2.1. Fano 3-folds

Any Fano $3$ -fold X comes with an intrinsic embedding $X\subset w\mathbb {P}$ in weighted projective space (up to automorphisms of $w\mathbb {P}$ ) as follows. The graded ring of X is

$$\begin{align*}R(X,-K_X) = \bigoplus_{m\in\mathbb N} H^0(X,-mK_X) \end{align*}$$

and $X\cong \operatorname {\mathrm {Proj}} R(X,-K_X)$ by ampleness. Any choice $f_0,\dots ,f_n$ of minimal homogeneous generators for $R(X,-K_X)$ has the same collection of weights $\{a_0,\dots ,a_n\}$ , where $a_i=\deg f_i$ , and we may suppose $a_0\le \cdots \le a_n$ . (Note the conventional abuse of notation: $\{a_0,\dots ,a_n\}$ is a list with possible repetitions, and similarly $\{a_0,\dots ,a_n\}\setminus \{a_3,a_7\}$ is the list obtained by removing one instance of each of $a_3$ and $a_7$ ). Thus, any choice of a minimal generating set determines a projectively normal embedding

(2.A) $$ \begin{align} X \subset \mathbb{P}(a_0,\dots,a_n), \end{align} $$

which we refer to as the anticanonical embedding of X. (This is not the same as the image $\Phi _{-K_X}(X)$ under the linear system $|{-}K_X|$ unless $-K_X$ is very ample, which is rarely the case here).

The Hilbert series $P_X$ of a Fano $3$ -fold X is simply that of $R(X,-K_X)$ , the formal power series

$$\begin{align*}P_X = P_X(t) = \sum_{m\in\mathbb N} h^0(X,-mK_X) t^m. \end{align*}$$

The Hilbert series $P_X$ of a Fano $3$ -fold X does not determine the weights $a_i$ of its anticanonical embedding—it not infrequently happens that $R(X,-K_X)$ has relations and generators in the same degree.

2.2. Numerical data of a Fano 3-fold

We write $\tfrac {1}r(a_1,a_2,a_3)$ to denote the singularity germ of the origin in $\mathbb C^3/\boldsymbol \mu _r$ , where $\boldsymbol \mu _r$ is the multiplicative group of rth roots of unity, and $\varepsilon \in \boldsymbol \mu _r$ acts on coordinates by $(\varepsilon ^{a_1}x_1,\varepsilon ^{a_2}x_2,\varepsilon ^{a_3}x_3)$ . Any Fano $3$ -fold has a basket of singularities ${\mathcal {B}}$ , which is a collection of terminal quotient singularities $\tfrac {1}{r}(1,a,-a)$ (possibly including repeats, where again we use set notation on this understanding). Following [Reference Reid89, §§8–10], in general, ${\mathcal {B}}$ is derived locally from the singularities of X by crepant blowup and Q-smoothing. When X has terminal singularities and lies in a weighted projective space as a quasismooth variety, then X is an orbifold and ${\mathcal {B}}$ is exactly the collection of singularities of X.

The genus $g=h^0(X,-K_X)-2\ge -2$ and basket ${\mathcal {B}}$ together determine the degree by the formula

(2.B) $$ \begin{align} {-}K_X^3 = 2g-2 + \sum_{\mathcal{B}} \frac{b(r-b)}{r}, \end{align} $$

where for each element $\tfrac {1}{r}(1,a,-a)$ of the basket ${\mathcal {B}}$ , we define $0<b<r$ by $ab\equiv 1\mod r$ .

Although they are not used for the initial construction of the Grdb, there are various different divisorial indices defined for a Fano $3$ -fold X that we consider later.

Definition 2.1. The Gorenstein index or singularity index of X is

$$\begin{align*}i_X = \min \{ n\in\mathbb Z_{>0} \mid nK_X \text{ is Cartier}\}. \end{align*}$$

The Fano index is the rational number $f_X = r/i_X$ , where $r>0$ is the largest integer for which $-i_X K_X \equiv rA$ for a Cartier divisor A. There are further types of Fano index:

$$ \begin{align*} q_X&= \max \{ q>0 \mid {-}K_X \sim qA, \text{ where }A \text{ is a } {\mathbb{Q}}\text{-Cartier Weil divisor} \}\\ q_{{\mathbb{Q}} X}&= \max \{ q>0 \mid {-}K_X \sim_{\mathbb{Q}} qA, \text{ where } A \text{ is a } {\mathbb{Q}}\text{-Cartier Weil divisor} \}. \end{align*} $$

These are natural generalizations of the divisibility of the anticanonical divisor of a nonsingular Fano $3$ -fold in its Picard group. For a Fano $3$ -fold, both q and $q_{\mathbb {Q}}$ are positive integers, and they are equal if $\operatorname {\mathrm {Cl}}(X)$ has no torsion [Reference Prokhorov83].

2.3. Effective Kawamata bounds for prime Fano 3-folds

We construct the set of genus–basket pairs $(g,{\mathcal {B}})$ that satisfy the constraints of [Reference Kawamata68].

2.3.1. Possible baskets

Controlling the possible baskets of Fano $3$ -folds can be done much more generally than the prime Fano case. Recall that a basket ${\mathcal {B}}$ is a set with possible repeats.

Definition 2.2. We fix a particular set $\mathbb B$ of baskets ${\mathcal {B}}$ as $\mathbb B = \left \{ {\mathcal {B}} \mid \sum _{{\mathcal {B}}} (r - \frac {1}{r}) < 24 \right \}$ , where each ${\mathcal {B}}$ is a basket and the sum is taken over the quotient singularities $\tfrac {1}r(1,a,-a)\in {\mathcal {B}}$ .

It is elementary and easy to check that $\#\mathbb B = 8{,}314$ . For any ${\mathcal {B}}\in \mathbb B$ , the defining bound of 24 implies first that $r\le 24$ for each $\tfrac {1}r(1,a,-a)\in {\mathcal {B}}$ , and moreover that $\#{\mathcal {B}}\le 15$ , since each $r-(1/r) \ge 3/2$ . Therefore, there are only finitely many possible collections of indices r appearing in ${\mathcal {B}}$ . The only terminal quotient singularity with $r=2$ is $\tfrac {1}2(1,1,1)$ , and for each index $r>2$ , there are $\phi (r)/2$ distinct terminal quotient singularities, namely, $\tfrac {1}r(1,a,-a)$ for $1\le a< r/2$ coprime to r. Thus, for each index r that appears in ${\mathcal {B}}$ , there are only finitely many singularities of index r that may occur. The precise bound 8,314 comes by enumerating all possible baskets satisfying these conditions (by computer, for example).

Theorem 2.3 [Reference Kawamata67, Lemmas 2.2 and 2.3] and [Reference Reid89, (10.3)]

Let X be a projective $3$ -fold with canonical singularities. Then

(2.C) $$ \begin{align} 24\chi(\mathscr O_X) = {-}K_Xc_2(X) + \sum_{{\mathcal{B}}} \left(r - \frac{1}r\right), \end{align} $$

where the sum is over the singularities $\tfrac {1}r(1,a,-a)$ of the basket ${\mathcal {B}}$ of X.

When X is a Fano $3$ -fold, $\chi (\mathscr O_X)=1$ and (2.C) simplifies.

Lemma 2.4. If X is a Fano $3$ -fold with canonical singularities which satisfies $-K_X c_2(X)> 0$ , then the basket ${\mathcal {B}}$ of X is in $\mathbb B$ .

Proof. The condition $-K_X c_2(X)> 0$ together with (2.C) implies that $\sum r - (1/r) < 24$ , where the sum is taken over the basket ${\mathcal {B}}$ .

Remark 2.5. The condition $-K_Xc_2(X)>0$ holds for prime Fano $3$ -folds by Theorem 2.7. Far more generally, any weak Fano $3$ -fold (i.e., $-K_X$ is only required to be nef and big) with terminal singularities satisfies $-K_Xc_2(X)\ge 0$ by [Reference Kollár, Miyaoka, Mori and Takagi69, 1.2(1)]. It is easy to check that relaxing the inequality provides (coincidentally) 24 additional cases with $\sum r-(1/r)=24$ , such as $16\times \tfrac {1}2(1,1,1)$ and $5\times \tfrac {1}5(1,2,3)$ .

2.3.2. Genus bounds

Controlling the possible values for the genus $g_X$ for each basket uses the bounds determined by [Reference Kawamata68].

Proposition 2.6. Let X be a Fano $3$ -fold with basket ${\mathcal {B}}$ and genus g that satisfies

(2.D) $$ \begin{align} {-}K_X^3 \le \unicode{x3ba} (-K_Xc_2(X)) \end{align} $$

for some $\unicode{x3ba}>0$ . Then $g_{\mathrm {min}} \le g \le g_{\mathrm {max}}$ where

$$\begin{align*}g_{\mathrm{min}} = \max\left\{ -2, \left\lfloor \tfrac{1}2 \left(2 - \sum_{{\mathcal{B}}} \tfrac{b(r-b)}r \right) \right\rfloor+1\right\} \end{align*}$$

and

(2.E) $$ \begin{align} g_{\mathrm{max}} = \left\lfloor \tfrac{1}2 \left(2 - \sum_{{\mathcal{B}}} \tfrac{b(r-b)}r + \unicode{x3ba} \left(24 - \sum_{{\mathcal{B}}} \left(r - \tfrac{1}r\right) \right)\right) \right\rfloor, \end{align} $$

where each sum is over $\tfrac {1}r(1,a,-a)\in {\mathcal {B}}$ and b is defined by $ab\equiv 1\mod r$ .

Proof. The lower bound is simply the condition that $-K_X^3>0$ in (2.B). For the upper bound, substituting (2.B) and (2.C) into (2.D) gives

$$\begin{align*}2g-2 + \sum_{\mathcal{B}} \tfrac{b(r-b)}{r} \le \unicode{x3ba}\left(24 - \sum_{{\mathcal{B}}} \left(r - \tfrac{1}r\right)\right) \end{align*}$$

and the upper bound follows.

Theorem 2.7 (Kawamata [Reference Kawamata68])

Let X be a prime Fano $3$ -fold. Then

$$\begin{align*}{-}K_X^3 \le \unicode{x3ba} (-K_Xc_2(X)) \end{align*}$$

for some real number $\unicode{x3ba}>0$ . In particular, $-K_X c_2(X)> 0$ .

If X is semistable, then the formula (2.E) holds with $\unicode{x3ba} =3$ . If $(\Omega _X^1)^{\vee \vee }$ has a rank $2$ maximal destabilizing subsheaf, then the formula holds with $\unicode{x3ba} =4$ . If $(\Omega _X^1)^{\vee \vee }$ has a rank $1$ maximal destabilizing subsheaf, then it holds with possibly larger $\unicode{x3ba}>0$ .

2.4. Proof of 1.2

The Hilbert series $P_X$ is equivalent to the data of genus–basket pair $(g,{\mathcal {B}})$ by the following Fletcher–Reid plurigenus formula together with (2.B) and the independence of basket contributions [Reference Fletcher52, 4.2]. We could instead use the integral orbifold Riemann–Roch formulas of [Reference Buckley, Reid and Zhou32], and for human calculation they are much preferred.

