consistency check of $\int_{X}c_{2}(X)\cdot H$.
It was numerically verified in \cite{van2} that the invariants computed from the differential equation $\mathscr{D}$
coinsides with the fundamental geometric invariants of the degree 13 pfaffian Calabi-Yau threefold $X$,
$\int_{X}H^{3}, \ \int_{X}c_{2}(X)\cdot H$ and $\int_{X}c_{3}(X)$.
One feature of mirror symmetry is that the interchange of Hodge numbers, but as $\check{X}$ admits no crepant resolution,
our claim that (a suitable resolution of) $\check{X}_{t}/\mathbb{Z}_{13}$ is a mirror partner for $X$
is based on the coincidence the fundamental geometric invariants mentioned above.
An alternative and preferable way of the verification is direct computation of the Gromov-Witten invariants of $X_{13}$, such as \cite{tjo}.
\begin{Conj}
The BPS invariants of the degree 13 pfaffian Calabi-Yau threefold $X_{13}$
coincides with the numbers $\{n_{d}^{g}\}_{d\in \N}$ we calculated above, as mirror symmetry predicts.
\end{Conj}

\section{Mirror Symmetry for Degree 5, 7, 10 Pfaffians}
\subsection{Mirror Partners}
Following the lead of the degree 13 case, we construct mirror families for the smooth Calabi-Yau threefolds we obtained in the section 2, except the degree 25 case.
Since we do not know a systematic way of finding an appropreate blow-down, we omit the finding procedure of mirror partners in this paper.
See Appendix for the conjectural BPS invariants computed via families of Calabi-Yau threefolds in this section. \\

\begin{Def}
Define $\check{\mathscr{X}}_{5}=\{\check{X}_{5,t}\}_{t\in \mathbb{P}^{1}}$ as the one-parameter family of degree 5 pfaffian Calabi-Yau threefolds $\check{X}_{5,t}$
associated to the following special skew-symmetric $5\times 5$ matrix $N_{5,t}$ parametrized by $t \in \mathbb{P}^{1}$.
$$
N_{5,t}= \begin{pmatrix}
                0           & tx_{6}    & x_{4}     & x_{0}x_{1}    & tx_{5} \\
                -tx_{6}  & 0           & t(x_{0}^{2}+x_{1}^{2})    & x_{5}   & x_{2}x_{3}\\
                -x_{4}  & -t(x_{0}^{2}+x_{1}^{2})& 0              & t(x_{2}^{2}+x_{3}^{2})   & x_{6} \\
                -x_{0}x_{1}  & -x_{5} & -t(x_{2}^{2}+x_{3}^{2})   & 0           &  tx_{4}\\
                -tx_{5} & -x_{2}x_{3} & -x_{6}    & -tx_{4}  & 0 \\
                \end{pmatrix}
$$
\end{Def}

This is a conjectural mirror family for $X_{5}$ and degenerates to a union of toric varieties with normal crossings at $t=0$.
In fact we choose $\check{X}_{5,0}$ as a candidate for the fiber over a maximally unipotent monodromy point and deform it so that the first order deformation is Fermat type.
Then the deformation automatically extends to higher orders, so long as it is a pfaffian Calabi-Yau threefold,  and we obtain $\check{\mathscr{X}}_{5}$. \\

$\mathbb{Z}_{10}$ acts on $\check{X}_{5,t}$ as follows
$$
\zeta_{10}\cdot [x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] =
[x_{0}:x_{1}:\zeta_{10}x_{2}:\zeta_{10}x_{3}:\zeta_{10}^{4}x_{4}:\zeta_{10}^{6}x_{5}:\zeta_{10}^{3}x_{6}],
$$
where $\zeta_{10} = e^{\frac{2\pi i}{10}}$.
There are 8 fixed points under this action,
$p_{\pm i}= \{x_{i} \pm x_{i+1} = 0, \ x_{i} \ne 0 \ x_{j}=0\ (j \ne i,i+1)\} \ (i=0,2)$ which are singular with multiplicity 2.
Each is locally isomorphic to the following cDV singularity
$$
f_{5}=x^{3}+y^{2}+zw=0, \ (x,y,z,w) \in \mathbb{C}^{4},
$$
where the action of $\mathbb{Z}_{10}$ is given by
$\zeta_{10}\cdot (x, y, z, w) = (\zeta_{10}^{4}x, \zeta_{10}^{6}y, \zeta_{10}z, \zeta_{10}w)$.
Since dim(Sing$(\check{X}_{5,t}))=0$ and deg(Sing$(\check{X}_{5,t}))=8$, there is no other singular point on $\check{X}_{5,t}$.\\

