3.1 Construction

Let $\bar {X} \subset w\mathbb {P}^7$ be a quasi-smooth codimension 4 $\mathbb {Q}$ -Fano 3-fold and $\bar {Z} \subset w\mathbb {P}^6$ a codimension 3 $\mathbb {Q}$ -Fano 3-fold, both of index $\iota =1$ , and suppose that $\bar {X}$ is obtained as Type I unprojection of Z at a divisor $D \cong w \mathbb {P}^2$ embedded as a complete intersection inside $\bar {Z}$ (for a detailed study of Type I unprojections; see [Reference Kustin and Miller27, Reference Papadakis and Reid35, Reference Papadakis32, Reference Brown, Kerber and Reid9]). For W a set of seven positive nonzero integers, call $x_0,x_1,x_2,y_1,y_2,y_3,y_4,s$ the coordinates of $w\mathbb {P}^7 = \mathbb {P}^7(1,W)$ , and consider $\gamma $ the $\mathbb {Z}/2\mathbb {Z}$ -action on $w\mathbb {P}^7$ that changes sign to the coordinate of $x_0$ of weight 1, that is,

(3.1) $$ \begin{align} \gamma \colon \left( x_0, x_1, x_2, y_1, y_2, y_3, y_4, s \right) \mapsto \left( -x_0, x_1, x_2, y_1, y_2, y_3, y_4, s \right) \;. \end{align} $$

Here, the divisor D is defined by the ideal . Provided that the equations of $\bar {X}$ are invariant under $\gamma $ , it is possible to perform the quotient of $\bar {X}$ by $\gamma $ . The 3-fold obtained as such is Fano, has terminal singularities, is quasi-smooth, has an ambient space that is $\mathbb {P}^7(2,W)$ and has index $\iota =2$ (cf [Reference Campo13, Lemmas 3.1, 3.2, 3.3, 3.4]). The coordinates of $\mathbb {P}^7(2,W)$ are $\xi ,x_1,x_2,y_1,y_2,y_3,y_4,s$ , where . This construction can be summarised by the following diagram.

(3.2)

The key point is to find an appropriate index 1 double cover $\bar {X}$ for X of index 2. The double cover is ramified on the half elephant $\left | -\frac {1}{2} K_X \right |$ , and the equations of X are inherited from the ones of $\bar {X}$ (see [Reference Campo13, Theorem 1.1]).

Recall from [Reference Brown, Kerber and Reid9] that there are between two and four different deformation families for $\bar {X}$ of index 1 sharing the same Hilbert series. They are derived from just as many so-called formats for the $5\times 5$ antisymmetric graded matrix M whose five maximal Pfaffians determine the equations of $\bar {Z}$ . These formats, defined by specific constraints on the polynomial entries of the matrix M (cf [Reference Brown, Kerber and Reid9, Definition 2.2]), are called Tom and Jerry formats. Accordingly, $\bar {X}$ is said to be either of Tom type or of Jerry type (cf [Reference Campo12, Definition 2.2]). Not all the formats are compatible with the double-cover construction, that is, not all formats descend to index 2. However, exactly one Tom format always does. A criterion to determine which formats in index 1 become formats in index 2 via the double-cover construction can be found in [Reference Campo13, Theorems 4.3, 5.1]; this exhausts all the Tom and Jerry formats.

When a Type I unprojection is employed, we only focus on $\bar {X}$ of Tom type, which represent 32 families out of the 34 we study in this paper. In this context, $\bar {X}$ is a general member in its Tom family, provided the $\gamma $ -invariance. Thus, X is general under the above constraints.

A close analysis of the unprojection equations of $\bar {X}$ , and therefore of X, gives crucial insights on the behaviour of the birational links from X. The unprojection equations of $\bar {X}$ are of the form $s y_i = g_i(x_0, x_1, x_2, y_1, \dots , y_4)$ for $1 \leq i \leq 4$ (cf [Reference Papadakis32, Theorem 4.3]). Consequently, four of the nine equations defining X are of the form

$$ \begin{align*} s y_i = g_i(\xi, x_1, x_2, y_1, \dots, y_4) \qquad \text{for }1 \leq i \leq 4 \,; \end{align*} $$

with a little abuse of notation, we call them unprojection equations as well. The point $\mathbf {p}_s \in X$ is a Type I centre ([Reference Brown, Kerber and Reid9, Theorem 3.2] and [Reference Campo13, Lemma 3.4]). Note that the unprojection variable s appears linearly in the unprojection equations of $\bar {X}$ and X, hence the point $\mathbf {p}_s \in X$ is called a linear cyclic quotient singularity in [Reference Duarte Guerreiro21, Definition 3.28]; we will occasionally use this nomenclature in the following. The equations of X are therefore of the form

$$ \begin{align*} \left( \operatorname{Pf}_i = sy_j-g_j=0, \,\, 1\leq i \leq 5,\, 1\leq j\leq 4 \right) \end{align*} $$

for $\operatorname {Pf}_i,\,g_j \in \mathbb {C}[\xi ,x_1,x_2,y_1,\ldots ,y_4]$ . From [Reference Campo12, Lemma 3.5], we have that each unprojection equation of the index 1 double cover $\bar {X}$ contains at least one monomial only in the orbinates $x_0,x_1,x_2$ . The following lemma is a consequence of the constraint of having $\bar {X}$ invariant under the action $\gamma $ .

