# Computation and Data in the Classification of Fano Varieties

Gavin Brown  Mathematics Institute
Zeeman Building
University of Warwick
Coventry
CV4 7AL
UK<EMAIL_ADDRESS>,  Tom Coates  Department of Mathematics
Imperial College London
180 Queen’s Gate
London
SW7 2AZ
UK<EMAIL_ADDRESS>,  Alessio Corti  Department of Mathematics
Imperial College London
180 Queen’s Gate
London
SW7 2AZ
UK<EMAIL_ADDRESS>,  Tom Ducat  Department of Mathematical Sciences
Durham University
Stockton Road
Durham
DH1 3LE
UK<EMAIL_ADDRESS>,  Liana Heuberger  Département mathématiques
Faculté des Sciences
2 Boulevard Lavoisier
Angers
France<EMAIL_ADDRESS>and  Alexander Kasprzyk  School of
Mathematical Sciences
University of Nottingham
Nottingham
NG7 2RD
UK<EMAIL_ADDRESS>
###### Abstract.

Fano varieties are ‘atomic pieces’ of algebraic varieties, the shapes that can
be defined by polynomial equations. We describe the role of computation and
database methods in the construction and classification of Fano varieties,
with an emphasis on three-dimensional Fano varieties with mild singularities
called $\mathbb{Q}$-Fano threefolds. The classification of $\mathbb{Q}$-Fano
threefolds has been open for several decades, but there has been significant
recent progress. These advances combine computational algebraic geometry and
large-scale data analysis with new ideas that originated in theoretical
physics.

###### Key words and phrases:

Fano variety, mirror symmetry, classification.

_Funding._ TC is supported by ERC Consolidator Grant 682603 and EPSRC
Programme Grant EP/N03189X/1. AC is supported by EPSRC Programme Grant
EP/N03189X/1. LH is supported by Projet Étoiles Montantes GeBi de la Région
Pays de la Loire. AK is supported by EPSRC Fellowship EP/N022513/1.

##  1\. Introduction

Fano varieties are an important class of geometric shapes. They are basic
building blocks in algebraic geometry, both via the Minimal Model Program [66,
67, 11, 45] and as a source of explicit constructions. For example, one can
construct three-dimensional Calabi–Yau manifolds (smooth varieties with zero
curvature) as anticanonical sections in smooth four-dimensional Fano
varieties; these are of particular importance in constructing models of
spacetime in Type II string theory [6, 72]. The classification of Fano
varieties is a fundamental problem in geometry yet, despite almost a century
of study, this classification is still far from understood. In this paper we
discuss the classification of Fano varieties over $\mathbb{C}$. In particular,
the $n$-dimensional Fano varieties that we consider can also be thought of, at
least when they are smooth, as $2n$-dimensional real manifolds with positive
curvature.

There are finitely many deformation families of Fano manifolds (that is,
smooth Fano varieties) in each dimension [61]. There is exactly $1$ one-
dimensional Fano manifold: the Riemann sphere $\mathbb{P}^{1}$. In two
dimensions there are $10$ deformation families, the del Pezzo surfaces:
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, $\mathbb{P}^{2}$, and $\mathbb{P}^{2}$
blown-up in at most eight points [37]. In dimension three the classification
is due to Fano [41], Iskovskikh [49, 50, 51], and Mori–Mukai [68, 69]: there
are $105$ deformation families111The three-dimensional classification can be
explored online via the excellent Fanography website [9].. Very little is
known about the classification of Fano manifolds in dimensions four or more.

When we allow mild singularities even less is known. There are finitely many
deformation families in each dimension with singularities of bounded
complexity [62, 10], but there are few explicit classification results. In
dimension three it is natural to focus on the _$\mathbb{Q}$ -Fano threefolds_:
Fano varieties with $\mathbb{Q}$-factorial terminal singularities – this is
the natural setting for the Minimal Model Program. Even here there is as yet
no classification, although several hundred deformation families have been
described; see e.g. [20, 80, 81, 48, 82, 55, 19, 75, 17, 40, 34]. What is
known, however, are the possible Hilbert series of $\mathbb{Q}$-Fano
threefolds222Strictly speaking, we mean here the possible Hilbert series of
Mori–Fano threefolds only – see [16, §5] for a precise discussion – but as
every known $\mathbb{Q}$-Fano threefold has a Hilbert series that is in this
list, we will blur the distinction in the rest of the paper. In any case, this
is a large and important part of the landscape of $\mathbb{Q}$-Fano
threefolds. [5, 16]. The Hilbert series of a Fano variety $X$ is the
generating series for the dimensions of the graded pieces of the anticanonical
ring

$R(X,{-K}_{X})=\bigoplus_{n=0}^{\infty}H^{0}(X,{-n}K_{X}).$ (1.1)

Hilbert series remain invariant under $\mathbb{Q}$-Gorenstein deformation of
$X$. The possible Hilbert series are recorded in the Graded Ring Database [13,
15]. One can think of this database as a first sketch, giving numerical
invariants only, of the possible ‘geography’ of $\mathbb{Q}$-Fano threefolds.

A Fano variety $X$ can be recovered as $\operatorname{Proj}$ of its
anticanonical ring (1.1). Choosing homogeneous generators for the
anticanonical ring defines an embedding of $X$ into a weighted projective
space $w\mathbb{P}$, and we can read off various measures of the complexity of
this embedding (or of the complexity of $R(X,{-K}_{X})$ as a ring) directly
from the Hilbert series. In particular, the Hilbert series can be used to
estimate a lower bound for the  _codimension_

$\operatorname{codim}{X}\coloneqq\dim{w\mathbb{P}}-\dim{X},$

that is, the number of independent equations required to define $X$ (locally)
inside $w\mathbb{P}$. The Hilbert series also determines the  _genus_ of the
embedding

$g(X)\coloneqq\dim H^{0}(X,{-K}_{X})-2.$

If $X$ is smooth then $g(X)$ is the genus of the curve defined by intersecting
$X$ with two generic hyperplanes in $w\mathbb{P}$; for the general case,
including an interpretation of the case where $g(X)<0$, see [5, §4]. The genus
increases with the complexity of the embedding. Figure 1 shows the range of
pairs $(g(X),\operatorname{codim}{X})$ for entries in the database of possible
Hilbert series of $\mathbb{Q}$-Fano threefolds. See [16] for precise details:
in particular, codimension here means estimated codimension as there.