Theorem 2.8 [Reference Fletcher51, Theorem 2.5] and [Reference Reid89, (10.3)]

Let X be a Fano $3$ -fold with canonical singularities and basket ${\mathcal {B}}$ . Then

(2.F) $$ \begin{align} P_X(t) = \frac{1+t}{(1-t)^2} - \frac{t(1+t)}{(1-t)^4}\,\frac{K_X^3}2 -\sum_{\mathcal{B}} \left(\frac{1}{(1-t)(1-t^r)} \sum\limits_{i=1}^{r-1} \frac{\overline{bi}(r-\overline{bi})t^i}{2r}\right), \end{align} $$

where for each element $\tfrac {1}{r}(1,a,-a)$ of the basket ${\mathcal {B}}$ , we fix any integer b that satisfies $ab\equiv 1\mod r$ , and set $\overline {bi}\in \{0,1,\ldots ,r-1\}$ to be the least residue of $(bi)$ modulo r.

Definition 2.9. For any $g\ge -2$ and ${\mathcal {B}}\in \mathbb B$ , the right-hand side of (2.F) determines a rational function that we denote $P_{g,{\mathcal {B}}}$ , where (2.B) expresses the $-K_X^3$ factor in terms of $g,{\mathcal {B}}$ . We may expand $P_{g,{\mathcal {B}}} = \sum P_nt^n$ as a power series, and conversely, given such a power series, we may recover $g,{\mathcal {B}}$ and the expression (2.F) as a rational function. For any $\unicode{x3ba}>0$ , define

$$\begin{align*}{\mathcal F}_\unicode{x3ba} = \left\{ (g,{\mathcal{B}},P_{g,{\mathcal{B}}}) \mid {\mathcal{B}}\in\mathbb B \text{ and } g_{\mathrm{min}}\le g\le g_{\mathrm{max}}, \text{ and if } P_1>0 \text{ then } P_n>0 \text{ for all } n\ge1 \right\}, \end{align*}$$

where $g_{\mathrm {min}}$ and $g_{\mathrm {max}}$ are defined for ${\mathcal {B}}$ by (2.E) with the given $\unicode{x3ba} $ . We treat ${\mathcal F}_\unicode{x3ba} $ as a set of genus–basket pairs, or equivalently as a set of rational functions they determine, and we write $(g,{\mathcal {B}})\in {\mathcal F}_\unicode{x3ba} $ or $P_{g,{\mathcal {B}}}\in {\mathcal F}_\unicode{x3ba} $ accordingly.

It follows from everything so far that if X is a Fano $3$ -fold with genus g and basket ${\mathcal {B}}$ that satisfies (2.D) for $\unicode{x3ba}>0$ , then $P_X=P_{g,{\mathcal {B}}}\in {\mathcal F}_{\unicode{x3ba} }$ . The condition on $P_n$ holds for any such $P_X$ : there cannot be a nonzero section $x\in H^0(X,-K_X)$ with $x^n=0$ for some $n>1$ .

Proof. Lemma 2.4 provides exactly $8{,}314$ possible baskets. Of these, (2.E) calculates $g_{\mathrm {max}}\ge -2$ in $7{,}492$ cases for $\unicode{x3ba} =3$ and in $7{,}683$ cases for $\unicode{x3ba} =4$ . Assembling pairs $(g,{\mathcal {B}})$ with ${\mathcal {B}}$ one of these baskets and $g_{\mathrm {min}}\le g\le g_{\mathrm {max}}$ bounded by Proposition 2.6 gives $39,\!558$ cases for $\unicode{x3ba} =3$ and $52,\!654$ cases for $\unicode{x3ba} =4$ .

For each genus–basket pair $(g,{\mathcal {B}})$ , expand the rational function $P_{g,{\mathcal {B}}}$ as a power series. In eight cases, this power series starts

$$\begin{align*}1+t+t^2+\cdots+t^n + O(t^{n+2}), \end{align*}$$

with either $n=2$ or $n=4$ , where the $t^{n+1}$ term has coefficient zero. We discard these eight series at this stage, to leave $39,\!550$ series in ${\mathcal F}_{3}$ and $52,\!646$ series in ${\mathcal F}_{4}$ .

Computer code to enumerate the Hilbert series of Proposition 2.10, either in the Magma system [Reference Bosma, Cannon and Playoust13] or in the Go-language [Reference Cox, Griesemer, Pike, Taylor and Thompson48] is available at [Reference Brown and Kasprzyk22], [Reference Brown and Kasprzyk24].

3.1. Hilbert series in low codimension

An embedding $X\subset \mathbb {P}(a_0,\dots ,a_n)$ is nondegenerate if there are no quasi-linear equations in the defining ideal of X, that is, no equations which contain a linear term and thereby may be used to eliminate a variable. Anticanonical embeddings (2.A) of Fano $3$ -folds are nondegenerate, as they are determined by the choice of a minimal generating set.

3.1.2. The graded ring game

Suppose $X\subset \mathbb {P}=\mathbb {P}(1^{m_1},2^{m_2},\dots )$ is a variety in a nondegenerate projectively normal embedding. Such X has a Hilbert series $P_X(t) = \sum _{i\ge 0} n_it^i$ , where $n_i = h^0(X,\mathscr O_X(i))$ . In general, knowing $P_X(t)$ alone is not enough to determine the $m_j$ , but it can provide estimates, even without information about the equations of X.

The following “graded ring game” is standard. Given a series $P_0 = 1+n_1t+n_2t^2+\cdots $ , we attempt to build some X as above, polarized by $\mathscr O_X(1)$ , with $P_X=P_0$ . By the assumptions, we must start with $m_1=n_1$ . Write $P_1 = (1-t)^{m_1}P_0 = 1 + n_2't^2 + \cdots $ . If $n_2'\ge 0$ , then there are necessarily at least $n_2'$ variables in weight $2$ , so we set $m_2=n_2'$ . (Of course, we may add more variables of weight 2, but each one would necessitate an equation of weight 2, and at least for a complete intersection general in its deformation family that would be excluded by nondegeneracy).

Continuing, write $P_2=(1-t^2)^{m_2}P_1=1+n_3"t^3 + \cdots $ . If now $n_3"\ge 0$ , then there are necessarily at least $n_3"$ variables in weight $3$ , so again we set $m_3=n_3"$ and we consider $P_3=(1-t^3)^{m_3}P_2$ . At some stage $n_{r+1}^{(r)}<0$ and the game ends: we can no longer infer that there must be additional generators in degree $r+1$ . Interpreting the Hilbert numerator according to (3.A), we may now attempt to construct $X\subset \mathbb {P}(1^{m_1},2^{m_2},\ldots ,r^{m_r})$ using suitably general polynomials of degrees $d_i$ , and check whether $P_X=P_0$ and whichever properties—irreducible, quasismooth, Fano, and so on—that we require do in fact hold.

As we saw, at any stage, there may be more generators in the degree we are considering, but there cannot be fewer. Indeed, at the ith stage,

$$\begin{align*}P_0 = \left(\tfrac{1}{(1-t)^{m_1}\dots(1-t^i)^{m_i}}\right)\cdot\left(1 + n_{i+1}^{(i)}t^{i+1} + \text{ higher order terms}\right), \end{align*}$$

where the first factor is the Hilbert series of $\mathbb {P}(1^{m_1},\dots ,i^{m_i})$ (with no relations), so in degree $i+1,$

$$\begin{align*}n_{i+1}^{(i)} = n_{i+1} - N_i, \end{align*}$$

where $N_i$ is the maximum possible span of lower degree terms; if $n_{i+1}^{(i)}>0,$ then the ring needs at least that many additional generators in degree $i+1$ .

When X is a complete intersection, this approach with a small twist determines the unique weights and degrees (cf. [Reference Iano-Fletcher56, §18]). To illustrate the twist, consider $C_{2,6}\subset \mathbb {P}(1^3,3)$ , a genus $5$ hyperelliptic curve polarized by $2A$ , where A is the hyperelliptic linear system $g^1_2$ . This polarized curve has Hilbert series $P=1+3t+5t^2+8t^3+\cdots $ , so the first step is to consider $(1-t)^3P = 1-t^2+t^3-t^5$ . The game is complete, but clearly there is no variety defined by a single quadric in $\mathbb {P}^2$ with a linear syzygy. Speculatively looking ahead for the next positive coefficient suggests considering a variable of weight $3$ , which recovers the numerical data of C as $(1-t^3)(1-t)^3P = 1-t^2-t^6+t^8$ . In practice, this phenomenon is rare, and when it does arise, the solution is as simple as this.

Even when $X\subset \mathbb {P}$ is not a complete intersection, this graded ring game provides a lower bound for the weights $m_i$ of a minimal generating set, and so provides both a lower bound on the actual codimension of X and a necessary subset of the weights.

Although this seems to give rather a lot away, when applied to the series $P_{g,{\mathcal {B}}}\in {\mathcal F}_{3}$ with low genus or small basket with small singularity indices it works well. For example, with

$$\begin{align*}g=-1\quad\text{and}\quad {\mathcal{B}} = \left\{\tfrac{1}2(1,1,1),\tfrac{1}3(1,1,2),\tfrac{1}5(1,1,4),\tfrac{1}5(1,2,3),\tfrac{1}7(1,1,6)\right\}, \end{align*}$$

this identifies necessary weights $1,5,6,7,8,9,10$ and Hilbert numerator $1-t^{16}-t^{17}-t^{18} - t^{19} - t^{20} + t^{25} + t^{26} + t^{27} + t^{28} + t^{29} - t^{45}$ . This numerical data provide enough information to write down a Fano $3$ -fold $X_{16\dots 20}\subset \mathbb {P}(1,5,6,7,8,9,10)$ as the vanishing of the five maximal Pfaffians (of degrees $16,\ldots ,20$ ) of a skew $5\times 5$ matrix of forms (cf. [Reference Brown, Kasprzyk and Zhu27], [Reference Buchsbaum and Eisenbud31], [Reference Corti and Reid46], [Reference Reid and Ohno91]).