Proof. Since $\iota _X=2$ , for terminality reasons the basket of singularities of X consists only of cyclic quotient singularities with odd order (cf [Reference Suzuki38, Lemma 1.2 (3)]). Consequently, the weight of the unprojection variable s is always odd. By hypotheses, the weights of $\xi $ and $x_1$ are even, so the orbinate $x_2$ has odd weight. By direct observation, at least three of the coordinates $y_1,\dots ,y_4$ have odd weight. Note that, if $s y_i$ for some $1 \leq i \leq 4$ has odd weight, the corresponding $g_i$ does not contain any pure monomial in $\xi , x_1$ ; that is, $\operatorname {wt}(y_i)$ must be odd for it to happen.

We briefly recall the notation of unprojection equations necessary to this proof; for the full details of the construction, we refer to [Reference Papadakis32, Section 5.3] and [Reference Campo12, Appendix]. To fix ideas, suppose that the matrix M is in Tom $_1$ format; the proof for the other Tom formats is analogous. Such matrix is of the form

$$ \begin{align*} M =\left( \begin{array}{c c c c} p_1 & p_2 & p_3 & p_4 \\ & q_1 & q_2 & q_3 \\ & & q_4 & q_5 \\ & & & q_6 \end{array} \right) \end{align*} $$

for $p_i \not \in I_D$ and $q_i \in I_D$ homogeneous polynomials in the given degrees of M. Without loss of generality, we can fill the entries of M with linear monomials when the Tom constraints and the degree prescription on M allow us to do so (for details, see [Reference Brown, Kerber and Reid9, Section 6.2]). In this context, at least three of the $q_i$ can be filled with one of the $y_1,\dots ,y_4$ . By homogeneity of the Pfaffians, at least two of the $p_i$ have even degree; thus, $\xi $ and $x_1$ can occupy those entries (not necessarily linearly). Define the matrices $N_j$ as

$$ \begin{align*} N_j = \left( \begin{array}{c c c c} p_1 & p_2 & p_3 & p_4 \\ & \alpha^j_{23} & \alpha^j_{24} & \alpha^j_{25} \\ & & \alpha^j_{34} & \alpha^j_{35} \\ & & & \alpha^j_{45} \end{array} \right), \end{align*} $$

where $\alpha ^j_{kl}$ is the coefficient of $y_j$ in $q_{kl}$ . Let Q be the $4 \times 4$ matrix $ Q = \left ( \operatorname {Pf}_i(N_j) \right )_{i,j=1 \dots 4}$ , where $\operatorname {Pf}_i(N_j)$ is calculated by excluding the $(i + 1)$ -th row and column of $N_j$ for $i=1, \dots 4$ . The polynomials $g_i$ in the right-hand side of the unprojection equations are given by $g_i = \frac {1}{p_i} \hat {Q}_i$ , where $\hat {Q}_i$ is the $3 \times 3$ matrix obtained deleting the i-th row and column of Q.

Since some of the polynomials $q_i$ are quasi-linear in the $y_i$ , the polynomials $p_i$ having even degrees are multiplied by 1 at least twice in the Pfaffians of the matrices $N_j$ (also note that the matrices we consider have always weights as in [Reference Campo12, Equation (4.2) and (4.3)]). Thus, at least three entries of each row of the $4\times 4$ matrix Q contain a monomial purely in $\xi , x_1$ . So, at least two of these entries are included in a $3\times 3$ minor of Q. Then, by taking the determinant of $\hat {Q}_i$ we have that $g_i$ has a monomial purely in $\xi , x_1$ up to a $p_i$ factor, which gets simplified in the definiton of $g_i$ thanks to [Reference Papadakis32, Lemma 5.3]. At worst, the two entries of $\hat {Q}_i$ containing pure monomials in $\xi , x_1$ are all concentrated in a $2\times 3$ block (or two $1\times 3$ blocks). This implies that only three of the $g_i$ have the desired monomials $f_i(\xi ,x_1)$ . Otherwise, all $g_i$ contain $f_i(\xi ,x_1)$ .

By Lemma 3.2, we have that the four unprojection equations of X are of the form

(3.3) $$ \begin{align} \begin{cases} sy_i = f_i(\xi,x_1)+h_i(\xi,x_1,x_2,y_1\ldots,y_4),\quad 1 \leq i \leq 3 \\ sy_4 = h_4(\xi,x_1,x_2,y_1\ldots,y_4), \end{cases} \end{align} $$

where $f_i$ is not identically zero by quasi-smoothness of X and $\operatorname {wt}(y_4)$ is even.

The remaining two families (GRDB ID #39569 and #39607) that we investigate here are constructed in a similar fashion as in diagram (3.2): In these cases, their respective double covers $\bar {X}$ only have Type II $_2$ centres, or worse (see [Reference Papadakis34, Section 1], [Reference Papadakis33, Definition 2.2, Theorem 2.15] for Type II unprojections, and [Reference Brown, Kerber and Reid9, Section 3.6], [Reference Reid and Ohno37] for Type II centres). Hence, the Tom and Jerry formats are not applicable as $\bar {X}$ is obtained via Type II $_2$ unprojections from a Fano hypersurface: See [Reference Campo13, Section 7] for the construction of these double covers.

Among the Hilbert series listed in the GRDB, there is also the smooth #41028; there are two distinguished deformation families associated to this Hilbert series: These are the classical examples of a divisor of bidegree $(1,1)$ in $\mathbb {P}^2 \times \mathbb {P}^2 \subset \mathbb {P}^8$ and $\mathbb {P}^1 \times \mathbb {P}^1 \times \mathbb {P}^1 \subset \mathbb {P}^7$ (cf [Reference Brown, Kerber and Reid9, Section 2]). These are rational and therefore nonsolid. This confirms the statement of Theorem 2.2 also in the case of #41028.