Figure 1. A sketch of the landscape of $\mathbb{Q}$-Fano threefolds, from the
Graded Ring Database [15].

The database [13, 15] is produced using combinatorial methods that do not
guarantee existence of a $\mathbb{Q}$-Fano threefold that realises any given
entry: there can be zero, one, or many deformation families of
$\mathbb{Q}$-Fano threefolds with a given candidate Hilbert series. In the
remainder of the paper, we discuss techniques for constructing and classifying
$\mathbb{Q}$-Fano threefolds, guided by the geography in Figure 1.

##  2\. Classifications in low codimension

The first examples of Fano varieties are projective spaces $\mathbb{P}^{n}$.
After these, projective hypersurfaces of low degree give the archetypal Fano
varieties: a cubic surface, a cubic or quartic threefold, and so on.
Broadening our view to orbifolds with cyclic quotient singularities greatly
extends the range of hypersurfaces that we can construct. For example, a
weighted homogeneous polynomial $f$ of degree $d$ in $n+1$ variables with
positive integer weights $a_{0},\dots,a_{n}$ defines a hypersurface $(f=0)$ in
weighted projective space $\mathbb{P}(a_{0},\dots,a_{n})$. If
$d<a_{0}+\cdots+a_{n}$ then this hypersurface is Fano. See [65, 38, 48].

In three dimensions, there are exactly $95$ families of $\mathbb{Q}$-Fano
threefolds that arise as quasismooth weighted projective hypersurfaces [78,
53, 14]. Here _quasismoothness_ ensures that all singularities of the
hypersurface arise from singularities of the ambient weighted projective space
[48]. The $95$ hypersurfaces are all of the form
$X_{d}\subset\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$, where
$d=a_{1}+a_{2}+a_{3}+a_{4}$, and correspond to the $95$ families of K3
surfaces $X_{d}\subset\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. Quasismooth
members $X$ of each family are birationally rigid [32, 22], meaning if $X$ is
birational to another $\mathbb{Q}$-Fano threefold or Mori fiber space $Y$ then
$X$ is in fact biregular isomorphic to $Y$. One may consider this to be a
strong form of non-rationality [52] since, in particular, they do not admit
birational maps to $\mathbb{P}^{3}$. The $95$ families have been studied from
many points of view, and serve as a testing ground for conjectures and
computations. For example, Kim–Okada–Won compute their log canonical
thresholds, and deduce $K$-stability results for $50$ of the $95$ families
[60]. In four dimensions, $\mathbb{Q}$-Fano quasismooth weighted projective
hypersurfaces have also been classified [7]: there are $11\,618$ families.

Hypersurfaces $(f=0)$ have codimension one, because they are defined
(globally) by a single equation. One can construct Fano varieties with higher
codimension by considering complete intersections

$(f_{1}=\cdots=f_{c}=0)$

in weighted projective spaces, where the $f_{i}$ are homogeneous polynomials
of sufficiently low degree. Iano-Fletcher describes $85$ families of
$\mathbb{Q}$-Fano threefolds that arise as codimension-two complete
intersections in weighted projective space [48]; this list is in fact complete
[23]. In higher codimension most Fano varieties are not weighted projective
complete intersections. In codimension three, for example, sections of
weighted Grassmannians $w\\!\operatorname{Gr}(2,5)$, defined by five equations
of Pfaffian type, are typical [33]. Codimension four embeddings are also
tractable [4, 19] but beyond codimension four, a comprehensive understanding
is restricted to special classes such as smooth Fano threefolds.

##  3\. Birational geometry and steps into higher codimension

Whilst the low codimension studies discussed in §2 are very successful and
comprehensive, they break down in higher codimension. There are several
approaches to this problem. One is to find simple examples whose anticanonical
embedding happens to be in high codimension: for example, $\mathbb{P}^{3}$
itself has $-K_{X}=4H$, for a linear hyperplane $H\subset\mathbb{P}^{3}$, so
that the anticanonical embedding is the 4th Veronese embedding
$\mathbb{P}^{3}\hookrightarrow\mathbb{P}^{34}$ in codimension $31$. Another
approach is to find varieties that naturally live in high codimension whose
sections are $\mathbb{Q}$-Fano threefolds [77, 18]. A more sophisticated
approach, the _Takeuchi program_ [83], analyses possible birational links
between Fano threefolds with the aim of constructing complicated examples from
simpler ones; cf. [3, 31]. This was exploited by Takagi [82] to classify all
$\mathbb{Q}$-Fano threefolds $X$ having only $\mathbb{Z}/2$ hyperquotient
singularities, Picard rank $\rho_{X}=1$, and $\dim H^{0}(X,-K_{X})\geq 4$; and
by Prokhorov [74] to analyse $\mathbb{Q}$-Fano threefolds with $-K_{X}$
divisible.

We discuss a fourth approach here, applying _unprojection_ [71, 79, 19]. This
is close in spirit to the Takeuchi program, and is rooted in Fano’s original
approach to the classification problem via projection. The idea is as follows.
If $X_{4}\subset\mathbb{P}^{4}$ is a terminal Fano quartic threefold that
contains a linear plane $\mathbb{P}^{2}\cong D\subset X_{4}$, then there is a
birational map $X_{4}\dashrightarrow Y=Y_{3,3}\subset\mathbb{P}(1,1,1,1,1,2)$.
In coordinates $x_{0},\dots x_{4}$ on $\mathbb{P}^{4}$, this is

$D=(x_{0}=x_{1}=0)\subset X_{4}=(x_{0}A-x_{1}B=0)\subset\mathbb{P}^{4}$

where $A$ and $B$ are cubic forms, and setting $y=A/x_{1}=B/x_{0}$ defines $Y$
and a morphism

$\begin{array}[]{r@{\ }c@{\
}l}X_{4}&\dashrightarrow&Y=(yx_{1}=A,yx_{0}=B)\subset\mathbb{P}(1,1,1,1,1,2)\\\
(x_{0},\dots,x_{4})&\mapsto&(x_{0},\dots,x_{4},y(x_{0},\dots,x_{4})).\end{array}$
(3.1)

Here $x_{0},\ldots,x_{4}$ and $y$ are the coordinates on
$\mathbb{P}(1,1,1,1,1,2)$ of weights $1,\ldots,1$ and $2$. The map (3.1) is
called the _unprojection_ of $D\subset X_{4}$, and conversely, elimination of
the variable $y$ from the equations of $Y$ recovers $X_{4}$ as a projection of
$Y$.