It is observed that $\check{\mathscr{X}}_{5}=\{\check{X}_{5,t}\}_{t\in \mathbb{P}^{1}}$ is not an effective family,
as $\check{X}_{5,t} \cong \check{X}_{5,\zeta_{10} t}$ for $\zeta_{10} = e^{\frac{2\pi i}{10}} $
via the map
$$
[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] \mapsto
[x_{0}:x_{1}:x_{2}:\zeta_{10}^{5}x_{3}:\zeta_{10}^{4}x_{4}:\zeta_{10}^{7}x_{5}:\zeta_{10}^{9}x_{6}].
$$

\begin{Prop}
The period integral of the mirror family $\{\check{X}_{5,t}\}_{t\in \mathbb{P}^{1}}$ is computed to be
$$
\Phi_{0}(\phi)=\sum_{n=0}^{\infty}\binom{2n}{n}\sum_{k}^{n}\binom{n}{k}\binom{n+k}{n}\binom{2n+2k}{n+k}\binom{2n+k}{2n-k}\phi^{n}
$$
and the Picard-Fuchs operator $\mathscr{D}_{5}$ is given by the following.
\begin{align}
\mathscr{D}_{5}
= & \Theta^{4}-2^{2}\phi(500\Theta^4+976\Theta^3+677\Theta^2+189\Theta+19) \notag \\
& +2^{4}\phi^{2}(3968\Theta^4+3968\Theta^3-1336\Theta^2-1164\Theta-177) \notag \\
& -2^{10}\phi^{3}(500\Theta^4+24\Theta^3-37\Theta^2+6\Theta+3)+2^{12}\phi^{4}(2\Theta+1)^{4}, \notag
\end{align}
where we put $\phi=t^{10}$.
\end{Prop}
\begin{proof}
The computation is almost identical to the degree 13 pfaffian case.
\end{proof}
This Picard-Fuchs equation $\mathscr{D}_{5}$ is the Calabi-Yau equation of No.302 listed in \cite{van1}.
The topological invariants computed from $\mathscr{D}_{5}$ coincide with those of $X_{5}$ as we expected.  \\

\begin{Cor}
Let $\alpha_{1},\alpha_{2}$ be the roots of $ 256\phi^{2}-1968\phi+1$, then Riemann's P-Scheme of $\mathscr{D}_{5}$ is
given by the following.
$$
\begin{Bmatrix}
 \begin{tabular}{c|ccccc}
 $\phi$  &  $0$  &  $\alpha_{1}$  &  $\alpha_{2}$   &  $1/16$  &  $\infty$ \\ \hline
 $\rho_{1}$         &  $0$  &  $0$  &  $0$  &  $0$  &  $1/2$  \\ \hline
 $\rho_{2}$               &  $0$  &  $1$  &  $1$  &  $1$  &  $1/2$  \\ \hline
 $\rho_{3}$               &  $0$  &  $1$  &  $1$  &  $3$  &  $1/2$  \\ \hline
 $\rho_{4}$              &  $0$  &  $2$  &  $2$  &  $4$  &  $1/2$
 \end{tabular}
 \end{Bmatrix}
$$
\end{Cor}

Interestingly enough, the Picard-Fuchs operator $\mathscr{D}_{5}$ has two special points, $0$ and $\infty$.
There is again a preferable new variable $\tilde{\phi}=\phi/2^{8}$ and
the Picard-Fuchs equation around $\infty$ with respect to this new variable is identical to the initial one.
Hence, both of the special points $0$ and $\infty$ seem to correspond to the Calabi-Yau threefold $X_{5}$ in this case.

\begin{Def}
Define $\check{\mathscr{X}}_{7}=\{\check{X}_{7,t}\}_{t\in \mathbb{P}^{1}}$ as the one-parameter family of degree 7 pfaffian Calabi-Yau threefolds $\check{X}_{7,t}$
associated to the following special skew-symmetric $5\times 5$ matrix $N_{7,t}$ parametrized by $t \in \mathbb{P}^{1}$.
$$
N_{7,t}= \begin{pmatrix}
                0           & tx_{2}^{3}    & x_{0}x_{1}     & x_{5}    & tx_{6} \\
                -tx_{2}^{3}  & 0           & tx_{5}    & x_{6}   & x_{3}x_{4}\\
                -x_{0}x_{1}  & -tx_{5}& 0              & t(x_{3}+x_{4})   & x_{2} \\
                -x_{5}  & -x_{6} & -t(x_{3}+x_{4})   & 0           &  t(x_{0}+x_{1})\\
                -tx_{6} & -x_{3}x_{4} & -x_{2}    & -t(x_{0}+x_{1})  & 0 \\
                \end{pmatrix}
$$
\end{Def}