3.2 Lift under the Kawamata blowup

We want to initiate a birational link by blowing up a cyclic quotient singularity on X. In order to understand what the equations of the blowup Y are, we first explain how the sections in $H^0(X,mA)$ lift to Y under a Kawamata blowup of $\mathbf {p_s} \in X$ , for $m\geq 1$ .

Recall that by [Reference Papadakis32, Theorem 4.3], [Reference Brown, Kerber and Reid9, Theorem 3.2] and [Reference Campo13, Lemma 3.4], $\mathbf {p_s}$ is a linear cyclic quotient singularity of X as in [Reference Duarte Guerreiro21, Definition 2.6.1]; locally, $\mathbf {p_s} \sim \frac {1}{a_s}(a_0,a_1,a_2)$ . In the notation introduced in Subsection 3.1, $a_0,a_1,a_2$ are the weights of the orbinates $\xi ,x_1,x_2$ .

We can assume (cf [Reference Suzuki38, Lemma 1.2 (3)]) that $a_0=2$ is equal to the Fano index of X. Moreover, since $\mathbf {p_s}$ is terminal, $\gcd (a_s,a_0a_1a_2)=1$ and, in particular, there is $k \in \mathbb {Z}$ such that $ka_0 \equiv 1 \pmod {a_s}$ . Denote by $\overline {a} < a_s$ the unique remainder of $a \bmod {a_s}$ . Then,

$$ \begin{align*} \mathbf{p_s} \sim \frac{1}{a_s}(1,\overline{ka_1},\overline{ka_2}) \sim \frac{1}{a_s}(1,\overline{ka_1},a_s-\overline{ka_1})=\frac{1}{a_s}\bigg(1,\frac{a_1}{2},a_s-\frac{a_1}{2}\bigg) \; .\ \end{align*} $$

Lemma 3.4. Let $\varphi \colon Y \rightarrow X$ be the Kawamata blowup centred at $\mathbf {p_s}$ . Then

$$ \begin{align*} &s \in H^0(Y,-m_1K_Y+m_2E) && \text{for some }m_1,\,m_2>0;\\ &\xi \in H^0(Y,-K_Y) , \qquad x_1 \in H^0(Y,-mK_Y) && \text{for some }m>0;\\ &z_i \in H^0(Y,-m_iK_Y-n_iE) && \text{for some }m_i,\,n_i>0, \end{align*} $$

where $z_i$ is one of $x_2,\, y_1,\,y_2,\,y_3,\,y_4$ .

Proof. Recall that for the Kawamata blowup $\varphi \colon Y \rightarrow X$ we have $K_Y=\varphi ^*(K_X)+\frac {1}{r}E $ , where $\frac {1}{r}$ is its discrepancy and r is the index of the cyclic quotient singularity. Let x be one of the sections above and $\nu $ be its vanishing order at E. Then,

$$ \begin{align*} x &\in H^0\,\left(Y,\frac{a_i}{2}\varphi^*(-K_X)-\nu E \right) \\[5pt] &= H^0\,\left(Y,\frac{a}{2}(-K_Y+\frac{1}{a_s}E)-\nu E \right)\\[5pt] &= H^0\,\left(Y,-\frac{a}{2}K_Y+\frac{a-2a_s\nu}{2a_s}E \right) \;. \end{align*} $$

By the description of $\mathbf {p_s}$ , it follows that $\xi $ vanishes at E with order $\nu =\frac {1}{a_s}$ . Similarly, the vanishing order of $x_1$ and $x_2$ at E is $\nu =\frac {a_1}{2a_s}$ and $\nu =\frac {2a_s-a_1}{2a_s}$ . Hence,

$$ \begin{align*} \xi \in H^0\,(Y,-K_Y), \quad x_1 \in H^0\,\left(Y,-\frac{a_1}{2}K_Y \right), \quad x_2 \in H^0\,\left(Y,-\frac{a_2}{2}K_Y-\frac{1}{2}E \right).\ \end{align*} $$

On the other hand, $ s \in H^0(Y,\frac {a_s}{2}\varphi ^*(-K_X))= H^0(Y,-\frac {a_s}{2}K_Y+\frac {1}{2}E)$ .

Starting from equation (3.3), we compute the vanishing order of $y_i,\, 1 \leq i\leq 3$ at E. The monomials of $f_i(\xi ,x_1)$ are of the form $\xi ^{\alpha }x_1^{\beta }$ and if $d_i$ is the homogeneous degree of the equation $sy_i-g_i=0$ , we have $\alpha a_0 + \beta a_1 = d_1$ . On the other hand such monomials vanish at E with order

$$ \begin{align*} \alpha\cdot \frac{1}{a_s} + \beta \cdot \frac{a_1}{2a_s} = \frac{d_i}{2a_s} < 1 \;. \end{align*} $$

Hence, $\xi ^{\alpha }x_1^{\beta }$ is pulled back by $\varphi $ to $\xi ^{\alpha }x_1^{\beta }t^{d_i/2a_s}$ , where is the exceptional divisor. Since $\frac {d_i}{2a_s}<1$ , when saturating the ideal of Y with respect to t, we find that $\xi ^{\alpha }x_1^{\beta }t^{d_i/2a_s}$ becomes $\xi ^{\alpha }x_1^{\beta }$ . Hence, $sy_i-g_i = 0$ is a divisor in $|-\frac {d_i}{2}K_Y|$ and we conclude that $ y_i \in H^0\,\left (Y, -\frac {d_i-a_s}{2}K_Y-\frac {1}{2}E\right ) $ .