Multiplying up the denominator in (3.1) of either expression for $y$ shows
that general points of $D$ are mapped to
$[0\\!:\\!0\\!:\\!0\\!:\\!0\\!:\\!0\\!:\\!1]\in Y$, and so $D$ is indeed
contracted. However, the map (3.1) is not a morphism. The geometry of the map
depends on the singularities that $X_{4}$ acquires at the points $(A=B=0)$ on
$D$. Typically these are ordinary double points, and the map consists of the
$D$-ample small resolution of each such point followed by the contraction of
$D$, but this requires detailed analysis in any given case. If one can repeat
this with two planes $D_{1}$, $D_{2}\subset X_{4}$, unprojecting them one
after the other, the result would be a Fano threefold embedded in codimension
three. It is easy to imagine continuing this process, but imposing successive
planes becomes an increasingly difficult geometrical and combinatorial
problem.

This approach works well into codimension four, with several hundred families
constructed by unprojecting weighted planes $D=\mathbb{P}(1,a,b)$ [19]. In the
example above, Brown–Kerber–Reid construct three different deformation
families of $\mathbb{Q}$-Fano threefolds, using different configurations of
planes. (Takagi’s classification describes two of these families; the third
has higher Picard rank.) Unprojection techniques have been extended into
codimension $\geq 5$ by Brown and Ducat [12]. The real power of this technique
becomes apparent when combined with ideas from mirror symmetry: see §9 below.

##  4\. Polytopes, fans, and toric varieties

Toric varieties are highly symmetrical algebraic varieties. An $n$-dimensional
normal projective variety $X$ is a  _toric variety_ if it contains a dense
algebraic torus $(\mathbb{C}^{\times})^{n}$ and the action of that torus on
itself extends to an action of the torus on all of $X$: see [35, 42]. For
example, projective spaces and weighted projective spaces are toric varieties.
In our context, toric varieties will play two different roles. Three-
dimensional toric varieties will occur as degenerations of $\mathbb{Q}$-Fano
threefolds; in general these toric varieties will be highly singular. Higher-
dimensional toric varieties will occur as ambient spaces when we construct
$\mathbb{Q}$-Fano threefolds as hypersurfaces or complete intersections; we
have already seen this in §2.

The study of toric varieties is underpinned by a rich ‘dictionary’ that
translates geometry into combinatorics. Let $M\cong\mathbb{Z}^{n}$ denote the
lattice of characters of $T=(\mathbb{C}^{\times})^{n}$, with dual lattice
$N\coloneqq\operatorname{Hom}(M,\mathbb{Z})$. A toric variety $X$ with dense
torus $T$ has a combinatorial description in terms of a fan $\Sigma$ in
$N_{\mathbb{Q}}$, and many geometric properties of $X$ can be rephrased in
terms of combinatorial statements about $\Sigma$. For example, let
$\rho_{1},\ldots,\rho_{k}$ be the one-dimensional cones of $\Sigma$. Each
$\rho_{i}$ is generated by a (unique) primitive lattice element $v_{i}\in N$,
and $X$ is Fano if and only if $v_{1},\ldots,v_{k}$ correspond to the vertices
of a convex lattice polytope in $N_{\mathbb{Q}}$ containing the origin in its
interior. Such a polytope is called a  _Fano polytope_. Conversely, given a
Fano polytope $P$ the fan $\Sigma$ (and hence the toric Fano variety $X$) can
be recovered by taking the collection of cones spanned by the faces of $P$; we
call this the  _spanning fan_ of $P$. Under this dictionary there is a
correspondence between the dual polytope

$P^{*}\coloneqq\\{u\in M_{\mathbb{Q}}\mid u(v)\geq-1\\}$

of $P$ and the anticanonical divisor ${-K}_{X}$ of $X$. Since $P^{*}$ always
contains the origin, it follows that $\dim{H^{0}(X,{-K}_{X})}=|P^{*}\cap M|$
is strictly positive.

##  5\. Mirror symmetry

A new approach to the classification of Fano varieties centres around a set of
ideas from string theory called  _mirror symmetry_ [44, 21, 47]. Mirror
symmetry gives a correspondence between Fano manifolds and certain Laurent
polynomials [24]. From this perspective, the key invariant of a Fano variety
$X$ is its  _regularised quantum period_

$\widehat{G}_{X}(t)=\sum_{d=0}^{\infty}c_{d}t^{d}.$ (5.1)

This is a power series with coefficients $c_{0}=1$, $c_{1}=0$, and
$c_{d}=r_{d}d!$, where $r_{d}$ is a certain Gromov–Witten invariant of $X$.
Intuitively speaking, $r_{d}$ is the number of rational curves in $X$ of
degree $d\geq 2$ that pass through a fixed generic point and have a certain
constraint on their complex structure. Under mirror symmetry, an
$n$-dimensional Fano manifold $X$ corresponds to a Laurent polynomial
$f\in\mathbb{C}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ if the regularised
quantum period $\widehat{G}_{X}$ agrees with the  _classical period_

$\pi_{f}(t)=\sum_{d=0}^{\infty}c^{\prime}_{d}t^{d}.$ (5.2)

Here $c_{d}^{\prime}$ is the coefficient of the unit monomial in $f^{d}$.

It is expected that if a Fano manifold $X$ corresponds under mirror symmetry
to a Laurent polynomial $f$, then there is a $\mathbb{Q}$-Gorenstein
degeneration of $X$ to a (singular) toric variety $X_{f}$. This toric variety
is defined by the spanning fan of the Newton polytope of $f$. One can hope,
therefore, to recover $X$ directly from its mirror partner $f$: by regarding
$f$ as encoding a log structure on $X_{f}$ that determines the smoothing from
$X_{f}$ to $X$, and then constructing $X$ as this smoothing.