$\mathbb{Z}_{7}$ acts on $\check{X}_{7,t}$ as
$$
\zeta_{7} \cdot [x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] =
[x_{0}:x_{1}:\zeta_{7}^{4}x_{2}:\zeta_{7}x_{3}:\zeta_{7}x_{4}:\zeta_{7}^{3}x_{5}:\zeta_{7}^{6}x_{6}],
$$
where $\zeta_{7} = e^{\frac{2\pi i}{7}}$.
This action has 6 fixed points, which do not depends on the value of parameter $t$.
We have dim(Sing$(\check{X}_{7,t}))=1$ and deg(Sing$(\check{X}_{7,t}))=4$.
Sing$(\check{X}_{7,t})$ passes through 2 of the above fixed points, namely $p_{i,i+1} = \{x_{i} + x_{i+1} = 0, \ x_{i} \ne 0 \ x_{j}=0\ (j \ne i,i+1)\} \ (i=0,3)$.\\

It is observed that $\check{\mathscr{X}}_{7}=\{\check{X}_{7,t}\}_{t\in \mathbb{P}^{1}}$ is not an effective family,
as $\check{X}_{7,t} \cong \check{X}_{7,\zeta_{9} t}$ for $\zeta_{9} = e^{\frac{2\pi i}{9}} $
via the map
$$
[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] \mapsto [x_{0}:x_{1},:\zeta_{9}^{2}x_{2}:x_{3}:x_{4}:\zeta_{9}^{8}x_{5}:\zeta_{9}^{8}x_{6}].
$$

\begin{Prop}
The period integral of the mirror family $\{\check{X}_{7,t}\}_{t\in \mathbb{P}^{1}}$ is computed to be
$$
\Phi_{0}(\phi)=\sum_{n=0}^{\infty}\binom{2n}{n}\sum_{k=0}^{2n}\binom{n+k}{k}\binom{2n}{k}^{2}\phi^{n}
$$
and the Picard-Fuchs operator $\mathscr{D}_{7}$ is given by the following.
\begin{align}
\mathscr{D}_{7}
= & 7^{2}\Theta^{4}-2\cdot3\cdot7\phi(1272\Theta^4+2508\Theta^3+1779\Theta^2+525\Theta+56) \notag \\
& +2^{2}3\phi^{2}(43704\Theta^4+38088\Theta^3-25757\Theta^2-20608\Theta-3360) \notag \\
& -2^{4}3^{3}\phi^{3}(2736\Theta^4-1512\Theta^3-1672\Theta^2-357\Theta-14)  \notag \\
& -2^{6}3^{5}\phi^{4}(2\Theta+1)^2(3\Theta+1)(3\Theta+2), \notag
\end{align}
where we put $\phi=t^{9}$.
\end{Prop}
\begin{proof}
The computation is almost identical to the degree 13 pfaffian case.
\end{proof}

This Picard-Fuchs equation $\mathscr{D}_{7}$ is the Calabi-Yau equation of No.109 listed in \cite{van1}.
The topological invariants computed from $\mathscr{D}_{7}$ coincide with those of $X_{7}$ as we expected. \\

\begin{Cor}Let $\alpha_{1},\alpha_{2}$ be the roots of $ 432\phi^{2}+1080\phi-1$, then Riemann's P-Scheme of $\mathscr{D}_{7}$ is
given by the following.
$$
\begin{Bmatrix}
 \begin{tabular}{c|ccccc}
 $\phi$  &  $0$  &  $\alpha_{1}$  &  $\alpha_{2}$   &  $7/36$  &  $\infty$ \\ \hline
 $\rho_{1}$         &  $0$  &  $0$  &  $0$  &  $0$  &  $1/2$  \\ \hline
 $\rho_{2}$               &  $0$  &  $1$  &  $1$  &  $1$  &  $1/2$  \\ \hline
 $\rho_{3}$               &  $0$  &  $1$  &  $1$  &  $3$  &  $1/3$  \\ \hline
 $\rho_{4}$              &  $0$  &  $2$  &  $2$  &  $4$  &  $2/3$
 \end{tabular}
 \end{Bmatrix}
$$
$0$ is the only maximally unipotent monodromy point of $\mathscr{D}_{7}$.
\end{Cor}