3.1.3. Polarizing baskets

We also add weights to realize the singularity basket correctly. For example, $g=2$ , ${\mathcal {B}} = \left \{ \tfrac {1}2(1,1,1), \tfrac {1}3(1,1,2) \right \}$ determines a series $P(t)$ that satisfies

$$\begin{align*}(1-t)^4(1-t^2)^2 P(t) = 1 - 3 t^4 + 3 t^6 - 2 t^7 + 3 t^9 - 3 t^{10} + \cdots. \end{align*}$$

Thus, the graded ring game only demands generators in degrees $1,1,1,1,2,2$ , whereas there must be some ambient orbifold locus with stabilizer $\boldsymbol \mu _3$ to accommodate the index $3$ orbifold point. That could be achieved with a generator in degree $3$ , or a combination of generators whose degrees have 3 as their greatest common divisor. Here, the simplest thing works:

$$\begin{align*}(1-t)^4(1-t^2)^2(1-t^3) P(t) = 1 - t^3 - 3 t^4 + 3 t^6 + t^7 - t^{10} \end{align*}$$

suggests a variety $X\subset \mathbb {P}(1,1,1,1,2,2,3)$ in codimension $3$ defined by five Pfaffians of degrees $3$ , $4$ , $4$ , $4,$ and $5$ , and one can use this numerical hint to construct such a Fano $3$ -fold. (Notice that the equation of degree $5$ is masked in the Hilbert numerator by a syzygy of degree $5$ : the Hilbert numerator is “really” $1 - t^3 -3t^4 - t^5 + t^5+3t^6+t^7-t^{10}$ . Knowing that codimension $3$ Gorenstein ideals have an odd number of generators defined as Pfaffians is extra information that comes from the Buchsbaum–Eisenbud theorem [Reference Buchsbaum and Eisenbud31]).

Other ways to introduce index $3$ points, such as introducing weights $6$ and $9$ , may work but increase codimension. In practice, by making this check in descending order of the singularity index, such questions are avoided, since additional high-degree variables may already polarize lower-degree singularities.

By this stage, for each Hilbert series, we have a collection of generator weights that are enough to span low-degree pieces of the graded ring and to polarize the singularities. This already determines the Grdb weights for codimension $\le 3$ , but is not always correct in high codimension. We fix this next.

3.2. Numerical unprojection Ansatz and weights

Type I Gorenstein unprojection [Reference Kustin and Miller71], [Reference Papadakis and Reid79] is a technique that takes as input a pair of Gorenstein schemes $D\subset X$ , with $D\subset X$ of codimension $1$ , and returns a new Gorenstein scheme Y. In applications to projective geometry, it often corresponds to a birational contraction of D to a point of Y, and that is how we wish to apply it.

3.2.1. The Type I unprojection Ansatz

We describe a model case of Kustin–Miller unprojection following Papadakis–Reid [Reference Papadakis and Reid79, 2.4–2.9]. Consider the following hypothetical input–output process:

Input: Let $X\subset \mathbb {P}(a_0,\dots ,a_n)$ be a Fano $3$ -fold with terminal singularities in its anticanonical embedding with basket ${\mathcal {B}}_X$ . Suppose $D\subset X$ for some coordinate plane $D=\mathbb {P}(a_i,a_j,a_k)$ with weights $(a_i,a_j,a_k)=(1,a,b)$ in some order and $\gcd (a,b)=1$ , and suppose moreover that X is quasismooth away from finitely many nodes $\Sigma \subset D$ .

Output: A quasismooth Fano $3$ -fold $Y\subset \mathbb {P}(a_0,\dots ,a_n,r)$ in its anticanonical embedding, with $r=a+b$ , such that:

1. 1. Y contains the point $P=(0:\dots :0:1)$ and $P\in Y$ is a terminal quotient singularity $\tfrac {1}r(1,a,b)$ .

2. 2. Y has the same genus as X. Moreover, if $a+b+c>d$ for every $c,d\in \{a_0,\ldots ,a_n\}\setminus \{a_i,a_j,a_k\}$ , then none of the equations of Y has nonzero constant or linear terms.

3. 3. The Gorenstein projection from $P\in Y$ is a birational map $Y\dashrightarrow X$ that factorizes into birational morphisms as follows:

   $$\begin{align*}\begin{array}{ccccc} &&Z&&\\ &\swarrow&&\searrow&\\ X&&&&Y \end{array} \end{align*}$$
   Starting from X with its unprojection divisor $D=\mathbb {P}(1,a,b)\subset X$ , the map $Z\rightarrow X$ is the small crepant D-ample resolution of the nodes of X (the contraction of finitely many $\mathscr O(-1)\oplus \mathscr O(-1)$ flopping curves, viewed from Z). The midpoint Z contains $\mathbb {P}(1,a,b)$ , the birational transform of D, meeting these exceptional curves transversally, and the morphism $Z\rightarrow Y$ contracts this $\mathbb {P}(1,a,b)$ to $P\in Y$ (which, viewed from Y, is the Kawamata blowup of $P\in Y$ ).

4. 4. The new basket ${\mathcal {B}}_Y$ of Y is obtained by replacing the two orbifold points $\frac {1}{a}(1,b,-b)$ and $\frac {1}{b}(1,a,-a)$ on D in X by the image point $\frac {1}{r}(1,a,b)$ of the contraction of D: that is,

   (3.C) $$ \begin{align} {\mathcal{B}}_X \cup \left\{\tfrac{1}{r}(1,a,b)\right\} = {\mathcal{B}}_Y \cup \left\{\tfrac{1}{a}(1,b,-b),\tfrac{1}{b}(1,a,-a)\right\}, \end{align} $$

   where $\tfrac {1}{a}(1,b,-b)$ is omitted if $a=1$ and $\tfrac {1}{b}(1,a,-a)$ is omitted if $b=1$ .

5. 5. The Hilbert series of Y is

   $$\begin{align*}P_Y = P_X + \tfrac{t^{a+b}}{(1-t)(1-t^a)(1-t^b)(1-t^r)}. \end{align*}$$

This process, part (3) in particular, is expressed as standard moves of the Mori MMP. In practice, it may often be described as a simple-minded geometric progression (or elimination of one variable), as in the next example.

Example 3.1 (cf. [Reference Reid and Ohno91, §2])

Consider $X=(x_0A-x_1B=0)\subset \mathbb {P}^4$ , where $A,B$ are general forms of degree 3 in coordinates $x_0,\ldots ,x_4$ of $\mathbb {P}^4$ . Any such X contains the plane $D=\mathbb {P}^2=(x_0=x_1=0)$ and has nine nodes on D. Now, with slight abuse of notation for the coordinates and using the same $A,B$ , construct $Y=(yx_0=A, yx_1=B)\subset \mathbb {P}(1^5,2)$ in coordinates $x_0,\ldots ,x_4,y$ . The point $P=(0:\ldots :0:1)\in Y$ is a quotient singularity $\tfrac {1}2(1,1,1)$ —the equations eliminate $x_0$ , $x_1$ near P—with nine smooth rational curves of degree $\tfrac {1}2$ defined by $A=B=0$ passing through it. The Gorenstein projection from $P\in Y$ is the elimination of s from the equations of Y, a birational map $Y\rightarrow X$ that factorizes as an extremal divisorial contraction $Z\rightarrow Y$ with center $P\in Y$ followed by the contraction of the birational transform in Z of the nine curves, which are now flopping curves orthogonal to $-K_Z$ . The Hilbert series of X and Y are related by

$$\begin{align*}\frac{1-2t^3+t^6}{(1-t)^5(1-t^2)} = \frac{1-t^4}{(1-t)^5} + \frac{t^2}{(1-t)^3(1-t^2)} \end{align*}$$

as in (5) (see [Reference Reid and Ohno91, §2] for more discussion).

In many cases, this process is a theorem (see, e.g., [Reference Brown, Kerber and Reid28, 3.2]). Indeed the setup $D\subset X$ satisfies the conditions for Kustin–Miller unprojection [Reference Prokhorov and Reid86, 2.4], giving a new variable s of degree $r=k_X-k_D=-1-(-1-a-b)=a+b$ and additional equations involving s of the form $sf_i=g_i$ , where $f_i$ form a basis of the ideal $I_D$ in the coordinate ring of X. One can see using a free resolution of the coordinate ring $\mathbb C[Y]$ over $\mathbb C[\mathbb {P}(a_0,\dots ,r)]$ [Reference Papadakis77] that $\mathscr O(-K_Y)=\mathscr O(1)$ , and the numerics of (1), (2), (4), and (5) follow (cf. [Reference Prokhorov and Reid86, 2.7–2.9]). Then given Y, the Kawamata blowup of $P\in Y$ is a weak Fano, and it has an anticanonical model $Z\rightarrow X$ . More complicated situations can arise—see (4.B), where $Z\rightarrow X$ makes both a crepant divisorial contraction and a disjoint flopping contraction—but since the assumption of nodes here already establishes that contracting the flopping curves result in a Fano, there can be no further contraction in the given situation.

The key point here is that we do not use the setup above as a theorem to be applied, rather we turn it around to act as an Ansatz, as follows.

Ansatz 3.2 (Type I unprojection)

Consider a genus–basket pair $(g,{\mathcal {B}}_Y)\in {\mathcal F}_{4}$ that is not among the $95+85$ cases assigned weights in codimension $1$ or $2$ by §3.1.1. Suppose that $(g,{\mathcal {B}}_X)$ is another genus–basket pair (with the same genus g) for some X which satisfies (3.C) with $r=a+b$ and $a, b$ coprime to r. If the weights $(a_0,\ldots ,a_n)$ of X listed in Grdb contain $(1,a,b)$ as a sublist, then we insist that the weights of Y in Grdb are $(a_0,\ldots ,a_n,r)$ .

3.3. Numerical corollaries

The crude classification by Theorem 1.2 already contains enough information to provide approximations to various strong and sharp theorems by elementary and easy means. For example, recall Prokhorov’s sharp bound on the degree.

Theorem 1.2 recovers the weaker bound $-K_X^3\le 72$ at once. Applying [Reference Karzhemanov63], [Reference Karzhemanov64] sharpens this to $-K_X^3\le 66\tfrac {1}2$ . The next case to exclude is $(g,{\mathcal {B}})=(34,\{\tfrac {1}2(1,1,1)\})$ , which we see (§4.2.1) is populated by the blowup $\operatorname {\mathrm {Bl}}_P\mathbb {P}(1,1,3,5)$ at a smooth point P, a Fano $3$ -fold with the non-terminal point $\tfrac {1}5(1,1,3)$ . The numerical methods alone cannot say more: instead, Prokhorov’s stronger geometric Sarkisov methods show that this case cannot be realized by a prime Fano $3$ -fold.

Another example is the following result of [Reference Chen, Chen and Chen36], proving [Reference Iano-Fletcher56, Conjecture 18.19(2)], which bounds the codimension of Fano complete intersections.

The proof in [Reference Chen, Chen and Chen36] is difficult and subtle, but with Theorem 1.2 and Corollary 1.4, it follows by inspection of the numerators of all Hilbert series in the database. Indeed, for a complete intersection, the process of determining generic weights and numerator (§3.1) is well defined. Conversely, given weights, the numerator determines a minimal set of degrees for equations. In most cases, this immediately rules out a complete intersection, as there are too many equations—and adding additional weights does not alter that. Of the remaining, there are cases of apparent complete intersections, but by equations whose degrees are too small to accommodate the high-degree variables: therefore, to be terminal, there must be further equations, and again complete intersection is ruled out.