For $y_4$ , since $sy_4-h_4=0$ contains no pure monomials in $\xi ,x_1$ we can only say that $y_4$ vanishes at E with order $\nu $ at least $\frac {\operatorname {wt}(y_4)+a_s}{2a_s}$ ; therefore, $\frac {\operatorname {wt}(y_4)-2a_s\nu }{2a_s} \leq -\frac {1}{2}$ with equality when the vanishing order is exactly $\frac {\operatorname {wt}(y_4)+a_s}{2a_s}$ . In other words,

$$ \begin{align*} y_4 \in H^0\,\left(Y,-\frac{\operatorname{wt}(y_4)}{2}K_Y-m_4E \right), \quad m_4 \geq \frac{1}{2}>0 \;. \\[-45pt] \end{align*} $$

The Lemma below follows from Lemma 3.4 and Lemma 3.2.

Biratonal links and toric varieties.

Let T be a rank 2 toric variety (up to isomorphism) for which the toric blowup $ \Phi \colon T \rightarrow \mathbb {P}$ restricts to the unique Kawamata blowup $Y \rightarrow X$ centred at $\mathbf {p_s} \in X$ . Then, $\operatorname {Cl}(T)= \mathbb {Z}[\Phi ^*H]+\mathbb {Z}[E]$ , where H is the generator of the class group of $\mathbb {P}$ and $E = \Phi ^{-1}(\mathbf {p_s})$ is the exceptional divisor. Notice that $\mathbf {p_s}$ is not in the support of H. Then, the Cox ring of T is

$$ \begin{align*} \operatorname{Cox}(T) = \bigoplus_{(m_1,m_2) \in \mathbb{Z}^2} H^0(T,m_1\Phi^*H+m_2E) \;. \end{align*} $$

Since T is toric, the Cox ring of T is isomorphic to a (bi)-graded polynomial ring, in this case,

$$ \begin{align*} \operatorname{Cox}(T) \simeq \mathbb{C}[t,\xi,x_1,x_2,y_1\ldots,y_4,s], \; \end{align*} $$

where the grading comes from the $\mathbb {C}^* \times \mathbb {C}^*$ -action defining $\mathbb {P}$ and $\Phi $ , respectively. We denote by $\mathbb {R}_+\Big [\Big (\begin {smallmatrix} \omega _1 \\ \omega _2 \end {smallmatrix}\Big )\Big ]$ the ray generated by a divisor D in the linear system $\Big |\mathcal {O}_T\Big (\begin {smallmatrix} \omega _1 \\ \omega _2 \end {smallmatrix}\Big )\Big |$ . Over $N^1(T)_{\mathbb {R}} \simeq \mathbb {R}^2$ , we can depict the rays generated by the sections of Lemma 3.4 as in Figure 1. The movable and effective cones of T in $N^1(T)_{\mathbb {R}}$ are

$$ \begin{align*} \operatorname{Mov}(T)=\mathbb{R}_+[M_1]+\mathbb{R}_+[M_2] \subset \operatorname{Eff}(T)= \mathbb{R}_+[E]+\mathbb{R}_+[E'] \;. \end{align*} $$

Notice that, since T is a normal simplicial toric variety, it follows from [Reference Cox, Little and Schenck20, Theorem 15.1.8 and Theorem 15.1.10], respectively, that both $\operatorname {Eff}(T)$ and $\operatorname {Mov}(T)$ are closed in the Euclidean topology. According to Lemma 3.4, notice that the rays $E,\, M_1$ and $-K_Y$ cannot coincide. On the other hand, it can happen that some of the other rays do coincide.

Figure 1 A representation of the Mori chamber decomposition of T. The outermost rays generate the cone of pseudo-effective divisors of T and in red it is represented the subcone of movable divisors of T.

We run several 2-ray games on T. We divide the construction of the elementary Sarkisov links in two main cases, depending roughly on the Mori chamber decomposition of T in Figure 1, namely the behaviour of its movable cone of divisors near the boundary of the effective cone of divisors of T. We consider three cases:

1. 1. Fibration: The class of $M_2$ is linearly equivalent to a rational multiple of $E'$ .

2. 2. Divisorial Contraction: The class of $M_2$ is not linearly equivalent to a rational multiple of $E'$ . Moreover, we contract the divisor $E'$ :

   1. (a) To a point: No generator class of the Cox ring of T is linearly equivalent to a rational multiple of $M_2$ , or

   2. (b) To a rational curve: There is one (and only one) generator class of the Cox ring of T which is linearly equivalent to a rational multiple of $M_2$ .

The diagram below sets the notation used in the rest of the paper. Note that we only consider Fano 3-folds not in Table 1. In each case, we have a birational link at the level of toric varieties

and $\dim \mathcal {F}' \leq \dim T'$ , with equality if and only if we are in the second case. We restrict the diagram above to the Kawamata blowup $ \varphi \colon Y \subset T \rightarrow X \subset \mathbb {P}$ in order to obtain a birational link between 3-folds.

It follows from [Reference Hu and Keel23, Proposition 2.11] that the birational contractions of a Mori dream space are induced from toric geometry. By [Reference Hu and Keel23, Proposition 1.11], one can always carry out a classical Mori program for any divisor on a Mori dream space Y. In particular, when $\rho (Y)=2$ it is called a 2-ray game. We refer the reader to [Reference Corti17] for the precise definition of 2-ray game.

The next three lemmas describe the nature of the restriction of the maps $\tau ,\tau _i$ to the birational link relative to $X \subset \mathbb {P}$ .

Proof. In this proof, call $z_i$ all variables that are not $t,s,\xi ,x_1$ . The small modification $\tau _0 \colon T \dashrightarrow T_1$ can be decomposed as

where

, in coordinates the map $\alpha _0$ is

$$ \begin{align*} \alpha_0 \colon T &\rightarrow \mathcal{F}_0 \\ (t,s,\xi,x_1, \ldots z_i \ldots) &\mapsto (\xi,x_1, \ldots u_j \ldots) \end{align*} $$

and $u_j$ is some monomial which is a multiple of $z_i$ and of either t or s.