\begin{Def}
Define $\check{\mathscr{X}}_{10}=\{\check{X}_{10,t}\}_{t\in \mathbb{P}^{1}}$ as the one-parameter family of degree 10 pfaffian Calabi-Yau threefolds $\check{X}_{10,t}$
associated to the following special skew-symmetric $5\times 5$ matrix $N_{10,t}$ parametrized by $t \in \mathbb{P}^{1}$.
$$
N_{10,t}= \begin{pmatrix}
                0           & tx_{4}^{2}   & x_{0}x_{1}     & x_{6}    & t(x_{2}+x_{3}) \\
                -tx_{4}^{2}  & 0           & tx_{6}    & x_{2}x_{3}   & x_{5}\\
                -x_{0}x_{1}  & -tx_{6}& 0              & tx_{5}^{2}   & x_{4} \\
                -x_{6}  & -x_{2}x_{3} & -tx_{5}^{2}   & 0           &  t(x_{0}+x_{1})\\
                -t(x_{2}+x_{3}) & -x_{5} & -x_{4}    & -t(x_{0}+x_{1})  & 0 \\
                \end{pmatrix}
$$
\end{Def}

$\mathbb{Z}_{10}$ acts on $\check{X}_{10,t}$ as
$$
\zeta_{10} \cdot [x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] =
[x_{0}:x_{1}:\zeta_{10}^{6}x_{2}:\zeta_{10}^{6}x_{3}:\zeta_{10}^{9}x_{4}:\zeta_{10}^{7}x_{5}:\zeta_{10}^{1}x_{6}],
$$
where $\zeta_{10} = e^{\frac{2\pi i}{10}}$.
There are 6 fixed points, all of which are singular and do not depends on the value of parameter $t$.
Among them, 4 points $p_{i} = \{x_{i} \ne 0, x_{j}=0\ (j \ne i) \} \ (i=0,1,2,3)$ appear with multiplicity 12
and other 2 $p_{i,i+1} = \{x_{i} + x_{i+1} = 0, \ x_{i} \ne 0 \ x_{j}=0\ (j \ne i,i+1)\} \ (i=0,2)$ with multiplicity 7.
The singularities are locally isomorphic to the following cDV singularities, for $(x,y,z,w) \in \mathbb{C}^{4}$.
$$
 f_{10}=x^{4}+y^{2}+z^{2}w+zw^{2}=0,
$$
where the action of $\mathbb{Z}_{10}$ is given by $\zeta_{10}\cdot (x, y, z, w) = (\zeta_{10}^{7}x, \zeta_{10}^{9}y, \zeta_{10}^{6}z, \zeta_{10}^{6}w)$,
and
$$
g_{10}= x^{8}+y^{2}+zw=0,
$$
where $\zeta_{10}\cdot (x, y, z, w) = (\zeta_{10}^{9}x, \zeta_{10}y, \zeta_{10}^{6}z, \zeta_{10}^{6}w)$.

Since dim(Sing$(\check{X}_{10}))=0$ and deg(Sing$(\check{X}_{10}))=62$, there is no other singular point on $\check{X}_{10,t}$.\\

It is observed that $\check{\mathscr{X}}_{10}=\{\check{X}_{10,t}\}_{t\in \mathbb{P}^{1}}$ is not an effective family,
as $\check{X}_{10,t} \cong X_{10,\zeta_{16}^{2} t}$ for $\zeta_{16} = e^{\frac{2\pi i}{16}} $
via the map
$$
[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}] \mapsto [x_{0}:x_{1}:x_{2}:x_{3}:\zeta_{16}^{3}x_{4}:\zeta_{16}^{3}x_{5}:\zeta_{16}^{7}x_{6}].
$$

\begin{Prop}
The period integral of the mirror family $\{\check{X}_{10,t}\}_{t\in \mathbb{P}^{1}}$ is computed to be
$$
\Phi_{0}(\phi)= \sum_{n=0}^{\infty}\binom{2n}{n}\sum_{k=0}^{2n}(-1)^{k+n}\binom{2n}{k}^{4}\phi^{n}
$$
and the Picard-Fuchs operator $\mathscr{D}_{10}$ is given by the following.
\begin{align}
\mathscr{D}_{10}
= & 5^{2}\Theta^4-2^{2}5\phi(688\Theta^4+1352\Theta^3+981\Theta^2+305\Theta+35) \notag \\
& +2^{4}\phi^{2}(5856\Theta^4+7008\Theta^3+96\Theta^2-1260\Theta-265) \notag \\
& -2^{10}\phi^{3}(176\Theta^4+120\Theta^3+69\Theta^2+30\Theta+5)+2^{12}\phi^{4}(2\Theta+1)^4, \notag
\end{align}
where we put $\phi=t^{8}$.
\end{Prop}
\begin{proof}
The computation is almost identical to the degree 13 pfaffian case.
\end{proof}