4.1. Nonsingular Fano 3-folds

The classification of 105 families of nonsingular Fano $3$ -folds [Reference Bayer, Kuznetsov and Macrì8], [Reference Iskovskikh58], [Reference Mori and Mukai75] is a cornerstone of higher-dimensional birational geometry. It is listed in [Reference Iskovskikh and Prokhorov59, Table 12.2] and online at [Reference Belmans11]. Each of the 105 families lies in its anticanonical embedding along the leading diagonal of Table 1: Table 2 and its explanation show how. We should expect similarly subtle behavior in hundreds of places throughout Table 1, so it is useful to absorb this familiar case first.

Table 2 uses the numbering convention of [Reference Belmans11] (which adapts [Reference Iskovskikh and Prokhorov59], [Reference Mori and Mukai75]) to list the 105 families: $\rho $ –n denotes the nth family of $3$ -folds with Picard rank $\rho $ . The table is indexed in columns by Picard rank $\rho =1,\dots ,10$ across the top (the sparse cases $\rho =5$ –10 are compressed into a single column). The entries of the table are the family numbers n (in full-size font) for the nth family with that $\rho $ ; these entries often have sub/superscript decorations that we will come to. Each row is indexed by genus $g_X=2,\dots ,33$ down the side, so that each row lists all families of a given genus g. Since for a nonsingular Fano $3$ -fold X the genus $g_X$ determines ${-}K_X^3$ and $P_X$ , one may imagine this table lying sideways along the leading diagonal of Table 1. For example, the nonsingular Hilbert series for $g=13$ is realized by the seven families 1–13, 2–17, 2–18, 3–7, 3–8, 4–1, and 7–1, and in Table 1, these all lie at the single position genus 13 and codimension 11 on the leading diagonal.

The decorations and positioning within each column indicate additional information. Fano manifolds X with $\rho _X=1$ , the first column, are divided with those of Fano index $f_X=1$ on the left-hand side of column $1$ –n, and $f_X\ge 2$ on the right-hand side. When $\rho \ge 2$ , many of these Fano $3$ -folds are constructed by extremal extractions from other nonsingular Fano $3$ -folds. The decorations on entries of the table include some of these extremal divisorial contractions by writing entries as $\rho $ – $n_m$ , abbreviated to $n_m$ in the table, to indicate a map from members of family $\rho $ –n to members of $(\rho -1)$ –m. Some codomains are very common, and we indicate these by the following special notation:

• • $2$ – $n^{*}\equiv 2$ – $n_{17}$ means map to $1$ – $17=\mathbb {P}^3$ .

• • $2$ – $n^{\dagger }\equiv 2$ – $n_{16}$ means map to $1$ – $16=X_2\subset \mathbb {P}^4.$

• • $3$ – $n^*\equiv 3$ – $n_{34}$ means map to $2$ – $34=\mathbb {P}^1\times \mathbb {P}^2.$

• • $3$ – $n^\dagger \equiv 3$ – $n_{35}$ means map to $2$ – $35=\operatorname {\mathrm {Bl}}_P\mathbb {P}^3.$

• • $4$ – $n^*\equiv 4$ – $n_{27}$ means map to $3$ – $27=\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1.$

Columns $\rho =2$ , $3$ are arranged to give a brief indication of the extremal contraction data, with varieties which admit maps to the common target listed on the left-hand side of each column, with those which only admit other morphisms to the right of the center, and finally those with no birational contractions to another nonsingular Fano $3$ -fold down the right-hand side; most of the latter are products or double covers, and their extremal rays are Mori fibrations. For example, the diagram

$$\begin{align*}\begin{array}{rcccl} &&3\text{--}15\\ &\swarrow&\downarrow&\searrow&\\ \mathbb{P}^1\times\mathbb{P}^2 = 2\text{--}34 && 2\text{--}29 && 2\text{--}31 \\ &&&\searrow&\ \ \downarrow\\ &&&&1\text{--}16 = (X_2\subset\mathbb{P}^4) \end{array} \end{align*}$$

indicates sequences of extremal contractions from 3–15 that one can read from Table 2 (see [Reference Belmans11] for full details). Finally, the right-hand column $\#\mathcal {T}$ lists the number of toric models with Gorenstein canonical, but not necessarily ${\mathbb {Q}}$ -factorial, singularities by genus (see §4.5).

Generalizing from smooth to Gorenstein terminal Fano $3$ -folds does not increase the number of families: any Gorenstein terminal Fano $3$ -fold may be smoothed [Reference Namikawa76] so it appears in Table 2. Moreover, the Picard rank does not change on smoothing (see [Reference Jahnke and Radloff61], which uses this to determine smoothing families). In contrast, there are Fano $3$ -folds with canonical Gorenstein singularities that do realize other families in the leading diagonal of Table 1, as we discuss next.

4.2. Gorenstein Fano 3-folds

When X is a Fano $3$ -fold with canonical singularities that has ${-}K_X$ Cartier, then its singularities are Gorenstein and its basket is empty. Therefore, its Hilbert series $P_X$ behaves as though X has no singularities at all, and so again lies on the leading diagonal of Table 1.

There is no classification of Fano $3$ -folds with Gorenstein canonical singularities, but there are results in many special cases: [Reference Cheltsov, Przhiyalkovskiĭ and Shramov35] settles the hyperelliptic and trigonal cases, [Reference Jahnke and Radloff60] for the case $|{-}K_X|$ has basepoints, and [Reference Bayer, Kuznetsov and Macrì7] for the terminal case with $g\in \{6,7,8,9,10,12\}$ . There are also precise results in certain extreme cases, which we describe now in relation to the Grdb geography. Recall that in the bottom-right block of Table 1 the number of Hilbert series in ${\mathcal F}_{3}$ by genus and codimension is

(4.A)

Prokhorov [Reference Prokhorov84, 1.5] proves that the largest possible genus for a Fano $3$ -fold with ${-}K_X$ Cartier is $g=37$ with degree ${-}K_X^3=2g-2=72$ (strengthening Cheltsov’s result $-K_X^3\le 184$ [Reference Cheltsov34]). Moreover, Prokhorov proves that $g=37$ is realized only by $\mathbb {P}(1,1,4,6)$ and $\mathbb {P}(1,1,1,3)$ . That completely clears up the final column.

Cheltsov and Karzhemanov extend this to Gorenstein $3$ -folds in genus $g\ge 33$ .

The point p in (1) is not a toric $0$ -stratum, but for a local toric model of $\operatorname {\mathrm {Bl}}_p$ blow up the toric cone $\left < (0,1,0), (1,0,1), (1,0,-1)\right>$ at $(1,1,0)$ (rather than the usual discrepancy 2 blowup at $(2,1,0)$ ).

4.2.1. Genus 34

We consider Karzhemanov’s degree $66$ example Z in more detail. The weighted projective space $X=\mathbb {P}(1^2,3,5)$ is a Fano $3$ -fold with a terminal singularity $\tfrac {1}3(1,1,2)$ , a canonical singularity $\tfrac {1}5(1,1,3)$ , genus $34,$ and degree $-K^3=66\tfrac {2}3$ . Its Hilbert series lies in ${\mathcal F}_{4}$ but not ${\mathcal F}_{3}$ , just outside the region of Table 1. It has anticanonical embedding $X\subset \mathbb {P}(1^{36},2,3)$ in the blank position $(34,34)$ in (4.A). It fits into a diagram of toric varieties and maps:

$$\begin{align*}\begin{array}{ccccccccccc} &&&&W\\ &&&\swarrow&&\searrow\\ U&\buildrel{\text{flip}}\over{\dashleftarrow}&Y&&&&V&&\buildrel{\text{flop}}\over{\dashrightarrow}&&V^+\\ \downarrow&&&\searrow&&\swarrow&&\searrow&&\swarrow\\ \mathbb{P}^1&&&&X&&&&Z \end{array} \end{align*}$$

where $Y\rightarrow X$ is the Kawamata $\tfrac {1}3(1,1,2)$ blowup of $P_3\in X$ (the index $3$ point), $Y\dashrightarrow U$ is the $(5,2,-1,-1)$ (canonical) flip to Karzhemanov’s bundle $U\rightarrow \mathbb {P}^1$ , $V\rightarrow X$ is the $\tfrac {2}3$ -discrepancy $\tfrac {1}3(2,2,1)$ blowup of $P_3\in X,$ and $V\dashrightarrow V^+$ is the $(5,1,-3,-3)$ (canonical) flop with base Z, while finally W is the resolution of $P_3\in X$ , which itself admits two flops to divisorial extractions of U and $V^+$ , respectively. In this picture,

(4.B) $$ \begin{align} \left( X\subset\mathbb{P}(1^{36},2,3) \right) \dashrightarrow \left( Y\subset\mathbb{P}(1^{36},2) \right) \dashrightarrow \left( Z\subset\mathbb{P}^{35} \right) \end{align} $$

is a sequence of projections between Fano $3$ -folds of degrees $66\tfrac {2}3$ , $66\tfrac {1}2$ , and $66$ , where $\rho _X=\rho _Z=1$ and $\rho _Y=2$ , and X and Y are ${\mathbb {Q}}$ -factorial while Z is not.

Loosely speaking, we see how knowing Karzhemanov’s example provides other Fano $3$ -folds by birational contractions; from the other end, knowing $\mathbb {P}(1^2,3,5)$ provides other Fano $3$ -folds by birational blow ups. We discuss this further in §4.5.4.

4.2.2. Genus 33

Karzhemanov [Reference Karzhemanov64] also classifies the case of degree $64$ : there is $\mathbb {P}^3$ and six other cases, all with strictly canonical singularities. The Grdb matches five toric cases, and we illustrate with a beautiful non- ${\mathbb {Q}}$ -factorial example

$$\begin{align*}\Phi_{{-}K_X}\colon X = \operatorname{\mathrm{TorVar}}_{\left(\begin{smallmatrix}5\\6\end{smallmatrix}\right)}\begin{pmatrix}1&1&5&3&0\\0&1&6&4&1\end{pmatrix} \hookrightarrow\mathbb{P}^{34}. \end{align*}$$

The notation $\operatorname {\mathrm {TorVar}}_\chi M_{2\times 5}$ denotes a toric variety expressed as the GIT quotient $\mathbb C^5 /\!\!/_\chi (\mathbb C^*)^2$ where the rows of M are the weights of the $(\mathbb C^*)^2$ action (see [Reference Brown, Corti and Zucconi18, §A.5]). The map $\Phi _D$ is the natural map to $\operatorname {\mathrm {Proj}}\oplus _{m\ge 0}H^0(mD)$ ; in this case, $D=-K_X$ has bidegree $(10,12)$ and, in coordinates $x,y,z,u,v$ of $\mathbb C^5$ , $\Phi _{-K_X}$ is given by the 35 monomials $z^2,xu^3,y^{10}v^2,\ldots $ of bidegree $(10,12)$ . This is the base of a $(6,1,-2,-5)$ (canonical) flop

$$\begin{align*}\begin{array}{ccccccccc} &&U&&\buildrel{\text{flop}}\over{\dashrightarrow}&&U^+\\ &\swarrow&&\searrow&&\swarrow&&\searrow\\ \mathbb{P}(1^2,4,6)&&&&X&&&&\mathbb{P}(1^2,3,5), \end{array} \end{align*}$$

where the map $U\rightarrow \mathbb {P}(1^2,4,6)$ is the blowup of a smooth point and $U^+\rightarrow \mathbb {P}(1^2,3,5)$ is a weighted $\tfrac {1}3(1,2,4)$ -blowup of the index $3$ point; that is,

$$\begin{align*}\text{blowup } (-1,-4,-6)\text{ in the cone }\left<(1,0,0),(0,0,1),(-1,-3,-5)\right>\kern-1pt. \end{align*}$$

(The numerics come from the observation that $(6,1,0,-2,-5)$ , $(-1,0,1,1,1),$ and $(4,1,2,0,-3)$ lie in the span of the rows of the defining matrix of X).