Gromov–Witten invariants are deformation invariant, so the regularised quantum
period $\widehat{G}_{X}$ is a deformation invariant of $X$. The classical
period $\pi_{f}$, on the other hand, is invariant under an equivalence
relation called  _mutation_ [2]. It is conjectured that mirror symmetry gives
a one-to-one correspondence between deformation-equivalence classes of
$\mathbb{Q}$-Fano varieties333That is, Fano varieties with
$\mathbb{Q}$-factorial terminal locally toric singularities. that admit a
toric degeneration, and mutation-equivalence classes of  _rigid maximally
mutable Laurent polynomials_ (rigid MMLPs) [28]. For the classification of
two- and three-dimensional Fano manifolds from this perspective, see [25, 1,
39, 59, 73].

If $X$ is a Fano projective hypersurface then the Givental/Hori–Vafa mirror
construction [43, 47] gives rise to a Laurent polynomial $f$ that corresponds
to $X$ under mirror symmetry. For example, if $X$ is a cubic surface in
$\mathbb{P}^{3}$ then the Givental/Hori–Vafa mirror is given by

$\displaystyle W=x_{0}+x_{1}+x_{2}+x_{3}$ where $\displaystyle
x_{0}x_{1}x_{2}x_{3}=1$ $\displaystyle x_{1}+x_{2}+x_{3}=1.$

Eliminating the second equation by setting

$\displaystyle x_{1}=\frac{1}{1+x+y}$ $\displaystyle x_{2}=\frac{x}{1+x+y}$
$\displaystyle x_{3}=\frac{y}{1+x+y}$

and solving for $x_{0}$ yields the Laurent polynomial

$f=W-7=\frac{(1+x+y)^{3}}{xy}-6.$

This is the Przyjalkowski method. It applies in greater generality to give
Laurent polynomials that correspond under mirror symmetry to complete
intersections in weighted projective spaces [76] and toric varieties [29, 39]
that satisfy mild combinatorial conditions.

The Givental/Hori–Vafa construction above can be reversed to recover a
presentation of $X$ as a toric complete intersection from the corresponding
Laurent polynomial $f$: this is  _Laurent inversion_ [30]. This process cannot
always work – for example, only $89$ of the $105$ smooth Fano threefolds are
known to be toric complete intersections – but nonetheless Laurent inversion
gives a powerful tool for constructing $\mathbb{Q}$-Fano varieties. In
outline: one starts with a Fano polytope $P$, searches for rigid MMLPs $f$
with Newton polytope $P$, and then applies Laurent inversion to construct a
corresponding Fano variety. Heuberger has used this approach to construct many
new $\mathbb{Q}$-Fano threefolds [46].

##  6\. Fano polytopes

The classification of Fano polytopes is an attractive combinatorial problem:
see [58] for an overview. Smooth Fano polytopes – corresponding to toric Fano
manifolds – can be classified efficiently using an algorithm of Øbro [70]. The
number grows slowly with dimension: just $18$ out of the $105$ three-
dimensional Fano manifolds are toric, and even by dimension eight there are
only $749\,892$ smooth Fano polytopes.  _Reflexive_ Fano polytopes –
corresponding to Gorenstein toric Fano varieties – are of particular
importance in mirror symmetry: topologically mirror-symmetric pairs of
Calabi–Yau varieties can be constructed as hypersurfaces in Gorenstein toric
Fano varieties, and this duality is reflected in the combinatorics of the
polytopes $P$ and $P^{*}$ [8]. There are $4319$ reflexive polytopes in
dimension three, and $473\,800\,776$ in dimension four [63, 64]. Toric Fano
varieties with only terminal singularities correspond to Fano polytopes
$P\subset N_{\mathbb{Q}}$ such that the only lattice points in $P$ are the
origin and the vertices. These have been classified in dimension three [55]:
there are $634$ cases. Toric Fano varieties with canonical singularities
correspond to Fano polytopes $P\subset N_{\mathbb{Q}}$ such that the only
interior lattice point of $P$ is the origin. These  _canonical Fano polytopes_
have also been classified in dimension three [56, 57]: there are $674\,688$
cases.

These polytope classifications provide a natural place to search for rigid
MMLPs. For example, an analysis of the $4319$ reflexive three-dimensional
polytopes was used in [2, 25] to recover the $98$ deformation families of
three-dimensional Fano manifolds with very ample anticanonical bundle.
Extending this idea, by constructing rigid MMLPs supported on Fano polytopes
more generally, we can begin to populate the geography of $\mathbb{Q}$-Fano
threefolds. This is developed further in [26, 46], but let us first consider
how much of the geography in Figure 1 we might hope to realise this way.

Genus and codimension are deformation invariants. We expect, as discussed
above, that a $\mathbb{Q}$-Fano threefold $X$ corresponding to a rigid MMLP
$f$ admits a degeneration to the toric variety $X_{f}$ defined by the spanning
fan of the Newton polytope $P\subset N_{\mathbb{Q}}$ of $f$. In this case we
can read off the genus and codimension of $X_{f}$ (and hence of $X$) from the
combinatorics of the dual polytope $P^{*}$ of $P$. Namely, the genus is given
by $|P^{*}\cap M|-2$, and the codimension is computed from the Hilbert basis
of the four-dimensional cone $\mathbb{R}_{\geq 0}\cdot(P^{*}\times\\{1\\})$.
Figure 2(a) shows the pairs $(g(Y),\operatorname{codim}{Y})$ realised by toric
Fano varieties $Y$ with at worst canonical singularities444See [16, Figure 6]
for a more detailed analysis.; here $Y$ is playing the role of $X_{f}$. It
should be emphasised that not every canonical Fano polytope will support a
rigid MMLP: initial experiments indicate ${\sim}15\%$ of the canonical Fano
polytopes support at least one rigid MMLP, resulting in approximately $8300$
mutation-equivalence classes of rigid MMLPs. The codimensions of the
hypothetical $\mathbb{Q}$-Fano threefolds discovered (or rather, suggested) in
this way are summarised in Figure 3: note that the majority of these have
quite high codimension.

(a) Canonical polytopes ($1$ interior point).

(b) Fano polytopes with $2$ interior points.