Notice that the irrelevant ideal of T is $(t,s) \cap (\xi , x_1, \ldots z_i, \ldots )$ . Hence, $\alpha _0$ contracts the locus $(z_i = 0) \subset T$ . This is indeed a small contraction since the ray $\mathbb {R}_+\Big [\Big (\begin {smallmatrix} 2 \\ 1 \end {smallmatrix}\Big )\Big ]$ is in the interior of $\operatorname {Mov}(T)$ . Now, we restrict this small contraction to Y. By Lemma 3.4, we know that $z_i$ are the sections in $H^0\Big (Y,-\frac {a_i}{2}K_Y-n_iE\Big )$ with $n_i> 0$ . On the other hand, by Lemma 3.5, the Pfaffian equations $\operatorname {Pf}_j$ must vanish identically and the remaining equations are $f_j(\xi ,x_1) = 0$ for $1 \leq j \leq 3$ . However, this is empty by Lemma 3.3.

Proof. By assumption, there are curves $C_i \subset Y_i$ and $C_{i+1} \subset Y_{i+1}$ such that the diagram

is a small $\mathbb {Q}$ -factorial modification. Clearly, we have $K_{Y_i}\cdot C_i < 0$ . Indeed, by Lemma 3.5, $C_i$ intersects $-K_{Y_i}$ transversely since $C_i \in \mathrm {Bs}|-m_i K_{Y_i}-n_iE|$ for some $m_i,\,n_i> 0$ ; then, the claim follows. On the other hand, there exists a divisor class

$$ \begin{align*} L \sim -m_1K_Y-m_2E \, \in \, \mathbb{R}_+[A_i] + \mathbb{R}_+[E'] \end{align*} $$

for which $L \cdot C_{i+1}>0$ , where $A_i$ and $E'$ are as in Figure 1 (implying, in particular, $m_i>0$ ). Moreover,

$$ \begin{align*} A_i \sim -n_1K_Y-n_2E,\,\, n_i>0 \end{align*} $$

and $A_i \cdot C_{i+1} = 0$ . Then, $m_2A_i-n_2L \sim (m_2n_1-m_1n_2)(-K_{Y_{i+1}})$ . Notice that $m_2n_1-m_1n_2=0$ if and only if $A_i \sim mL$ for some nonzero $m \in \mathbb Q$ , which is not possible since they have different intersections with $C_{i+1}$ . Additionally, $m_2n_1-m_1n_2>0$ since L is in the cone $\mathbb {R}_+[A_i]+\mathbb {R}_+[E']$ but is not linearly equivalent to any nonzero rational multiple of $A_i$ . In particular,

$$ \begin{align*} -K_{Y_{i+1}} \cdot C_{i+1} = \frac{1}{m_2n_1-m_1n_2}(m_2A_i\cdot C_{i+1}-n_2L\cdot C_{i+1}) <0 \end{align*} $$

since $A_i$ and L were chosen so that $A_i\cdot C_{i+1} = 0$ and $L\cdot C_{i+1}>0$ .

Lemma 4.8 [Reference Duarte Guerreiro21, Lemma 2.5.7].

Let $\sigma \colon X \dashrightarrow X'$ be an elementary birational link between $\mathbb {Q}$ -Fano 3-folds initiated by a divisorial extraction $\varphi \colon E \subset Y \rightarrow X$ . Then, there is a birational map $\Psi \colon Y \dashrightarrow X'$ which is the composition of small $\mathbb {Q}$ -factorial modifications followed by a divisorial contraction $\varphi ' \colon E' \subset Y' \rightarrow X'$ with discrepancy

$$ \begin{align*} a= \frac{m_2}{m_1n_2-m_2n_1}, \end{align*} $$

where $m_i$ and $n_i$ are positive rational numbers such that $\varphi ^{\prime *}(-K_{X'}) \sim -m_1K_Y-m_2E$ and ${E' \sim -n_1K_Y-n_2E}$ .

In practice, to successfully run this game we need to guarantee that each step $\tau _i \colon T_i \dashrightarrow T_{i+1}$ of the birational link contracts finitely many curves, with the exception of $\tau _i$ an isomorphism on $Y_i \subset T_i$ . However, in each case it is possible to retrieve explicitly the loci contracted and extracted by the maps $\tau _i$ .

In the rest of this section, we will present several tables containing information about the links for each family examined. We specify the Type I centre whose blowup initiates the link in cases where there is more than one Type I centre; for instance, we write “#39993 $1/5$ ” instead of just “#39993” if the link starts with the Kawamata blowup of a $1/5$ singularity.

Figure 2 The possible effective cones of T ending with a fibration to a rational curve $B' = \operatorname {Proj} \mathbb {C}[z_4,z_5]$ . The variables $z_1,\ldots ,z_5$ are $y_1,\ldots ,y_4,x_2$ up to permutation.