4.3. Index 2 singularities

The list ${\mathcal F}_{4}$ contains $360$ pairs $(g,{\mathcal {B}})$ , where ${\mathcal {B}}=\{N\times \tfrac {1}2(1,1,1)\}$ with $N\ge 1$ of which $272$ lie in ${\mathcal F}_{3}$ . Restricting to $g\ge 2$ , these numbers reduce to $325$ and $238,$ respectively. Sano [Reference Sano93], [Reference Sano94] and Campana–Flenner [Reference Campana and Flenner33] classify such terminal Fano $3$ -folds X under the assumption Fano index $f_X\ge 1$ . For non-empty baskets $\{N\times \tfrac {1}2(1,1,1)\}$ and $f_X>1$ , these are [Reference Sano94] (with their anticanonical embeddings mapping to the right)

$$\begin{align*}\begin{array}{rclcl} \mathbb{P}(1^2,2^2,3) \supset X_6&\hookrightarrow&\mathbb{P}(1^8,2^3)&\qquad&\text{(}N = 3, g=6, \text{ codim 7)}\\ \mathbb{P}(1^3,2^2) \supset X_4&\hookrightarrow&\mathbb{P}(1^{16},2^2)&&\text{(}N = 2, g=14, \text{ codim 14)}\\ \mathbb{P}(1^4,2) \supset X_3&\hookrightarrow&\mathbb{P}(1^{23},2)&&\text{(}N = 1, g=21, \text{ codim 20)}\\ \mathbb{P}(1^3,2)&\hookrightarrow&\mathbb{P}(1^{34},2)&&\text{(}N = 1, g=32, \text{ codim 31).} \end{array} \end{align*}$$

For the case $f_X=1$ , [Reference Sano93, 1.1] presents a classification as follows: there are 14 families of nonsingular Fano $3$ -folds, particular members Y of which admit a $\boldsymbol \mu _2$ action with quotient $X=Y/\boldsymbol \mu _2$ having index 2 singularities and $f_X=1$ . The theorem constructs representative cases of each type, again all in rather high anticanonical codimension, but does not enumerate all possible Y and $\boldsymbol \mu _2$ actions.

4.3.1. Prime cases

The remaining cases for index $2$ baskets satisfy $i_X=2$ and $f_X=1/2$ . Takagi [Reference Takagi98] classifies prime Fano $3$ -folds with such genus–basket pairs (imposing also $g\ge 2$ ) into $35$ families matching 23 elements of ${\mathcal F}_{3}$ . They are presented in Tables 1–5 of [Reference Takagi98], with the individual families numbered 1.1, 1.2, …, 5.5. These are listed in Table 3, ranging from Family $3.1$ , $X_5\subset \mathbb {P}(1^4,2)$ , to Family 1.14, $X\subset \mathbb {P}(1^{10},2^2)$ in codimension $8$ , with Type I projections going up the columns. (The remarkable division into two cases in codimension 4 is the starting point for [Reference Brown, Kerber and Reid28]).

Table 3

Families of Tables $n.m$ of [Reference Takagi98] as they appear in Table 1, arranged by genus g and codimension c. The generic member of each family is embedded as $X\subset \mathbb {P}(1^{g+2},2^N)$ with basket ${\mathcal {B}}=\{N\times \tfrac {1}2(1,1,1)\},$ where $N=c-g+2$

Table 3 Long description

The table organizes mathematical families n dot m by Genus g (columns) and Codimension c (rows).

* Genus values range from 2 to 8 across the top.

* Codimension values range from 1 to 8 down the left side.

Data entries by Codimension (c):

* c equals 1: Genus 2 contains 3.1.

* c equals 2: Genus 2 contains 3.2; Genus 3 contains 5.2.

* c equals 3: Genus 2 contains 2.1; Genus 3 contains 5.3; Genus 4 contains 4.3.

* c equals 4: Genus 2 contains 2.2 and 3.3; Genus 3 contains 4.1 and 5.4; Genus 4 contains 1.1 and 4.4; Genus 5 contains 1.4.

* c equals 5: Genus 2 contains 2.3, 3.4, and 5.1; Genus 3 contains 4.2 and 5.5; Genus 4 contains 1.2, 1.3, and 4.5; Genus 5 contains 1.5 and 1.6; Genus 6 contains 1.9 and 1.10.

* c equals 6: Genus 2 contains 2.4; Genus 4 contains 4.6; Genus 5 contains 1.7 and 1.8; Genus 6 contains 1.11; Genus 7 contains 1.12.

* c equals 7: Genus 4 contains 4.7; Genus 8 contains 1.13 and 4.8.

* c equals 8: Genus 8 contains 1.14.

The comparison with the geography in Table 1 is striking. The $272$ pairs $(g,{\mathcal {B}})\in {\mathcal F}_{3}$ are spread over most of the table, away from the top diagonal line of Gorenstein pairs $(g,\emptyset )$ , and Takagi’s result shows that most are not realized by prime Fano $3$ -folds. Many of the remaining $272-23=249$ pairs are realized by more general Fano $3$ -folds (cf. §§4.3.2 and 4.5), though the complete classification of Fano $3$ -folds of Gorenstein index $2$ remains unknown.

4.3.2. Non-prime cases

The general hypersurface $X_5\subset \mathbb {P}(1,1,1,2,3)$ has $-K_X=\mathscr O(3)$ and $g=12$ , with ${\mathcal {B}}=\{\tfrac {1}2(1,1,1)\}$ . Its actual singularities are the expected $\tfrac {1}2(1,1,1)$ together with an additional canonical singularity $\tfrac {1}3(1,1,1)$ which has a crepant resolution and so does not contribute to the basket. The anticanonical embedding $X\subset \mathbb {P}(1^{14},2)$ lies in codimension 11, and so X occupies position of Table 1 that Takagi’s classification does not fill. There are many other similar examples (cf. [Reference Brown and Kasprzyk23]).

Venkatachalam [Reference Venkatachalam110] classifies 251 families of terminal ${\mathbb {Q}}$ -factorial Fano 3-folds X of Picard rank $\rho _X>1$ that lie in rank 2 toric variety $\mathbb F$ . (More precisely, the hypothesis is that the pair $(\mathbb F,X)$ is a Fano terminal pair, with $\mathbb F$ simplicial and X not a cone). This provides other families of non-prime Fano $3$ -folds with index 2 singularities. For example, #61 of [Reference Venkatachalam111, §2.2, Table 1] (= #29394 in [Reference Brown and Kasprzyk22])

$$\begin{align*}X_{\left(\begin{smallmatrix}2\\4\end{smallmatrix}\right)} = \operatorname{\mathrm{TorVar}}_{\left(\begin{smallmatrix}1\\2\end{smallmatrix}\right)}\begin{pmatrix}1&1&1&0&0&0\\0&1&1&1&1&2\end{pmatrix} \quad\text{with } g=6 \text{ and } {\mathcal{B}}=\{2\times\tfrac{1}2(1,1,1)\} \end{align*}$$

has anticanonical embedding into $\mathbb {P}(1^8,2^2)$ in codimension $6$ . (Recall the notation for toric varieties from §4.2.2; $X_D$ denotes a member of the linear system $|D|$ , i.e., $f=0$ for a polynomial f of the given bidegree D). This gives a different family having the same $g,{\mathcal {B}}$ and codimension as Family 1.11 of [Reference Takagi98].

Similarly, #68 of [Reference Venkatachalam111, §2.2, Table 1] (= #29696 in [Reference Brown and Kasprzyk22])

$$\begin{align*}X_{\left(\begin{smallmatrix}3\\6\end{smallmatrix}\right)} = \operatorname{\mathrm{TorVar}}_{\left(\begin{smallmatrix}2\\3\end{smallmatrix}\right)}\begin{pmatrix}1&1&1&1&1&0\\0&1&2&2&3&1\end{pmatrix} \quad\text{with } g=6 \text{ and } {\mathcal{B}}=\{4\times\tfrac{1}2(1,1,1)\} \end{align*}$$

has anticanonical embedding into $\mathbb {P}(1^8,2^4)$ in codimension $8$ . There is no prime Fano $3$ -fold that matches this Hilbert series.

4.4. Higher Fano index

Among all Fano $3$ -folds X, there are some that have a divisible canonical class, and we indicate the Hilbert series of those in Table 4. The notion of divisibility we consider here is the following: $-K_X = \iota A$ for a maximal integer $\iota \ge 2$ and ample integral Weil divisor A. (Note that $\iota $ is quite different from the Gorenstein index $i_X$ and the Fano index $f_X$ ). The graded ring $R(X,A) = \oplus _{m\ge 0} H^0(X,mA)$ is Gorenstein as $H^0(X,-K_X)\subset R(X,A)$ [Reference Goto and Watanabe54, 5.1.9]. Suzuki [Reference Brown and Suzuki29], [Reference Brown and Suzuki30], [Reference Suzuki96] carries out the analysis to find a set of possible baskets ${\mathcal {B}}_\iota $ for each $\iota $ , with additional genus information when $\iota \le 2$ . Again, we impose the stability condition $-K_X^3\le -3K_Xc_2$ . The number of baskets (or basket–genus pairs for $\iota \le 2$ ) per index $\iota \ge 1$ is

$$\begin{align*}\begin{array}{r|c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c} \hline \iota&1&2&3&4&5&6&7&8&9&11&13&17&19\\ {\#{\mathcal{B}}_\iota}&39550&1413&181&82&34&6&12&4&2&3&2&1&1\\ \hline \end{array} \end{align*}$$

There are no prime Fano $3$ -folds of indices $\iota =12$ , 14, 15, 16, or 18 by [Reference Suzuki96], and by applying more sophisticated techniques [Reference Prokhorov83] also proves that there is no prime Fano $3$ -fold of index; cf. also [Reference Prokhorov and Reid86] $10$ .

Table 4

Number of Hilbert series of semistable Fano $3$ -folds of index $\iota \ge 2$ listed by genus and estimated minimal codimension

Table 4 Long description

The table maps Genus (horizontal axis) against Codimension (vertical axis). Genus values range from negative 2 to 37, with vertical separators after Genus 2, 7, 12, 17, 22, 27, 32, and 37. Codimension values range from 1 to 35.

Key data clusters include:

* Low Codimension (2 to 6): Values are concentrated in Genus negative 1 to 6. For example, Codimension 6 has 5 at Genus negative 1, 7 at Genus 0, and 5 at Genus 1.