This Picard-Fuchs equation $\mathscr{D}_{10}$ is the Calabi-Yau equation of No.263 listed in \cite{van1}.
The topological invariants computed from $\mathscr{D}_{10}$ coincide with those of $X_{10}$ as we expected. \\

\begin{Cor}Let $\alpha_{1},\alpha_{2}$ be the roots of $ 256\phi^{2}-544\phi+1$, then Riemann's P-Scheme of $\mathscr{D}_{10}$ is
$$
\begin{Bmatrix}
 \begin{tabular}{c|ccccc}
 $\phi$  &  $0$  &  $\alpha_{1}$  &  $\alpha_{2}$   &  $5/16$  &  $\infty$ \\ \hline
 $\rho_{1}$               &  $0$  &  $0$  &  $0$  &  $0$  &  $1/2$  \\ \hline
 $\rho_{2}$               &  $0$  &  $1$  &  $1$  &  $1$  &  $1/2$  \\ \hline
 $\rho_{3}$               &  $0$  &  $1$  &  $1$  &  $3$  &  $1/2$  \\ \hline
 $\rho_{4}$              &  $0$  &  $2$  &  $2$  &  $4$  &  $1/2$ \\
 \end{tabular}
 \end{Bmatrix}.
$$
\end{Cor}

The Picard-Fuchs operator $\mathscr{D}_{10}$ has two special points, $0$ and $\infty$.
The Picard-Fuchs operator $\tilde{\mathscr{D}}_{10}$ around $\infty$ with respect to the new variable $\tilde{\phi}=1/(\phi2^{12})$ is
\begin{align}
\tilde{\mathscr{D}}_{10}
=&\tilde{\Theta}^4-2^{4}\tilde{\phi}(704\tilde{\Theta}^4+928\tilde{\Theta}^3+612\tilde{\Theta}^2+148\tilde{\Theta}+13)\notag \\
&+2^{12}\tilde{\phi}^{2}(5856\tilde{\Theta}^4+4704\tilde{\Theta}^3-1632\tilde{\Theta}^2-972\tilde{\Theta}-121)\notag \\
&-2^{20}5\tilde{\phi}^{3}(2752\tilde{\Theta}^4+96\tilde{\Theta}^3-60\tilde{\Theta}^2+24\tilde{\Theta}+7)
+2^{28}5^{2}\tilde{\phi}^{4}(2\tilde{\Theta}+1)^{4}. \notag
\end{align}
This is the Calabi-Yau equation of No.271 listed in \cite{van1}.
The existence of a Calabi-Yau threefold with the corresponding topological invariants was predicted in \cite{van2}.
It would be interesting to find the Calabi-Yau threefold and study the relationship with $X_{10}$.

 \subsection{Conclusion}
As a generic member of the mirror family is quite singular just as the degree 13 case
and we don't know if it admits any crepant resolution,
the verification of the mirror phenomena is again based on the coincidence of
the fundamental geometric invariants of the initial Calabi-Yau threefold
with the invariants obtained from monodromy calculation of the Picard-Fuchs equation of the mirror family \cite{van2}.
\begin{Conj}
The BPS invariants of the pfaffian Calabi-Yau threefold $X_{i} \ (i=5,7,10)$
coincides with the numbers $\{n_{d}^{g}\}_{d\in \N}$ in Appendix as mirror symmetry predicts.
\end{Conj}

\section{Another Example}
Although we could not find any other new smooth pfaffian Calabi-Yau threefolds in weighted projective spaces,
there is an interesting example $X_{9}$.
\begin{Def}
Set a weighted projective space $\mathbb{P}_{\mathbf{w}_{9}}=\mathbb{P}_{(1^{6},2)}$.
Define the degree $9$ Calabi-Yau threefold $X_{9}$ as a pfaffian variety associated to the locally free sheaf
$\mathscr{E}_{9}=\mathscr{O}_{\mathbb{P}_{\mathbf{w}_{9}}}(2)\oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{9}}}(1)^{\oplus2}\oplus \mathscr{O}_{\mathbb{P}_{\mathbf{w}_{9}}}^{\oplus2}$.
 \end{Def}

$X_{9}$ turns out to be isomorphic to $\mathbb{P}^{5}_{3^{2}}$.
Therefore $X_{9}$ twofold interpretation.
If we regard $X_{9}$ as a pfaffian Calabi-Yau threefold, we can apply the mirror construction we did to the previous pfaffian cases.