Figure 2. The portions of the landscape of $\mathbb{Q}$-Fano threefolds that
might be realised by three-dimensional Fano polytopes with one and two
interior lattice points, overlaid on Figure 1. Figure 3. Known mutation-
equivalence classes of rigid MMLPs supported on $3$-dimensional canonical Fano
polytopes, by codimension.

We expect that increasing the number of interior lattice points allowed in the
Newton polytope (equivalently, allowing log-terminal singularities in the
corresponding toric Fano varieties) will reveal new mutation-equivalence
classes of rigid MMLPs. For example, Fano polytopes with two interior lattice
points are classified in [7]. The pairs $(g(Y),\operatorname{codim}{Y})$ for
the resulting toric Fano varieties $Y$ are shown in Figure 2(b). Once again we
expect only a fraction of these Fano polytopes to support a rigid MMLP, and
hence degenerate to a $\mathbb{Q}$-Fano threefold. In particular, we expect
that none of the two-point polytopes that fall outside the gray shaded region
in Figure 2(b) – or equivalently the region shown in Figure 1 – support a
rigid MMLP. Figure 2 suggests that increasing the number of interior lattice
points might allow us to find $\mathbb{Q}$-Fano threefolds in both higher
codimension and lower genus. The latter point is intuitively plausible: as the
Newton polytope $P$ grows larger, so the dual polytope $P^{*}$ grows smaller
and so we expect $|P^{*}\cap M|$ and hence the corresponding genus to
decrease.

##  7\. The $95$ hypersurfaces

We return to the $95$ $\mathbb{Q}$-Fano threefold hypersurfaces $X_{d}$ in
weighted projective space discussed in §2. As in §5, we can apply the
Givental/Hori–Vafa construction to obtain a Laurent polynomial mirror for
$X_{d}$. Write

$\displaystyle W=x_{0}+x_{1}+x_{2}+x_{3}+x_{4}$ where $\displaystyle
x_{0}x_{1}^{a_{1}}x_{2}^{a_{2}}x_{3}^{a_{3}}x_{4}^{a_{4}}=1$ $\displaystyle
x_{1}+x_{2}+x_{3}+x_{4}=1.$

Eliminating the second equation by setting

$\displaystyle x_{1}=\frac{1}{1+x+y+z}$ $\displaystyle
x_{2}=\frac{x}{1+x+y+z}$ $\displaystyle x_{3}=\frac{y}{1+x+y+z}$
$\displaystyle x_{4}=\frac{z}{1+x+y+z}$

and solving for $x_{0}$ yields the Laurent polynomial

$\displaystyle f=W-1-c=\frac{(1+x+y+z)^{d}}{x^{a_{2}}y^{a_{3}}z^{a_{4}}}-c$
where $\displaystyle c=\frac{d!}{a_{1}!a_{2}!a_{3}!a_{4}!}$

Here the constant $c$ is chosen to ensure that the coefficient
$c^{\prime}_{1}$ of the classical period (5.2) is zero. The corresponding
Newton polytope $P$ is a simplex; the number of interior lattice points is
given by the tetrahedral number $(d-3)(d-2)(d-1)/6$.

The hypersurface with the largest value for $d$ is
$X_{66}\subset\mathbb{P}(1,5,6,22,33)$. It has genus $-1$, and gives a simplex
with $43\,680$ interior lattice points. Experiments mutating the Laurent
polynomial show that we cannot expect to substantially decrease the number of
interior lattice points. This agrees with our intuition in §6 that
constructing low-genus $\mathbb{Q}$-Fano threefolds requires Fano polytopes
with large numbers of interior points. Classifying Fano polytopes with tens of
interior lattice points, let alone tens of thousands, is well beyond the reach
of current methods, so it is implausible that we could find examples such as
the 95 hypersurfaces using mirror symmetry methods alone.

##  8\. An example in codimension two

As discussed, we expect that if a $\mathbb{Q}$-Fano threefold $X$ corresponds
under mirror symmetry to a Laurent polynomial $f$, then $X$ admits a
degeneration to the toric variety $X_{f}$ defined by the spanning fan of the
Newton polytope $P\subset N_{\mathbb{Q}}$ of $f$. If this is the case, then
only those $\mathbb{Q}$-Fano threefolds that admit toric degenerations can be
detected using mirror symmetry methods. For example, consider the codimension
two $\mathbb{Q}$-Fano threefold $X_{12,14}\subset\mathbb{P}(2,3,4,5,6,7)$,
which is part of the Iano-Fletcher classification [48]. This has
$\dim{H^{0}(X,{-K}_{X})}=0$, and so can never deform to a toric variety555It
would be interesting to know if in general the vanishing of
$H^{0}(X,{-K}_{X})$ is the only obstruction to $X$ admitting a toric
degeneration.. So mirror symmetry methods alone will not be enough to complete
the classification.

##  9\. A two-pronged approach

Classical construction methods (§2, §7–8) and the mirror symmetry approach
(§5–6) complement each other well, with the classical methods most effective
for $\mathbb{Q}$-Fano threefolds of low codimension and mirror symmetry giving
remarkable coverage in high codimension. Mirror symmetry also gives new
insight at one of the most difficult points of the analysis: determining
whether a given set of possible numerical invariants is realised by a
$\mathbb{Q}$-Fano variety. Conjecturally this problem is equivalent to
determining the existence (or non-existence) of rigid MMLPs with appropriate
Newton polytopes: a challenging but tractable combinatorial problem, which is
amenable to machine computation. Traditionally there are no systematic methods
available here.

This suggests a new approach – a hybrid of classical and mirror symmetry
methods – to populate and analyse the landscape of $\mathbb{Q}$-Fano
threefolds. First, use graded ring methods [13, 5, 16] to determine an
explicit list of possible Hilbert series. Then, for each candidate Hilbert
series $H(t)$ on that list, classify rigid MMLPs $f$ supported on Fano
polytopes $P$ with dual Ehrhart series $\operatorname{Ehr}(P^{*},t)$ equal to
$H(t)$, where $P$ lies in an appropriate combinatorial class. This predicts
the number of deformation families, as well as specific
$\mathbb{Q}$-Gorenstein toric degenerations $X_{f}$ of these families.
Finally, use deformation, Laurent inversion, or unprojection methods to
construct each family. Meaningful analysis beyond particular cases will
require cluster-scale computation and database methods, because of the numbers
of cases involved (thousands of deformation families, hundreds of thousands of
polytopes), but this is well within what is now possible. For a substantial
step in this direction, see [26, 46, 27].