* Mid Codimension (10 to 20): Data spreads wider across Genus values. Codimension 14 peaks with 50 at Genus 1 and 31 at Genus 0. Codimension 18 shows a high concentration with 67 at Genus 2 and 44 at Genus 1.

* High Codimension (21 to 35): Values become sparser and shift toward higher Genus. Codimension 21 has 24 at Genus 7 and 19 at Genus 9 and 11. Codimension 31 has 1 at Genus 32 and 2 at Genus 33. The highest Genus value of 37 contains a single entry at Codimension 35.

* Empty Regions: Codimension 1 and 33 to 34 are entirely empty across all Genus values. Genus negative 2 is empty for all Codimension values.

The point here is that if $-K_X=\iota A$ , then the A-embedded model $\operatorname {\mathrm {Proj}} R(X,A)$ is usually much simpler than the anticanonical ring. The numbers in low A-embedding codimension are

(4.C) $$ \begin{align} \begin{array}{r|c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c|c} \hline \iota&2&3&4&5&6&7&8&9&11&13&17&19&\text{total}\\ \hline\text{codim 0}&0&0&1&1&0&1&0&0&1&1&1&1&7\\ \text{codim 1}&8&7&2&5&1&4&3&2&2&1&&&35\\ \text{codim 2}&26&6&7&1&&&&&&&&&40\\ \hline \end{array} \end{align} $$

Example 4.2. The codimension $0$ row of (4.C) contains the seven weighted projective spaces

$$\begin{align*}\begin{array}{r|c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c} \hline &\mathbb{P}^3&\mathbb{P}(1^3,2)&\mathbb{P}(1^2,2,3)&\mathbb{P}(1,2,3,5)&\mathbb{P}(1,3,4,5)&\mathbb{P}(2,3,5,7)&\mathbb{P}(3,4,5,7)\\ \hline {-}K^3&64&125/2&343/6&1331/30&2197/60&4913/210&6859/420\\ g&33&32&29&22&18&11&7\\ \text{cod}&31&31&30&27&25&22&20\\ W&1^{35}&1^{34},2&1^{31},2^2,3& 1^{24},2^4,3^2,5& 1^{20},2^4,3^3,4,5& 1^{13},2^5,3^4,4^2,5,7& 1^{9},2^7,3^4,4,5^2,7\\ \hline \end{array} \end{align*}$$

and these embed anticanonically in $\mathbb {P}(W)$ as the model given in Grdb.

Example 4.3. Many of the A-hypersurfaces embed in a bigger space under their anticanonical $\iota $ -Veronese embedding than Grdb suggests. For example, $X_{10}\subset \mathbb {P}(1^2,2,3,5)$ has $\iota =2$ , so $-K_X=2A$ and the anticanonical embedding is

$$\begin{align*}X\buildrel{\cong}\over{\longrightarrow}\Phi_{2A}(X)\subset\mathbb{P}(1^4,2^2,3^3,4) \end{align*}$$

in codimension $6$ . Its ( $-K_X$ -polarized) Hilbert series, however, admits a simpler model $Y_{4,4}\subset \mathbb {P}(1^4,2,3)$ , and this is the one given in Grdb.

There is a small twist in the Grdb numerics in this case. For varieties with $\iota \ge 2$ , the Gorenstein projection does not work in the same way as the $\iota =1$ case (the image of projection has non-isolated singularities). Therefore, rather than using projection to identify weights, the Grdb compares with the K3 database [Reference Brown15]: a degree $\iota $ slice $S\in |\iota A|$ has the numerical properties of a K3 surface polarized by $A_{|S}$ , so a choice of weights is determined by [Reference Brown15]; then including a variable of degree $\iota $ (noting that it may then be eliminated by an equation of degree $\iota $ ) gives a choice of weights for X.

4.5. Toric Fano 3-folds

Kasprzyk [Reference Kasprzyk65] classifies toric Fano $3$ -folds with canonical singularities as a list ${\mathcal T}_{\mathrm {can}}$ of $674,\!688$ lattice polytopes; this classification is available from [Reference Kasprzyk66]. These varieties are all ${\mathbb {Q}}$ -Gorenstein, and $12,\!190$ of them are ${\mathbb {Q}}$ -factorial.

The Grdb contains ${\mathcal T}_{\mathrm {can}}$ , and it can be analyzed online [Reference Brown and Kasprzyk22]. More importantly, the Grdb connects ${\mathcal T}_{\mathrm {can}}$ with the Fano $3$ -fold database: each polytope $\Delta \in {\mathcal T}_{\mathrm {can}}$ has a Hilbert series $P_\Delta \in {\mathcal F}_{4}$ , all but $12$ of which lie in ${\mathcal F}_{3}$ . These absent $12$ all have isolated, ${\mathbb {Q}}$ -factorial, strictly canonical singularities, and are either one of the six weighted projective spaces

$$\begin{align*}\mathbb{P}(1, 1, 3, 5),\ \mathbb{P}(1, 2, 5, 7),\ \mathbb{P}(1, 3, 7, 10),\ \mathbb{P}(1, 3, 7, 11),\ \mathbb{P}(1, 5, 7, 13),\ \mathbb{P}(3, 5, 11, 19) \end{align*}$$

or a rank $2$ blowup of one of these. The Grdb links $\Delta $ to $P_\Delta $ , and conversely for any $P\in {\mathcal F}_{3}$ reports all those $\Delta $ with $P_\Delta =P$ . This matching is significant in different ways, as we discuss next.

4.5.1. Location in geography

The toric Fano $3$ -folds in ${\mathcal T}_{\mathrm {can}}$ populate large areas of Table 1. Table 6 is the submap of Hilbert series that are realized by at least one toric Fano $3$ -fold.

Only $8$ of the toric cases are prime Fano $3$ -folds: the seven weighted projective spaces of Example 4.2 together with the magical fake weighted projective space of degree $64/5$ and genus $5$

$$\begin{align*}\mathbb{P}^3/\boldsymbol\mu_5(1,2,3,4) \hookrightarrow \mathbb{P}(1^7, 2^8, 3^4, 5^4) \end{align*}$$

in codimension $19$ . Including these, there are $634$ terminal cases, of which $233$ are terminal ${\mathbb {Q}}$ -factorial.

For some Hilbert series, there are many matching toric $3$ -folds, and this is recorded in the Grdb, with an idea of the multiplicities in Table 7. It seems amazing to us that two different polytopes can contain the same number of lattice points at all dilations—but of course this is exactly what happens whenever two polytopes are mutation equivalent [Reference Akhtar, Coates, Galkin and Kasprzyk1].

There are $4{,}319$ varieties $X\in {\mathcal T}_{\mathrm {can}}$ that have ${-}K_X$ Cartier (cf. [Reference Kreuzer and Skarke70]); of these, $194$ are ${\mathbb {Q}}$ -factorial. The number of these is listed by genus in the column $\#\mathcal {T}$ of Table 2. We have not checked whether any satisfy Petracci’s non-smoothability condition [Reference Petracci80, 1.1], nor whether any lie at the intersection of multiple smooth families (see §4.5.3).

4.5.3. Deformation of toric varieties and intersections of families

Following Altmann’s local analysis [Reference Altmann5], [Reference Altmann6], a lot is known about how toric varieties deform. For toric Fano $3$ -folds with isolated singularities, global deformations surject onto local deformations [Reference Petracci81, 2.3], so understanding the deformation theory of singularities on toric Fano $3$ -folds is a powerful tool.

Example 4.4. The first element $X_1\in {\mathcal T}_{\mathrm {can}}$ has the Hilbert series of some $X\subset \mathbb {P}(1^{7},2^4)$ with a basket $4\times \tfrac {1}2(1,1,1)$ . However, $X_1$ is not quasismooth: it has six Gorenstein facets that are the cone on the del Pezzo surface of degree $6$ and four cones of type $\tfrac {1}2(1,1,1)$ . Each of these del Pezzo cone singularities has two smoothing components locally, so since deformations of toric Fano $3$ -folds surject onto local deformations, there are at least seven distinct quasi-smoothing components that contain different small deformations of $X_1$ (cf. Example 4.6 below, which suggests that there are at least 24 families).

Extending this to Gorenstein-index- $2$ toric Fano $3$ -folds that also have isolated cone over del Pezzo degree $6$ singularities gives three more examples with two distinct quasi-smoothing families:

$$\begin{align*}\begin{array}{ccccccc} \hline {\mathcal T}_{\mathrm{can}}(\text{id})&\#\tfrac{1}2&\#\text{dP}_6&{\mathcal F}_{3}(\text{id})&X\subset w\mathbb{P}&g&\text{codim}\\ \hline 1&4&6&27334&\mathbb{P}(1^7,2^4)&5&7\\ 254482&3&1&38250&\mathbb{P}(1^{17},2^3)&15&16\\ 254485&6&1&36639&\mathbb{P}(1^{13},2^6)&11&15\\ 254810&3&1&38935&\mathbb{P}(1^{20},2^3)&18&19\\ \hline \end{array} \end{align*}$$

4.5.4. High codimension representatives and cascades

Many of the weights in Grdb are constructed inductively by considering a single projection, but varieties frequently arise in sequences, or cascades, of projections: famously $X = \Phi _{-K}(\mathbb {P}^2)\subset \mathbb {P}^9$ has sequences of projections from points (which in this case are blowups) that recover elements of most families of del Pezzo surfaces (see [Reference Reid and Suzuki92] for extensions). It seems typical that the ends of such cascades are simpler to describe than the middles: things like toric varieties live at the top, while hypersurfaces live at the bottom. Thus, any $X\in {\mathcal T}_{\mathrm {can}}$ of high codimension for its genus may be a good candidate for the head of a cascade.

For example, in $g=8$ , the highest codimension ${\mathbb {Q}}$ -factorial terminal Fano $3$ -fold is

$$\begin{align*}{\mathcal T}_{\mathrm{can}}(544385)\colon \mathbb{P}^1\times\mathbb{P}^2 / {\tfrac{1}3(0,1,0,1,2)} \subset\mathbb{P}(1^{10},2^6,3^6) \end{align*}$$

of Picard rank $2$ in codimension $18$ , with $6\times \tfrac {1}3(1,1,2)$ singularities at the 6 index 3 toric 0-strata. (This has id #33967 in ${\mathcal F}_{3}$ ). The Fano polytope is the simplicial decomposition on vertices

$$\begin{align*}(1,0,0), (0,1,0), (-1,-1,0), (1,2,3), (-1,-2,-3) \end{align*}$$

with six index $3$ cones meeting at a central “ $\mathbb {P}^2$ ” triangular equator with a cycle of three northern cones and three southern cones with $\operatorname {\mathrm {Sym}}_3$ symmetry. (This is a lattice triangular bipyramid and is fun to draw on a blackboard). The Kawamata blowup of any one of the index $3$ points is equivalent to any other and gives the first projection. There are four ways to project from a pair of index $3$ points, depending on adjacency; in the case of the blowup of a northern cone and the adjacent southern cone, the equator becomes a flopping curve that is contracted to an ordinary node. Continuing with different configurations of projections, we end up at the projection from all index $3$ points, which is a variety $Y\subset \mathbb {P}(1^{10},2^6)$ in codimension $12$ with $6\times \tfrac {1}2(1,1,1)$ singularities and 3 nodes. This variety admits a quasi-smoothing, so is a Gorenstein-index- $2$ Fano $3$ -fold that does not appear in Takagi’s classification Table 3, as it has $\rho _Y>1$ . Further projections from index $2$ points give more Gorenstein-index- $2$ varieties that extend Table 3 in genus $8$ to Fano $3$ -folds that are not prime Fano.