\begin{Def}
Define $\check{\mathscr{X}}_{9}=\{\check{X}_{9,t}\}_{t\in \mathbb{P}^{1}}$ as the one-parameter family of degree 9 pfaffian Calabi-Yau threefolds $\check{X}_{9,t}$
associated to the following special skew-symmetric $5\times 5$ matrix $N_{9,t}$ parametrized by $t \in \mathbb{P}^{1}$.
$$
 N_{9,t}= \begin{pmatrix}
                0           & x_{0}x_{1}x_{2}          & 0     & tx_{6}    & x_{3}x_{4} \\
                -x_{0}x_{1}  & 0                            & x_{6}    & t(x_{3}+x_{4})   & tx_{5}\\
                0                     & -x_{6}                  & 0              & x_{5}   & t(x_{0}+x_{1}+x_{2}) \\
                -tx_{6}           & -t(x_{3}+x_{4})   & -x_{5}   & 0           &  1\\
                x_{3}x_{4}    & -tx_{5}                  & -(x_{0}+x_{1}+x_{2})    & -1  & 0 \\
                \end{pmatrix}
 $$
 \end{Def}
Indeed $X_{9}$ is globally defined by complete intersection of $P_{0},P_{1}$ and $P_{2}$.
The period integral and the Picard-Fuchs operator $\mathscr{D}_{9}$ of the conjectural mirror family $\check{\mathscr{X}}_{9}$
coincides with well-known one
$$
\Phi_{0}(\phi)= \sum_{n=0}^{\infty}\binom{3n}{n}^{2}\binom{2n}{n}^{2}\phi^{n}, \ \ \
\mathscr{D}_{9}=\Theta^{4}-3^{2}\phi(3\Theta+1)^2(3\Theta+2)^2,
$$
where we put $\phi=t^{8}$.
Interesting observation is that this one-parameter family $\check{\mathscr{X}}_{9}$ is not isomorphic to the conventional mirror family for $\mathbb{P}^{5}_{3^{2}}$ defined by
 \begin{gather}
 x_{0}x_{1}x_{2}+t(x_{3}^{3}+x_{4}^{3}+x_{5}^{3}) \notag\\
 x_{3}x_{4}x_{5}+t(x_{0}^{3}+x_{1}^{3}+x_{2}^{3}). \notag
 \end{gather}
This is a family of smooth Calabi-Yau threefolds, while a general member of $\check{\mathscr{X}}_{9}$ has singularities along a curve.
These two families may bridge two mirror constructions. \\

It is classically known that a mirror family for a given family Calabi-Yau threefolds can be constructed by taking special loci of the initial family,
which are not necessarily on the Fermat points emphasized by the initial construction inspired by the conformal field theories.
For more details, see \cite{dgj}, and the reference therein.\\

\section*{Appendix}

\begin{center}
 \begin{tabular}{| c | l | l |}
 \hline
 \multicolumn{3}{| c |}{$X_{5}$}\\
  \hline
 $d$  &  $n_{d}^{0}$ &  $ n^{1}_{d}$  \\ \hline
 1      &   2220                                                                    &   0\\
 2      &   285520                                                               &   460\\
 3      &    95254820                                                         &   873240\\
 4      &   47164553340                                                  &   1498922677\\
 5      &   28906372957040                                          &   2306959237408\\
\hline
 \end{tabular}
  \end{center}

 \begin{center}
 \begin{tabular}{| c | l | l |}
 \hline
 \multicolumn{3}{| c |}{$X_{7}$}\\
 \hline
 $d$  &  $n_{d}^{0}$  &  $n_{d}^{1}$ \\ \hline
 1      &   1434   &   0 \\
 2      &   103026   &   26  \\
 3      &   18676572  &    53076  \\
 4      &   4988009280  &   65171063  \\
 5      &   1646787631350     &   63899034076  \\
 \hline
 \end{tabular}
  \end{center}

  \begin{center}
 \begin{tabular}{| c | l | l |}
 \hline
 \multicolumn{3}{| c |}{$X_{10}$}\\
 \hline
 $d$  &  $n_{d}^{0}$ &  $n_{d}^{1}$\\ \hline
 1      &   888                                                      &  0 \\
 2      &  33084                                                  &  1\\
 3      &  3003816                                             &   2496\\
 4      &  399931068                                        &   2089393\\
 5      &   65736977760                                  &   1210006912\\
  \hline \hline

 $d$  &  $\tilde{n}_{d}^{0}$ &  $\tilde{n}_{d}^{1}$\\ \hline
 1      &  2400$a$        & 40\\
 2      &  1829880$a$         & 138040 \\
 3      &  2956977632$a$     &  687719624\\
 4      &  7117422755016$a$   &  3822563543952\\
 5       &  21319886408804640$a$    &  21893828822263288\\
  \hline
 \end{tabular}
 \end{center}
 \footnote{$a$ is expected to be $2$ in \cite{van2}.}

config: arxiv


TITLE: de morgan's law for greater and less than
QUESTION [1 upvotes]: Suppose that we let $!$ mean not (negation) and let
$a,z,p$ be integer variables. We have the expression.
$$! ( (a>7) \& (z<=p) ) $$
and we can solve it by De Morgan's laws to yield:
$$(a<=7) \text{OR} (z>p)$$
Firstly, is this the correct answer? And secondly, in De Morgan laws, when do we negate the variable it self, like $A$ becomes $A'$, do we negate the variable just in binary variables?