We close by discussing a classical example from this new point of view.
Consider a sequence of $\mathbb{Q}$-Fano threefolds
$Y^{(i)}\subset\mathbb{P}(1,1,1,1,1,2,\ldots,2)$ in codimension $i+1$ that
arise by successively unprojecting $i$ planes inside a quartic hypersurface
$X_{4}\subset\mathbb{P}^{3}$. The case when $i=1$ is discussed in §3. When
$i\geq 3$, there may be more than one configuration of planes: see [12]. By
determining mutation-equivalence classes of rigid MMLPs, we predict numbers of
distinct deformation families as shown in Table 1. There remains much detailed
analysis still to do, but at a coarse level this prediction matches an
observation of Brown–Ducat [12]: that a quartic threefold can contain at most
seven planes meeting only pairwise in points. Indeed a quartic $X$ that
contains a plane $\Pi$ is birational to a cubic del Pezzo fibration, obtained
as the projection away from $\Pi$. Any other plane in $X$ that meets $\Pi$
only in points corresponds to a section of $(-1)$-curves in this fibration,
and thus there are at most six such planes, corresponding to a maximal set of
six disjoint lines in the general fibre of this cubic del Pezzo fibration; cf.
[54]. Moreover a set of seven such planes can be explicitly realised inside
the Burkhardt quartic $X_{4}^{\textrm{Bur}}\subset\mathbb{P}^{4}$, which is
the unique quartic having the maximal number of $45$ ordinary double points
[36] and contains a configuration of $40$ planes. Therefore, if we constrain
the unprojection plane configurations to meet as simply as possible, there is
no configuration of eight planes to unproject into codimension nine, in
accordance with the conjecture.

$\begin{array}[]{rccccccccc}\hline\cr\hline\cr i&0&1&2&3&4&5&6&7&8\\\
\hline\cr\text{GRDB ID}&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=20521&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=20522&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=20543&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=20652&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=20938&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=21436&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=22093&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=22801&\href
http://grdb.co.uk/search/fano3?id_cmp=in&id=23386\\\
\text{Codimension}&1&2&3&4&5&6&7&8&9\\\ \text{Number of
families}&1&1&1&3&7&11&11&4&0\\\ \hline\cr\hline\cr\end{array}$ Table 1. The
predicted number of deformation families of $\mathbb{Q}$-Fano threefolds
$Y^{(i)}$ from §9.