The highest-codimension toric Fano $3$ -folds with terminal singularities by genus—that is, the toric candidates for the top of terminal cascades—are listed in Table 5.

Table 5

High-codimension non-Gorenstein ${\mathbb {Q}}$ -factorial terminal toric Fano $3$ -folds, sorted by increasing genus g

Table 5 Long description

The table contains 22 rows of data with 8 columns. The headers are script upper T sub can (id), g, rho sub X, script upper F sub 3 (id), X subset of w double struck upper P, script upper B, codim, and an unnamed final column for additional geometric notes.

* Row 1: id 547383, g 5, rho 1, F3 29211, X in P(1 super 7, 2 super 8, 3 super 4, 5 super 4), B 4 times 2/5, codim 19, note P cubed / 1/5 (1, 2, 3, 4).

* Row 2: id 547379, g 7, rho 1, F3 32734, X in P(1 super 9, 2 super 7, 3 super 4, 4, 5 super 2, 7), B 1/3, 1/4, 2/5, 3/7, codim 20, note P(3, 4, 5, 7).

* Row 3: id 544385, g 8, rho 2, F3 33967, X in P(1 super 10, 2 super 6, 3 super 6), B 6 times 1/3, codim 18.

* Row 4: id 547380, g 11, rho 1, F3 36623, X in P(1 super 13, 2 super 5, 3 super 4, 4 super 2, 5, 7), B 1/2, 1/3, 1/5, 3/7, codim 22, note P(2, 3, 5, 7).

* Row 5: id 430483, g 12, rho 4, F3 36948, X in P(1 super 14, 2 cubed, 3 cubed), B 3 times 1/3, codim 16.

* Row 6: id 520102, g 13, rho 3, F3 37585, X in P(1 super 15, 2 super 4, 3 cubed, 4), B 1/2, 2 times 1/3, 1/4, codim 19.

* Row 7: id 544370, g 14, rho 2, F3 38020, X in P(1 super 16, 2 super 4, 3 cubed, 4, 5), B 1/3, 1/4, 2/5, codim 21.

* Row 8: id 544376, g 15, rho 2, F3 38404, X in P(1 super 17, 2 super 5, 3 squared, 5), B 2 times 1/2, 1/3, 2/5, codim 21.

* Row 9: id 430473, g 16, rho 4, F3 38533, X in P(1 super 18, 2 cubed, 3), B 2 times 1/2, 1/3, codim 18.

* Row 10: id 520107, g 17, rho 3, F3 38760, X in P(1 super 19, 2 cubed, 3), B 2 times 1/2, 1/3, codim 19.

* Row 11: id 520148, g 17, rho 2, F3 38760, same as 520107.

* Row 12: id 547382, g 18, rho 1, F3 39006, X in P(1 super 20, 2 super 4, 3 cubed, 4, 5), B 1/3, 1/4, 2/5, codim 25, note P(1, 3, 4, 5).

* Row 13: id 520124, g 19, rho 3, F3 39052, X in P(1 super 21, 2 squared, 3), B 1/2, 1/3, codim 20.

* Row 14: id 520103, g 20, rho 3, F3 39192, X in P(1 super 22, 2 cubed, 3), B 2 times 1/2, 1/3, codim 22.

* Row 15: id 544394, g 21, rho 1, F3 39278, X in P(1 super 23, 2 cubed, 3), B 2 times 1/2, 1/3, codim 23, note X sub 4 subset of P(1 squared, 2 squared, 3).

* Row 16: id 547381, g 22, rho 1, F3 39368, X in P(1 super 24, 2 super 4, 3 squared, 5), B 1/2, 1/3, 2/5, codim 27, note P(1, 2, 3, 5).

* Row 17: id 520128, g 24, rho 3, F3 39416, X in P(1 super 26, 2), B 1/2, codim 23.

* Row 18: id 520131, g 24, rho 3, F3 39416, same as 520128.

* Row 19: id 544383, g 25, rho 2, F3 39457, X in P(1 super 27, 2 squared, 3), B 1/2, 1/3, codim 26.

* Row 20: id 544389, g 26, rho 2, F3 39476, X in P(1 super 28, 2), B 1/2, codim 25.

* Row 21: id 544388, g 28, rho 2, F3 39510, X in P(1 super 30, 2), B 1/2, codim 27.

* Row 22: id 547384, g 29, rho 1, F3 39526, X in P(1 super 31, 2 squared, 3), B 1/2, 1/3, codim 30, note P(1 squared, 2, 3).

* Row 23: id 547385, g 32, rho 1, F3 39541, X in P(1 super 34, 2), B 1/2, codim 31, note P(1 cubed, 2).

Table 6

The 5760 Hilbert series of (canonical) toric Fano $3$ -folds that lie in ${\mathcal F}_{3}$ , listed by genus and codimension. Note that 12 other toric Fano Hilbert series lie in ${\mathcal F}_{4}$ outside ${\mathcal F}_{3}$

Table 6 Long description

The table maps Codimension (vertical axis, rows 1 to 35) against Genus (horizontal axis, columns negative 2 midline horizontal ellipsis 2, and 3 through 37).

* The Genus header contains values: negative 2 midline horizontal ellipsis 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, and 37.

* The Codimension column contains values from 1 to 35.

* Data distribution highlights:

- At Codimension 1, Genus 3 has a value of 1.

- At Codimension 12, Genus negative 2 midline horizontal ellipsis 2 has a value of 1.

- The highest frequency counts are concentrated in the upper-left quadrant, such as Codimension 16, Genus 4 with 161; Codimension 15, Genus 4 with 148; and Codimension 18, Genus 7 with 131.

- As Codimension increases beyond 20, the values become sparser and shift toward higher Genus values.

- The table concludes at Codimension 35, Genus 37 with a value of 1.

Table 7

Number of (canonical) toric Fano $3$ -folds that satisfy the semistable numerical condition, listed by genus and codimension

Table 7 Long description

The table organizes data with Genus as the horizontal header and Codimension as the vertical index.

* Header: The Genus values range from negative 2 midline horizontal ellipsis 2, followed by integers 3 through 37.

* Vertical Index: Codimension values range from 1 to 35.

* Data Distribution:

- Codimension 1: Genus 3 has 1 entry.

- Codimension 2: Genus 3 has 1, Genus 4 has 7.

- Codimension 5: Peak values occur at Genus 5 (307) and Genus 6 (182).

- Codimension 11: High density across Genus 5 (21,656), Genus 6 (22,519), and Genus 7 (19,992).

- Codimension 12: Contains a unique entry of 7 at Genus negative 2 midline horizontal ellipsis 2.

- Codimension 20: Data shifts rightward, with Genus 15 (285), Genus 16 (557), and Genus 17 (441).

- Codimension 35: The final entry is 2 at Genus 37.

The table is sparse in the upper-right and lower-left quadrants, with the highest concentrations of 3-folds appearing between Codimension 8 and 13 for Genus 4 through 10.

Beyond toric, [Reference Bechtold, Hausen, Huggenberger and Nicolussi9], [Reference Bechtold, Hausen, Huggenberger and Nicolussi10] initiates the analysis of low-complexity Fano varieties with the classification of prime Fano $3$ -folds of complexity $1$ —that is, Fano $3$ -folds whose automorphism group has a maximal torus of rank 2. The result is a list of 46 varieties and one 1-parameter family. The complexity condition enforces very particular trinomial equations. As in Example 4.3, these are almost all toric hypersurfaces with high-codimension anticanonical embedding. For example, the family of hypersurfaces $X_6\subset \mathbb {P}(1,2,3,4,5)$ has $\mathscr O_X(-K_X) = \mathscr O_X(9)$ , genus $g=18,$ and anticanonical embedding in codimension 24. It contains two particular members, $(x_0x_4 + x_1x_3 + x_2^2=0)$ and $(x_0x_4 + x_1^3 + x_2^2=0)$ , which have complexity 1 (the latter is not quasismooth, but has an exceptional terminal singularity of type $\tfrac {1}4(1,1,2,3;2)$ ).

4.6. Formats and low codimension

All $95+85$ Hilbert series in ${\mathcal F}_{4}$ whose Grdb model is in codimension $1$ or $2$ actually lie in ${\mathcal F}_{3}$ and may be constructed by hand as complete intersections as proposed; these are the first two rows of Table 1. These varieties all have Picard rank $1$ by the Lefschetz hyperplane theorem (cf. [Reference Corti, Pukhlikov and Reid45, 3.5]).

The same is true of all $70$ Hilbert series with codimension $3$ models, which occupy the third row of Table 1. In that case, only $X_{2,2,2}\subset \mathbb {P}^6$ is a complete intersection. The remaining $69$ cases are cut out by the five maximal Pfaffians of a skew $5\times 5$ matrix. Corti and Reid [Reference Corti and Reid46], following Grojnowski, explain this as a pullback from a weighted Grassmannian $w\!\operatorname {\mathrm {Grass}}(2,5)$ in a precise sense, which informally we may treat as saying that the Plücker embedding $\operatorname {\mathrm {Grass}}(2,5)\subset \mathbb {P}^9$ is described by the Pfaffians of a generic skew $5\times 5$ matrix of linear forms, and we may specialize these forms as we please, taking care with homogeneity. Again, these varieties have Picard rank $1$ by [Reference Brown and Fatighenti21, 3].

This idea leads to the general idea of “format” [Reference Brown, Kasprzyk and Zhu27], where the equations (and syzygies, and indeed the whole minimal free resolution) of a “key variety” (i.e., of any variety you like) are used as a model for the equations of other varieties by graded pullback. This idea is implemented in several independent programs: for example, early cases include [Reference Corti and Reid46], [Reference Qureshi and Szendrői87], Coughlan–Ducat apply cluster varieties [Reference Coughlan and Ducat47], and Takagi gives a sequence of key varieties in higher codimension [Reference Takagi99], [Reference Takagi100], [Reference Takagi102]–[Reference Takagi105]. These case construct prime Fano $3$ -folds, but other format constructions such as [Reference Brown, Kasprzyk and Qureshi26] may lead to higher Picard rank.