REPLY [2 votes]: De Morgan's laws are functioning as normal in this situation; it is true that
$$\neg((a>7)\wedge(a\leq p))$$
is equivalent to
$$(\neg (a>7))\vee (\neg (a\leq p)).$$
Your question is then how we manipulate things like $\neg(a>7)$ - well this is somewhat beyond just De Morgan's laws, since that treats $(a>7)$ just as some proposition that might be true or false, and doesn't care about what it might mean. However, if we, for instance, looked at a number line (or used the properties of orders), we would find that $\neg(a>7)$ is equivalent to changing the direction of the $>$ to be $\leq$ instead - to we get $a\leq 7$. In general, to negate a statement involving order, just change the direction (and whether it's strict (like $<$ and $>$) or non-strict (like $\leq$ and $\geq$)). We don't manipulate the integer variables, since negation of an integer is a very different operation from negation of a boolean.


TITLE: Optimization Algorithm
QUESTION [1 upvotes]: I have encountered an optimization problem that I would like advice on what types of methods I could use to solve it.
Let's suppose a town is represented by a Polygon. The town contains n points (addresses) according to the two parameters latitude and longitude. The goal is to determine a "hotspot" by selecting the best array of addresses.
Every address broadcasts a 300m signal, so the algorithm must find the ideal combination of points while minimizing its number.
For example:
Town and points
So every point has a range of 300m, thus for this dataset we have:
points and range
Here, I've made a classification so useless zones are deleted. If I proceed to a simple calculation we have:
Final representation
With:

Covered Area: 0.0005055475481536275
Total area: 0.0007334658607563916
Relative coverage: 68.92584579632353
Computation delay: 96.36 sec

So, the project is to consider all the addresss from a town, then finding an optimisation algorithm that selects the best hotspots that cover 100% of the area.
I hope that my explanation was clear enough. What are methods that could assist me in this? Some algorithms that could be adapted to my problem? I have attended optimization and operations research courses, but for this problem, can't find a shoe fitting on my foot.
Thank you for you help!

REPLY [0 votes]: Suppose that the total area of the greyed land is denoted by the constant LA.
Let the collection of addresses be denoted as a list of elements $n_{ij}$ where $i$ corresponds to the latitude of each address, and $j$ corresponds to the longitude of each address. We’ll also need to represent $n_{ij}$ as a binary value on whether the model selects the address $n$ or not.
In addition, let $m_{a_{ij}}$ denote the euclidean points in terms of (lat, long) of each address.
The area of each signal from an address is $300^2\pi=90000\pi$. Of course the address near the boarder of the map will need to be accounted for and have their total areas reduced by the amount they overflowed over the edge. We can then denote the total area of the signal of each address by an list of elements $A_{ij}$ where $i$ denotes the latitude of the address and $j$ denotes the longitude of an address.
Let $N$ be a constant integer value for the allowable total amount of hotspots to be turned on at once. This will be allowed to change per iteration, as the first goal of this model is to cover the most area, and the next goal of the model is to use the minimal amount of addresses to cover that same amount of area.
Then we can proceed to find the minimum amount of addresses by using a rough template minimization model of the sort:
$$\text{Minimize } z = LA -\sum^{|n_i|}_{i=0}\sum^{|n_j|}_{j=0}A_{ij}\cdot n_{ij}$$
Subject to:
$$\sum^{|n_i|}_{i=0}\sum^{|n_j|}_{j=0}n_{ij} \le N $$
$$\sqrt{m_{a_{ij}} ^2+ m_{a_{ij}} ^2}\ge300\text{ }\forall ij\text{ where } i\ne j$$
$$\text{where } n_{ij}\in\{0,1\}\text{ }\forall ij$$
However, this is definitely not the best way to do this, but it would be a start in the right direction towards what you want to do. A reason why this model is not the best is due to the second constraint. There, it’s saying that the distance per address must be greater or equal to the radius of a signal to prevent overlapping. This will cause issues if the address is a boarder address, or that the optimal solution will probably have overlap. However, this would produce an "acceptable" solution. In other words, this will give you a platform to start.