## References

  * [1] Mohammad Akhtar, Tom Coates, Alessio Corti, Liana Heuberger, Alexander M. Kasprzyk, Alessandro Oneto, Andrea Petracci, Thomas Prince, and Ketil Tveiten.  Mirror symmetry and the classification of orbifold del Pezzo surfaces.  Proc. Amer. Math. Soc., 144(2):513–527, 2016.
  * [2] Mohammad Akhtar, Tom Coates, Sergey Galkin, and Alexander M. Kasprzyk.  Minkowski polynomials and mutations.  SIGMA Symmetry Integrability Geom. Methods Appl., 8:Paper 094, 17, 2012.
  * [3] Valery Alexeev.  General elephants of ${\bf Q}$-Fano 3-folds.  Compositio Math., 91(1):91–116, 1994.
  * [4] Selma Altınok.  Graded Rings Corresponding to Polarised K3 Surfaces and $\mathbb{Q}$-Fano 3-folds.  PhD thesis, University of Warwick, 1998.
  * [5] Selma Altınok, Gavin Brown, and Miles Reid.  Fano 3-folds, $K3$ surfaces and graded rings.  In Topology and geometry: commemorating SISTAG, volume 314 of Contemp. Math., pages 25–53. Amer. Math. Soc., Providence, RI, 2002.
  * [6] Paul S. Aspinwall, Brian R. Greene, and David R. Morrison.  Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory.  Nuclear Phys. B, 416(2):414–480, 1994.
  * [7] Gabriele Balletti and Alexander M. Kasprzyk.  Three-dimensional lattice polytopes with two interior lattice points.  arXiv:1612.08918 [math.CO], 2016\.
  * [8] Victor V. Batyrev.  Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties.  J. Algebraic Geom., 3(3):493–535, 1994.
  * [9] Pieter Belmans.  Fanography: a tool to visually study the geography of Fano 3-folds.  https://www.fanography.info.
  * [10] Caucher Birkar.  Singularities of linear systems and boundedness of Fano varieties.  Ann. of Math. (2), 193(2):347–405, 2021.
  * [11] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan.  Existence of minimal models for varieties of log general type.  J. Amer. Math. Soc., 23(2):405–468, 2010.
  * [12] Gavin Brown and Tom Ducat.  Double unprojection of Fano 3-folds.  In preparation.
  * [13] Gavin Brown and Alexander M. Kasprzyk.  The Graded Ring Database.  http://www.grdb.co.uk.
  * [14] Gavin Brown and Alexander M. Kasprzyk.  Four-dimensional projective orbifold hypersurfaces.  Exp. Math., 25(2):176–193, 2016.
  * [15] Gavin Brown and Alexander M. Kasprzyk.  The Fano 3-fold database.  _Zenodo_ https://doi.org/10.5281/zenodo.5820338, 2022.
  * [16] Gavin Brown and Alexander M. Kasprzyk.  Kawamata boundedness for Fano threefolds and the Graded Ring Database.  arXiv:2201.07178 [math.AG], 2022\.
  * [17] Gavin Brown, Alexander M. Kasprzyk, and Muhammad Imran Qureshi.  Fano 3-folds in $\mathbb{P}^{2}\times\mathbb{P}^{2}$ format, Tom and Jerry.  Eur. J. Math., 4(1):51–72, 2018.
  * [18] Gavin Brown, Alexander M. Kasprzyk, and Lei Zhu.  Gorenstein formats, canonical and Calabi–Yau threefolds.  Experimental Mathematics, pages 1–19, 2019.
  * [19] Gavin Brown, Michael Kerber, and Miles Reid.  Fano 3-folds in codimension 4, Tom and Jerry. Part I.  Compos. Math., 148(4):1171–1194, 2012.
  * [20] F. Campana and H. Flenner.  Projective threefolds containing a smooth rational surface with ample normal bundle.  J. Reine Angew. Math., 440:77–98, 1993.
  * [21] Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes.  A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory.  Nuclear Phys. B, 359(1):21–74, 1991.
  * [22] Ivan Cheltsov and Jihun Park.  Birationally rigid Fano threefold hypersurfaces.  Mem. Amer. Math. Soc., 246(1167):v+117, 2017.
  * [23] Jheng-Jie Chen, Jungkai A. Chen, and Meng Chen.  On quasismooth weighted complete intersections.  J. Algebraic Geom., 20(2):239–262, 2011.
  * [24] Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, and Alexander M. Kasprzyk.  Mirror symmetry and Fano manifolds.  In European Congress of Mathematics, pages 285–300. Eur. Math. Soc., Zürich, 2013.
  * [25] Tom Coates, Alessio Corti, Sergey Galkin, and Alexander M. Kasprzyk.  Quantum periods for 3-dimensional Fano manifolds.  Geom. Topol., 20(1):103–256, 2016.
  * [26] Tom Coates, Liana Heuberger, and Alexander M. Kasprzyk.  Mirror symmetry, Laurent inversion and the classification of $\mathbb{Q}$-Fano threefolds.  arXiv:2210.07328 [math.AG], 2022\.
  * [27] Tom Coates, Alexander M. Kasprzyk, and Giuseppe Pitton.  Computing maximally mutable Laurent polynomials.  In preparation.
  * [28] Tom Coates, Alexander M. Kasprzyk, Giuseppe Pitton, and Ketil Tveiten.  Maximally mutable Laurent polynomials.  Proc. of the Royal Society A., 477(2254):Paper No. 20210584, 21, 2021.
  * [29] Tom Coates, Alexander M. Kasprzyk, and Thomas Prince.  Four-dimensional Fano toric complete intersections.  Proc. of the Royal Society A., 471(2175):Paper No. 20140704, 14, 2015.
  * [30] Tom Coates, Alexander M. Kasprzyk, and Thomas Prince.  Laurent inversion.  Pure Appl. Math. Q., 15(4):1135–1179, 2019.
  * [31] Alessio Corti.  Factoring birational maps of threefolds after Sarkisov.  J. Algebraic Geom., 4(2):223–254, 1995.
  * [32] Alessio Corti, Aleksandr Pukhlikov, and Miles Reid.  Fano $3$-fold hypersurfaces.  In Explicit birational geometry of 3-folds, volume 281 of London Math. Soc. Lecture Note Ser., pages 175–258. Cambridge Univ. Press, Cambridge, 2000.
  * [33] Alessio Corti and Miles Reid.  Weighted Grassmannians.  In Algebraic geometry, pages 141–163. de Gruyter, Berlin, 2002\.
  * [34] Stephen Coughlan and Tom Ducat.  Constructing Fano 3-folds from cluster varieties of rank 2.  Compos. Math., 156(9):1873–1914, 2020.
  * [35] V. I. Danilov.  The geometry of toric varieties.  Uspekhi Mat. Nauk, 33(2(200)):85–134, 247, 1978.
  * [36] A. J. de Jong, N. I. Shepherd-Barron, and A. Van de Ven.  On the Burkhardt quartic.  Math. Ann., 286(1-3):309–328, 1990.
  * [37] Pasquale Del Pezzo.  Sulle superficie dell’$n^{\text{mo}}$ ordine immerse nello spazio ad $n$ dimensioni.  Rend. del Circolo Mat. di Palermo, 1:241–255, 1887.
  * [38] Igor Dolgachev.  Weighted projective varieties.  In Group actions and vector fields (Vancouver, B.C., 1981), volume 956 of Lecture Notes in Math., pages 34–71. Springer, Berlin, 1982.
  * [39] Charles F. Doran and Andrew Harder.  Toric degenerations and Laurent polynomials related to Givental’s Landau-Ginzburg models.  Canad. J. Math., 68(4):784–815, 2016.
  * [40] Tom Ducat.  Constructing $\mathbb{Q}$-Fano 3-folds à la Prokhorov & Reid.  Bull. Lond. Math. Soc., 50(3):420–434, 2018.