Example 4.5. In [Reference Brown, Kasprzyk and Qureshi26, 1.2], the variety $\mathbb {P}^2\times \mathbb {P}^2\subset \mathbb {P}^8$ in its Segre embedding is used as a key variety to model some Fano $3$ -folds in codimension $4$ that have Picard rank $2$ . That analysis constructs examples of deformation families in different codimension for the same Hilbert series. For example, ${\mathcal F}_{3}(548)$ is presented in Grdb as

$$\begin{align*}X\subset\mathbb{P}(1,3,4,5,6,7,10) \end{align*}$$

a codimension $3$ Pfaffian that is easy to construct, but there is another family

$$\begin{align*}X'\subset\mathbb{P}(1,3,4,5,6,7,9,10), \end{align*}$$

which arises from the unprojection of a degeneration. One may suspect that such Fano $3$ -folds are degenerations of the Pfaffian model, but this is not the case: quasismooth members of the two families have different Picard rank $h^{1,1}(X)$ and Euler characteristic $e_X$ , so do not lie in a common flat family.

Different formats cover everything in codimension $\le 3$ , much in codimension $4$ (see [Reference Coughlan and Ducat47, 5.3], where a cluster variety format describes certain subfamilies of deformations), and some in codimension $5$ . Although there are examples in higher codimension, they seem to realize only a small part of the classification there (see [Reference Qureshi and Szendrői87, 4.4, 5.2]), where flag variety formats recover classical nonsingular Fano $3$ -folds but no other element of ${\mathcal F}_{3}$ has suitable Hilbert series, or [Reference Coughlan and Ducat47, 5.8], where a codimension $6$ format realizes no Fano $3$ -folds. (Of course, such failures could be because there are few Fano $3$ -folds in high codimension—it seems unlikely, but we simply do not know).

4.7. Codimension 4: Unprojection

The unprojection Ansatz in §3.2.1 can sometimes be realized. For example, given a coordinate plane $D=\mathbb {P}(a_i,a_j,a_k)\subset \mathbb {P}(a_0,\dots ,a_n)$ , one may be able to construct a Fano $3$ -fold X that contains D with X quasismooth away from finitely many nodes on D. In this model case, the Type I unprojection constructs a quasismooth Fano $3$ -fold $Y\subset \mathbb {P}(a_0,\dots ,a_n,r)$ (the mild numerical conditions of §3.2 are only to exclude quasi-linear equations in the ideal of Y).

This is carried out in [Reference Brown, Kerber and Reid28] in the case $X\subset w\mathbb {P}^6$ lies in codimension $3$ , so that $Y\subset w\mathbb {P}^7$ lies in codimension $4$ , with further cases in [Reference Brown, Coughlan and Ducat20]. Type II unprojections are considered in [Reference Papadakis78], [Reference Reid and Ohno91], [Reference Taylor109], which construct other cases in codimension $4$ . Using this and §4.6, for every Hilbert series of ${\mathcal F}_{3}$ whose Grdb model is in codimension $4$ , there is a construction of a variety that matches that model, and in the majority of cases, there are 2 or more distinct deformation families.

4.8. The Fanosearch programme

Coates et al. [Reference Coates, Corti, Galkin, Golyshev and Kasprzyk38] and others, following ideas of Golyshev [Reference Golyshev53], provide an alternative approach to the Fano classification problem. The idea is that, via Mirror Symmetry, Fano classification can be rephrased as a fundamentally combinatorial problem of identifying suitable Laurent polynomials whose periods generate solutions to Picard–Fuchs equations on the other side of the mirror. Although the combinatorial problems seem hard, and the required mirror theorems are not wholly in place, [Reference Coates, Corti, Galkin and Kasprzyk39] confirms that the two sides of the mirror agree in the smooth case, and in doing so provides a host of tools for constructing Fano varieties and passing through the mirror. The Grdb is a key tool (see, e.g., [Reference Brown, Coates, Corti, Ducat, Heuberger, Kasprzyk, He, Ge, Bai, Bao and Hirst17], [Reference Coates, Heuberger and Kasprzyk40], [Reference Heuberger55]).

5.1. Review of guiding examples

The Fano $3$ -fold database consists of two lists of Hilbert series ${\mathcal F}_{3}\subset {\mathcal F}_{4}$ (defined in Definition 2.9) together with the estimated weights $X\subset \mathbb {P}(a_0,\dots ,a_n)$ that the Grdb assigns to each one (§§3.1 and 3.2). It is intended as a first coarse approximation to the classification of Fano $3$ -folds. However, it is certainly nowhere near to a final classification. The following remarks and examples are intended as quick reminders to help avoid misunderstandings.

1. 1. Overview of the Fano 3-fold database:

   1. (a) We distinguish between Fano $3$ -folds (with canonical singularities; Definition 1.1) and the much smaller class of prime Fano $3$ -folds (outcomes of the MMP; Definition 1.3).

   2. (b) The set ${\mathcal F}_{3}$ contains all rational functions that satisfy necessary conditions to be the Hilbert series of a Fano $3$ -fold X with $-K_X^3\le -3K_Xc_2$ . The geography of Table 1 is of ${\mathcal F}_{3}$ only.

   3. (c) By [Reference Kawamata68], the inequality in (b) is satisfied by semistable prime Fano $3$ -folds (§2) and [Reference Liu and Liu73] proves it holds for all prime Fano $3$ -folds (1.4).

   4. (d) A series $P\in {\mathcal F}_{3}$ may be realized by many deformation families (Tables 2 and 3) or by none (4.1(2)): ${\mathcal F}_{3}$ does not count the number of deformation families. This is a basic part of the classification problem, and it is fully understood only in the case of nonsingular Fano $3$ -folds and some specific cases with only $\tfrac {1}2(1,1,1)$ singularities.

2. 2. Existence and non-existence:

   1. (a) Proven cases of prime Fano $3$ -folds are sparse: the nonsingular Fano $3$ -folds of Picard rank $1$ (§4.1), Takagi’s Gorenstein-index- $2$ classification (§4.3), and a range of cases in low anticanonical codimension (§4.6) or of high Fano index (§4.4).

   2. (b) There is no reason why the Hilbert series of more general Fano $3$ -folds should appear in ${\mathcal F}_{3}$ or ${\mathcal F}_{4}$ , yet every Fano $3$ -fold we know does in fact appear on these lists.

   3. (c) Many locations in Table 1 are realized by a Fano $3$ -fold but not by a prime Fano $3$ -fold (Table 2 and §§4.2, 4.3.2, and 4.5.4).

   4. (d) We expect that many of the high codimension, lower genus Hilbert series are not realized by any Fano $3$ -fold. However, we only know one place in Table 1 where this is proven: Karzhemanov’s nonexistence result for genus $35$ (§4.2).

3. 3. The estimated anticanonical embedding $X\subset \mathbb {P}(a_0,\dots ,a_n)$ :

   1. (a) The Hilbert series $P_X$ of a Fano $3$ -fold X does not determine the weights $a_0,\dots ,a_n$ of its anticanonical embedding (Example 4.5).

   2. (b) The embedding weights given to each $P\in {\mathcal F}_{4}$ in the Grdb are only a suggestion. They are derived from known examples in low codimension (§3.1.1), an analysis of conjectured Gorenstein projections (§3.2) and, in harder cases, an analysis of singularities or the linear systems on possible K3 sections. Although the weights are often right, there is no reason why your X should be embedded in this way.

   3. (c) Even if the general member of a deformation family is $X\subset \mathbb {P}(a_0,\dots ,a_n)$ , there are likely to be degenerations in higher codimension (§3.1.1).

   4. (d) It can happen that $P\in {\mathcal F}_{3}$ is realized by two or more distinct deformation families whose general members embed in different codimensions (Example 4.5).

   5. (e) Some higher index Fano $3$ -folds lie in higher codimension than Grdb suggests (Example 4.3).

5.2. Nonexistence and other challenges

The Grdb is simply one way of assembling and presenting the vast amount of data associated with the classification of Fano $3$ -folds, and as such it naturally invites more questions than it answers. A selection of topics:

1. 1. Find $P\in {\mathcal F}_{3}$ not realized by a Fano $3$ -fold, or not realized by a prime Fano $3$ -fold.

2. 2. Can one show that each model for $P\in {\mathcal F}_{3}$ in codimension $5$ or $6$ is realized by a Fano $3$ -fold using similar birational methods as in codimension $4$ ?

3. 3. There are toric Fano $3$ -folds in high codimension: can projection from these realize Fano $3$ -folds for sequences of Hilbert series?

4. 4. Fano $3$ -folds with $|{-}K_X|$ empty are rare. The Grdb has $264$ semistable Hilbert series with linear coefficient zero (the left-hand column $g=-2$ of Table 1). The first are Iano–Fletcher’s example $X_{12,14}\subset \mathbb {P}(2,3,4,5,6,7)$ and the three families in codimension $4$ , studied by [Reference Altınok and Reid4], [Reference Coughlan and Ducat47], [Reference Papadakis78], [Reference Taylor109], though they remain to be fully understood. The next candidate lies in codimension 6, namely, $X\subset \mathbb {P}(2,3,4^2,5^2,6^2,7^2)$ , and has no Type I or Type II projection attack.

5. 5. The Grdb is constructed with Gorenstein projection in mind, and the Fano $3$ -fold database includes that data; click [Reference Brown and Kasprzyk22]. Sarkisov links provide another connection between Fano $3$ -folds, and projection is often the first step of a Sarkisov link. Classification attacks often exploit such links, [Reference Takeuchi108] for example, but does it even make sense to describe a web of Sarkisov links overlaying the Grdb? (cf. [Reference Brown, Buczyński and Kasprzyk16]).

6. 6. The bounds of [Reference Kawamata68] work over any field k of characteristic $0$ , not necessarily algebraically closed. Thus, for example, the Grdb makes sense over $k={\mathbb {Q}}$ (the Type I Ansatz 3.2 is defined over ${\mathbb {Q}}$ , but that is not clear for more complicated unprojection constructions), and the generic fibers of relatively three-dimensional Mori fiber spaces also have relative Hilbert series in the Fano $3$ -fold database. Compare the general conditions and classifications of [Reference Codogni, Fanelli, Svaldi and Tasin42], [Reference Codogni, Fanelli, Svaldi and Tasin43].

5.3. Closing repetition of the main warning

It is easy to mistake the Fano $3$ -fold database for a classification of prime Fano $3$ -folds. It is not that classification: that classification does not yet exist. It is instead the classification of genus–basket pairs that satisfy certain conditions of geometric origin, or equivalently it is a list of the rational functions they determine by the plurigenus formula.

The confusion arises in part because each rational function is presented as though it is the Hilbert series of a Fano $3$ -fold $X\subset \mathbb {P}(a_0,\ldots ,a_n)$ embedded by its total anticanonical ring in weighted projective space with given weights. These weights are sometimes accurate, and in all known examples, they appear at least as a subset of the generators, but they are not proven to be necessary and should be treated only as a convenient informed estimate. There is no claim that a Fano $3$ -fold exists with this data, nor that a particular one you may be considering is necessarily embedded anticanonically as indicated here. Moreover, a single Hilbert series may be realized by more than one family of varieties, and these multitudes may lie in different ambient weighted projective spaces.