TITLE: Sketch the graph of the following peridoic function and find its Fourier series
QUESTION [0 upvotes]: I think I've the graph wrong, just struggling with the Fourier series.

REPLY [0 votes]: Your function is even on $[-\pi,\pi]$. So you end up with a Fourier cosine series only. For $n=1,2,3,\cdots$,
$$
         a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(n x)dx \\= \frac{1}{\pi}\int_{-\pi}^{0}(\pi +x)\cos(n x)dx + \frac{1}{\pi}\int_{0}^{\pi}(\pi-x)\cos(n x)dx
$$
For $n=0$,
$$
        a_0 = \frac{1}{2\pi}\int_{-\pi}^{0}(\pi +x)dx+\frac{1}{2\pi}\int_{0}^{\pi}(\pi-x)dx.
$$
The Fourier series is
$$
      f \sim \sum_{n=0}^{\infty}a_n \cos(n\pi x).
$$


TITLE: How to find Triangular Numbers
QUESTION [2 upvotes]: I read that Gauss's Eureka Theorem says that any positive integer can be represented by at most 3 triangular numbers.  So say I have some positive integer X, how do I find which 3 triangular numbers added together make that Number X?
edit: if there isn't a way to do this without just guessing and combining all combinations of 3 triangular numbers less than X, then is there a program that can do it for me?
edit 2: It looks like for some integers that there is more than 1 way to describe them using 3 triangular numbers. The way to calculate the possible number of ways is using this formula:
http://www.3quarksdaily.com/3quarksdaily/2015/03/last-month-at-3qd-we-discovered-that-while-it-was-invented-to-solve-problems-in-counting-and-probability-pascals-triangle-c.html
I still don't know if there is a formula to find which triangular numbers though :(

REPLY [2 votes]: ( partial answer/hint) A few things, will help along the way, to make guessing easier:

odd+odd=even
even+even= even
odd+even= odd
squares are the sum of two consecutive triangular numbers
At most: means it may not always be necessary, for that upper bound to occur in any given example.

If we allow 0 to be counted as a triangular number,we can just add 0 until we hit 3.
1 is odd. From 1,2 and 3 above, we can deduce that, at least an odd number of the triangular numbers in the sum, will be odd (which happens to mean it's of odd index if you count 0 as the 0th triangular number).
2 is even. From 1,2,and 3 above, we can deduce that, There are an even number of odd index triangular numbers, in the sum ( includes the possibility of 0 of them).
we can work out the highest index a triangular number could be using $\lfloor sqrt(2X)\rfloor$. So, in conclusion, you can find which three allow you to solve for X, by using restrictions at the very least.


TITLE: Variation of Heron's problem (maximize absolute difference of distances)
QUESTION [1 upvotes]: Given two points $A(4,1)$ and $B(0,4)$, find a point $P(x,y)$ on the line $ l_1: 3x-y-1=0$, so that $abs(AP-BP)$ is the maximal possible [the absolute value of the difference between the distances from A to P and from B to P].
Such a point $P (2,5)$ belongs to the line $l_2: y=-2x+9$  where also the points $A(4,1)$ and $B'(3,3)$ are placed. The latter is the reflection of $B(0,4)$ below the line $l_1$.
Apparently $BP$ and $B'P$ are equal and offset each other in the difference of interest. So, it is $B'A$ that represents the absolute difference of distances $AP$ and $BP$.
How to prove that with $P$ that belongs to the line $AB'$ the absolute difference of distances $AP$ and $BP$ is maximized?

REPLY [1 votes]: Note that for all $P \in  {\ell }_{1} , \  \left|A P-B P\right| = \left|A P-{B'} P\right|$ because $B$ and $B'$ are orthogonally symmetrical with respect to $\ell_1$. By the reverse triangle inequality, the absolute difference of the norms of two vectors is smaller than the norm of the difference of these two vectors:
\begin{equation}\left|\left\|\overrightarrow{u}\right\|-\left\|\overrightarrow{v}\right\|\right|  \leqslant  \left\|\overrightarrow{u}-\overrightarrow{v}\right\|\end{equation}
Apply this with the vectors $\overrightarrow{u} = \overrightarrow{P {B'}}$ and
$\overrightarrow{v} = \overrightarrow{P A}$, one gets that
\begin{equation}\left|P {B'}-P A\right|  \leqslant  A {B'}\end{equation}
For the point $P$ drawn on the figure, the difference is exactly
$A {B'}$, hence it is  the maximum.