  * [41] Gino Fano.  Nuove ricerche sulle varietà algebriche a tre dimensioni a curve-sezioni canoniche.  Pont. Acad. Sci. Comment., 11:635–720, 1947.
  * [42] William Fulton.  Introduction to toric varieties, volume 131 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 1993.  The William H. Roever Lectures in Geometry.
  * [43] Alexander Givental.  A mirror theorem for toric complete intersections.  In Topological field theory, primitive forms and related topics (Kyoto, 1996), volume 160 of Progr. Math., pages 141–175. Birkhäuser Boston, Boston, MA, 1998.
  * [44] B. R. Greene and M. R. Plesser.  Duality in Calabi-Yau moduli space.  Nuclear Phys. B, 338(1):15–37, 1990.
  * [45] Christopher D. Hacon and James McKernan.  Existence of minimal models for varieties of log general type. II.  J. Amer. Math. Soc., 23(2):469–490, 2010.
  * [46] Liana Heuberger.  $\mathbb{Q}$-Fano threefolds and Laurent inversion.  arXiv:2202.04184 [math.AG], 2022\.
  * [47] Kentaro Hori and Cumrun Vafa.  Mirror symmetry.  arXiv:hep-th/0002222, 2000\.
  * [48] A. R. Iano-Fletcher.  Working with weighted complete intersections.  In Explicit birational geometry of 3-folds, volume 281 of London Math. Soc. Lecture Note Ser., pages 101–173. Cambridge Univ. Press, Cambridge, 2000.
  * [49] V. A. Iskovskih.  Fano threefolds. I.  Izv. Akad. Nauk SSSR Ser. Mat., 41(3):516–562, 717, 1977.
  * [50] V. A. Iskovskih.  Fano threefolds. II.  Izv. Akad. Nauk SSSR Ser. Mat., 42(3):506–549, 1978.
  * [51] V. A. Iskovskih.  Anticanonical models of three-dimensional algebraic varieties.  In Current problems in mathematics, Vol. 12 (Russian), pages 59–157, 239 (loose errata). VINITI, Moscow, 1979.
  * [52] V. A. Iskovskih and Ju. I. Manin.  Three-dimensional quartics and counterexamples to the Lüroth problem.  Mat. Sb. (N.S.), 86(128):140–166, 1971.
  * [53] Jennifer M. Johnson and János Kollár.  Fano hypersurfaces in weighted projective 4-spaces.  Experiment. Math., 10(1):151–158, 2001.
  * [54] Anne-Sophie Kaloghiros.  The topology of terminal quartic 3-folds.  PhD thesis, University of Cambridge, 2007.
  * [55] Alexander M. Kasprzyk.  Toric Fano three-folds with terminal singularities.  Tohoku Math. J. (2), 58(1):101–121, 2006.
  * [56] Alexander M. Kasprzyk.  Canonical toric Fano threefolds.  Canad. J. Math., 62(6):1293–1309, 2010.
  * [57] Alexander M. Kasprzyk.  The classification of toric canonical Fano 3-folds.  _Zenodo_ https://doi.org/10.5281/zenodo.5866330, 2010.
  * [58] Alexander M. Kasprzyk and Benjamin Nill.  Fano polytopes.  In Strings, gauge fields, and the geometry behind, pages 349–364. World Sci. Publ., Hackensack, NJ, 2013.
  * [59] Alexander M. Kasprzyk, Benjamin Nill, and Thomas Prince.  Minimality and mutation-equivalence of polygons.  Forum Math. Sigma, 5:Paper No. e18, 48, 2017.
  * [60] In-Kyun Kim, Takuzo Okada, and Joonyeong Won.  K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces.  arXiv:2011.07512 [math.AG], 2020\.
  * [61] János Kollár, Yoichi Miyaoka, and Shigefumi Mori.  Rational connectedness and boundedness of Fano manifolds.  J. Differential Geom., 36(3):765–779, 1992.
  * [62] János Kollár, Yoichi Miyaoka, Shigefumi Mori, and Hiromichi Takagi.  Boundedness of canonical $\mathbb{Q}$-Fano 3-folds.  Proc. Japan Acad. Ser. A Math. Sci., 76(5):73–77, 2000.
  * [63] Maximilian Kreuzer and Harald Skarke.  Classification of reflexive polyhedra in three dimensions.  Adv. Theor. Math. Phys., 2(4):853–871, 1998.
  * [64] Maximilian Kreuzer and Harald Skarke.  Complete classification of reflexive polyhedra in four dimensions.  Adv. Theor. Math. Phys., 4(6):1209–1230, 2000.
  * [65] Shigefumi Mori.  On a generalization of complete intersections.  J. Math. Kyoto Univ., 15(3):619–646, 1975.
  * [66] Shigefumi Mori.  Threefolds whose canonical bundles are not numerically effective.  Ann. of Math. (2), 116(1):133–176, 1982.
  * [67] Shigefumi Mori.  Flip theorem and the existence of minimal models for $3$-folds.  J. Amer. Math. Soc., 1(1):117–253, 1988.
  * [68] Shigefumi Mori and Shigeru Mukai.  Classification of Fano $3$-folds with $B_{2}\geq 2$.  Manuscripta Math., 36(2):147–162, 1981/82.
  * [69] Shigefumi Mori and Shigeru Mukai.  Erratum: “Classification of Fano 3-folds with $B_{2}\geq 2$”.  Manuscripta Math., 110(3):407, 2003.
  * [70] Mikkel Øbro.  An algorithm for the classification of smooth Fano polytopes.  arXiv:0704.0049 [math.CO], 2007\.
  * [71] Stavros Argyrios Papadakis and Miles Reid.  Kustin-Miller unprojection without complexes.  J. Algebraic Geom., 13(3):563–577, 2004.
  * [72] Joseph Polchinski.  String theory. Vol. II.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2005.  Superstring theory and beyond, reprint of 2003 edition.
  * [73] Thomas Prince.  Smoothing toric Fano surfaces using the Gross-Siebert algorithm.  Proc. Lond. Math. Soc. (3), 117(3):617–660, 2018.
  * [74] Yuri Prokhorov.  $\mathbb{Q}$-Fano threefolds of large Fano index, I.  Doc. Math., 15:843–872, 2010.
  * [75] Yuri Prokhorov and Miles Reid.  On $\mathbb{Q}$-Fano 3-folds of Fano index 2.  In Minimal models and extremal rays (Kyoto, 2011), volume 70 of Adv. Stud. Pure Math., pages 397–420. Math. Soc. Japan, [Tokyo], 2016\.
  * [76] Victor Przyjalkowski.  Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models.  Cent. Eur. J. Math., 9(5):972–977, 2011.
  * [77] Muhammad Imran Qureshi and Balázs Szendrői.  Constructing projective varieties in weighted flag varieties.  Bull. Lond. Math. Soc., 43(4):786–798, 2011.
  * [78] Miles Reid.  Canonical $3$-folds.  In Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pages 273–310. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980.
  * [79] Miles Reid.  Graded rings and birational geometry.  In Proceedings of algebraic geometry symposium (Kinosaki, Oct 2000), K. Ohno (Ed.), pages 1–72. 2002.  Available from http://www.warwick.ac.uk/~masda/3folds/.
  * [80] Takeshi Sano.  On classifications of non-Gorenstein ${\bf Q}$-Fano $3$-folds of Fano index $1$.  J. Math. Soc. Japan, 47(2):369–380, 1995.
  * [81] Takeshi Sano.  Classification of non-Gorenstein ${\bf Q}$-Fano $d$-folds of Fano index greater than $d-2$.  Nagoya Math. J., 142:133–143, 1996.
  * [82] Hiromichi Takagi.  On classification of $\mathbb{Q}$-Fano 3-folds of Gorenstein index 2\. I, II.  Nagoya Math. J., 167:117–155, 157–216, 2002.
  * [83] Kiyohiko Takeuchi.  Some birational maps of Fano $3$-folds.  Compositio Math., 71(3):265–283, 1989.