4.1 Case I: fibrations

In each case in Figure 2, whose notation we refer to, the movable cone of T is not strictly contained in the effective cone of T. Indeed, the rays generated by $z_4$ and $z_5$ are both in $\partial \operatorname {Mov}(T)$ and $\partial \operatorname {Eff}(T)$ . Hence, we have a diagram of toric varieties

where $\Phi $ is a divisorial contraction, $\Phi '$ is a fibration into $\mathcal {F}'\simeq \operatorname {Proj}\mathbb {C}[z_4,z_5] \simeq \mathbb {P}^1$ . The map $\tau $ is a small $\mathbb {Q}$ -factorial modification. We restrict $\Phi \colon T \rightarrow \mathbb {P}$ to be the unique Kawamata blowup $\varphi \colon E \subset Y \rightarrow \mathbf {p_s} \in X$ . By Lemmas 4.5 and 4.7, the map $\tau |_Y \colon Y \dashrightarrow Y' $ is an isomorphism followed by a finite sequence of isomorphisms or flips. Referring to the notation in Figure 3, by assumption the rays $\mathbb {R}_+[z_4]$ and $\mathbb {R}_+[z_5]$ are given by linearly dependent vectors $v_4$ and $v_5$ in $\mathbb {Z}^2$ . Let B be the matrix whose columns are the vectors $v_4$ and $v_5$ . Then, there is $A \in \operatorname {GL}(2,\mathbb {Z})$ such that

$$ \begin{align*} A\cdot B = \begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix} \;. \end{align*} $$

Then, performing a row operation on the grading of $T'$ via the matrix A, $T'$ is isomorphic to a toric variety with $\mathbb {C}^* \times \mathbb {C}^*$ -action given by

Figure 3 The possible effective cones of T ending with a contraction of the divisor $D_{z_5} \colon (z_5=0)$ . In cases II.a.1, II.a.2, II.a.3 the divisor $D_{z_5}$ is contracted to the point $\mathcal {F}'=\operatorname {Proj} \mathbb {C}[z_4]$ and in cases 3d and 3e to the rational curve $\mathcal {F}'=\operatorname {Proj} \mathbb {C}[z_3,z_4]$ . The variables $z_1,\ldots ,z_5$ are $y_1,\ldots ,y_4,x_2$ up to permutation. The exceptional divisor of the Kawamata blowup $\varphi \colon Y \rightarrow X$ is $E\colon (t=0)$ .

where the column vectors are ordered clockwise. The irrelevant ideal of $T'$ is $(t , s, \xi , x_1, z_1, z_2, z_3) \cap (z_4, z_5)$ . The map $\Phi '$ can be realised as

$$ \begin{align*} \Phi' \colon T' &\longrightarrow \mathcal{F}' \\ (t , s, \xi, x_1, z_1, z_2, z_3, z_4, z_5) & \longmapsto (z_4,z_5) \end{align*} $$

and the fibre of $\Phi '$ over each point is isomorphic to $\mathbb {P}(\lambda _0,\ldots ,\lambda _6)$ . The following Table 2 shows what the restrictions of $\Phi '$ to $Y'$ are.

Table 2 Elementary birational links to a fibration (cases in Table 1 excluded). The family #39607 is embedded in the weighted projective space $\mathbb {P}^7(2,3,3,4,5,5,6,7)$ with coordinates $\xi ,u,z,y,v,s_0,s_1,s_2$ .

Example 4.9. Let be a quasi-smooth member of the family #39961, with $x_2, \xi , y_4, y_1, x_1, y_2, y_3, s$ the homogeneous variables of $\mathbb {P}$ .

We take the toric blowup $\Phi \colon T \rightarrow \mathbb {P}$ centred at $\mathbf {p_s}=(0: \cdots : 0 :1) \in \mathbb {P}$ and restrict it to the unique Kawamata blowup $\varphi \colon Y \rightarrow X$ centred at $\mathbf {p_s}$ . The point $\mathbf {p_s}$ is a cyclic quotient singularity of type $\frac {1}{5}(1,2,3)$ and local analytical variables $\xi , x_1, x_2$ called the orbinates. By Lemma 3.4, in order to restrict $\Phi $ to the the Kawamata blowup of X at $\mathbf {p_s}$ we need for T to have a certain bigrading: The one relative to #39961 is listed in Table 2. In particular,

$$ \begin{align*} \operatorname{Mov}(T)=\langle \big(\begin{smallmatrix} 7 \\ 4 \end{smallmatrix}\big),\big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) \rangle \subset \langle \big(\begin{smallmatrix} 0 \\ 1 \end{smallmatrix}\big),\big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) \rangle = \operatorname{Eff}(T) \subset \mathbb{R}^2.\ \end{align*} $$

Now, we run the 2-ray game on T following the movable cone in Case I.b of Figure 2, and then we restrict it to Y. After saturation with respect to the new variable t, the equations defining $Y:=\Phi ^{-1}_* X \subset T$ are the following:

$$ \begin{align*} \begin{cases} x_2 \xi y_1 - y_1^2 - y_4 x_1 + x_2 y_2 =0\\ - x_2^7 t^3 - x_2^3 \xi^2 t - x_2 \xi^3 + x_2^2 \xi y_1 t + x_2 \xi x_1 - x_2^2 y_2 t - y_1 x_1 + y_4 s =0\\ \xi^3 y_4 - y_4^4 t^3 + x_2 \xi y_2 - y_1 y_2 + x_2 y_3 =0\\ x_2^2 \xi^3 t + x_2 y_4^2 y_1 t^3 + x_2^2 \xi x_1 t + x_2^2 y_4 x_1 t^2 - x_1^2 - y_1 s =0\\ x_2^6 y_1 t^3 - x_2 y_4^4 t^4 + \xi^3 y_1 - y_4^3 y_1 t^3 - x_2 y_4^2 x_1 t^2 + x_2^2 y_3 t + x_1 y_2 =0\\ x_2^5 y_4 y_1 t^3 - y_4^5 t^4 - y_4^3 x_1 t^2 + x_2 y_4 y_3 t + y_2^2 - y_1 y_3 =0\\ x_2^7 y_1 t^4 - x_2^6 x_1 t^3 + x_2^2 \xi^4 t - x_2^2 y_4^4 t^5- x_2 y_4^3 y_1 t^4 + x_2^2 \xi y_4 x_1 t^2\\ - x_2^2 y_4^2 x_1 t^3 - \xi^3 x_1 + y_4^3 x_1 t^3 - x_2 y_4 y_1 x_1 t^2 - x_2 y_4^2 y_2 t^3 + x_2^3 y_3 t^2 + y_2 s =0\\ x_2^5 y_4 x_1 t^3 - x_2^6 y_2 t^3 - x_2 \xi y_4^4 t^4 + y_4^4 y_1 t^4 - x_2 \xi y_4^2 x_1 t^2 \\+ y_4^2 y_1 x_1 t^2 - \xi^3 y_2 + y_4^3 y_2 t^3 + x_2^2 \xi y_3 t - x_2 y_1 y_3 t - x_1 y_3 =0\\ - x_2^6 \xi^3 t^3 + x_2^6 y_4^3 t^6 - x_2^6 \xi x_1 t^3 + x_2^2 \xi^2 y_4^3 t^4 + x_2^5 y_1 x_1 t^3 - \xi^6 + \xi^3 y_4^3 t^3 - x_2 \xi y_4^3 y_1 t^4 \\+ x_2^2 \xi^2 y_4 x_1 t^2 - y_4^4 x_1 t^4 - x_2 \xi y_4 y_1 x_1 t^2 - x_2 \xi y_4^2 y_2 t^3 + x_2 y_4^3 y_2 t^4 \\- y_4^2 x_1^2 t^2 + y_4^2 y_1 y_2 t^3 + x_2 y_4 x_1 y_2 t^2 + x_2 x_1 y_3 t - y_3 s =0. \end{cases} \end{align*} $$

These consist of the five maximal Pfaffians $\operatorname {Pf}_i=0$ together with the four unprojection equations $sy_j-g_j=0$ . Also, Y is inside the rank 2 toric variety T whose irrelavant ideal is $(t,s) \cap (\xi ,x_1,y_3,y_2,y_1,x_2,y_4)$ . By Lemma 4.5, the map $\tau _0 \colon T \dashrightarrow T_1$ restricts to an isomorphism on Y and therefore we can assume $Y\subset T_1$ , where $T_1$ has irrelevant ideal $I_1=(t,s,\xi ,x_1) \cap (y_3,y_2,y_1,x_2,y_4)$ . Crossing the $y_3$ -wall contracts the locus $\mathbb {V}(y_2,y_1,x_2,y_4) \subset T_1$ . Its restriction to $Y_1$ is $\mathbb {V}(y_3,y_2,y_1,x_2,y_4)\subset \mathbb {V}(I_1)$ , where $I_1$ is the ideal defining $Y_1 \subset T_1$ . Hence, the contraction happens away from $Y_1$ , so $\tau _1$ restricts to an isomorphism $Y_1 \cong Y_2$ . Just as before we just set $Y_1 \cong Y_2 \subset T_2$ , where $I_2=(t,s,\xi ,x_1,y_3) \cap (y_2,y_1,x_2,y_4)$ . Crossing the $y_2$ -wall restricts to a contraction of

and an extraction of

. Hence, the map $\tau _2$ corresponds to a toric flip over a point, denoted by $(-7,-1,1,5)$ and $Y_3 \subset T_3$ with irrelevant ideal $I_3=(t,s,\xi ,x_1,y_3,y_2) \cap (y_1,x_2,y_4)$ . The map $\tau _3$ restricts to

$$ \begin{align*} (x_2=y_4=0)|_{Y_3}= (y_1=x_2=y_4=0) \subset \mathbb{V}(I_3), \end{align*} $$

where $I_3$ is the ideal defining $Y_3 \subset T_3$ . Therefore, the small contraction $\tau _3 \colon T_3 \dashrightarrow T_4$ happens away from $Y_3$ . Finally, we have a map $\varphi ' \colon Y_4 \rightarrow \mathcal {F}' = \operatorname {Proj} \mathbb {C}[x_2,y_4]$ . A generic fibre is a surface S given by

$$ \begin{align*} t\xi^3 + t^2y_2 + t\xi y_2 - y_2^2 + t^2\xi y_1 - \xi^3y_1 - ty_2y_1 - \xi y_1y_2 - t^2y_1^2 + y_2y_1^2 =0 \end{align*} $$

inside $\mathbb {P}(1_t,1_{\xi },1_{y_1},2_{y_2})$ . Hence, S is a del Pezzo surface of degree 2. Therefore, we have the diagram

where $\varphi ' \colon Y_4 \rightarrow \mathbb {P}(1_{x_2},2_{y_4})$ is a del Pezzo fibration of degree two.

4.2 Case II: divisorial contractions

In each case in Figure 3, whose notation we refer to, the movable cone of T is strictly contained in the effective cone of T. Hence, we have a diagram of toric varieties

where $\Phi $ and $\Phi '$ are divisorial contractions and $\tau $ is a small $\mathbb {Q}$ -factorial modification. As usual, we restrict $\Phi \colon T \rightarrow \mathbb {P}$ to be the unique Kawamata blowup $\varphi \colon E \subset Y \rightarrow \mathbf {p_s} \in X$ . By Lemmas 4.5 and 4.7, the map $\tau |_Y \colon Y \dashrightarrow Y' $ is an isomorphism followed by a finite sequence of isomorphisms or flips. Referring to the notation in Figure 3, by assumption the rays $\mathbb {R}_+[z_4]$ and $\mathbb {R}_+[z_5]$ are given by linearly independent vectors $v_4$ and $v_5$ in $\mathbb {Z}^2$ . Let $-d \not = 0$ be the determinant of the matrix B whose columns are $v_4$ and $v_5$ ; without loss of generality, we can assume that $d>0$ (cf [Reference Ahmadinezhad2, Lemma 2.4]). Then, there is $A \in \operatorname {GL}(2,\mathbb {Z})$ such that

$$ \begin{align*} A\cdot B = \begin{pmatrix} 0 & -d \\ d & 0 \end{pmatrix} \;. \end{align*} $$

After a row operation on the grading of $T'$ via the matrix A, $T'$ is isomorphic to a toric variety with $\mathbb {C}^* \times \mathbb {C}^*$ -action given by

(4.1)

where every $2 \times 2$ minor is nonpositive. In cases II.a, the irrelevant ideal of $T'$ is $(t , s, \xi , x_1, z_1, z_2, z_3) \cap (z_4, z_5)$ , and $(t , s, \xi , x_1, z_1, z_2) \cap (z_3, z_4, z_5)$ in cases II.b.

Cases II.a. Divisorial contractions to a point.

We have $\kappa _6 \not = 0$ by assumption. The map $\Phi '$ is given by the sections multiples of $D_{z_4}$ (in the notation of [Reference Brown and Zucconi10, Section 4.1.7]) and is a divisorial contraction to a point in $\mathcal {F}'$ . This can be realised as

$$ \begin{align*} \Phi' \colon T' &\longrightarrow \mathcal{F}' \\ (t , s, \xi, x_1, z_1, z_2, z_3, z_4, z_5) & \longmapsto (tz_5^{\frac{\kappa_0}{d}},sz_5^{\frac{\kappa_1}{d}},\xi z_5^{\frac{\kappa_2}{d}},x_1 z_5^{\frac{\kappa_3}{d}}, z_1z_5^{\frac{\kappa_4}{d}}, z_2 z_5^{\frac{\kappa_5}{d}}, z_3z_5^{\frac{\kappa_6}{d}},z_4) \;. \end{align*} $$

Assume X is not the family #39660. Then, $\mathcal {F}'= \mathbb {P}(\lambda _0, \ldots , \lambda _6, d)$ and $\Phi '$ contracts the divisor $E' \colon (z_5=0) \simeq \mathbb {P}(\kappa _0, \ldots ,\kappa _6)$ to the point $\mathbf {p}=(0:\cdots : 0: 1) \in \mathcal {F}'$ . In particular, $\mathbf {p}$ is smooth whenever $d=1$ . The contraction $\Phi '$ restricts to a weighted blowup $\varphi ' \colon Y' \rightarrow X'$ as in Table 3.

Table 3 We list the restriction $\Phi '|_{Y'}=\varphi '$ and the model to which $\varphi '$ contracts to. In each case, $\varphi '$ is a weighted blowup with weights $\frac {1}{r}(a_1,\ldots ,a_l)$ with $r\geq 1$ . For case #39890, $\varphi '$ is a contraction to a hyperquotient singularity; in the other instances, $\varphi '$ is a contraction to a Gorenstein point. The family #39569 is embedded in the weighted projective space $\mathbb {P}^7(2,3,5,6,7,7,8,9)$ with coordinates $\xi ,z,u,y,v,s_0,s_1,s_2$ .

The following explicit example illustrates a link that terminates with a 3-fold in a fake weighted projective space.

Example 4.10. Consider the deformation family with ID #39660 in Tom format. This is

with homogeneous coordinates $\xi ,\,y_4,\,y_1,\,x_2,\,y_2,\,y_3,\,x_1,\,s$ . By Lemma 3.4, we know that the weighted blowup of $\mathbf {p_s}=(0:\ldots : 0 : 1) \in \mathbb {P}$ restricts to the Kawamata blowup $\varphi \colon Y \subset T \rightarrow X \subset \mathbb {P}$ provided that the bigrading of T is

up to multiplication by a matrix in $\operatorname {GL}(2,\mathbb {Z})$ (see [Reference Ahmadinezhad2, Lemma 2.4 and Example 2.13]). We have a sequence of small $\mathbb {Q}$ -factorial modifications $\tau \colon T \dashrightarrow T'$ where $T'$ has the same Cox ring as T but whose irrelevant ideal is $(t,s,\xi ,x_1,y_3,x_2,y_2) \cap (y_1,y_4)$ . Let $T^{y_4}$ be the same rank 2 toric variety as $T'$ , except that $y_4$ has bidegree $(-1,0)$ instead of $(-2,0)$ . Then, there is a map $q \colon T^{y_4} \rightarrow T'$ given by $y_4 \mapsto y_4^2$ and $T^{y_4}$ can be seen as a double cover of $T'$ . On the other hand, $T'=\operatorname {Proj} \mathbb {C}[t,s,\xi ,x_1,y_3,x_2,y_2,y_1,y_4]^{\boldsymbol {\mu }_2}$ , where $\boldsymbol {\mu }_2$ is the multiplicative cyclic group of order 2 acting on $\mathbb {C}[t,s,\xi ,x_1,y_3,x_2,y_2,y_1,y_4]$ via

$$ \begin{align*} \epsilon \cdot (t,s,\xi,x_1,y_3,x_2,y_2,y_1,y_4) = (t,s,\xi,x_1,y_3,x_2,y_2,y_1,\epsilon y_4) \;. \end{align*} $$

Hence, q is the quotient map of $T^{y_4}$ under this group action. Consider the map

$$ \begin{align*} \Phi^{y_4} \colon T^{y_4} &\longrightarrow \mathbb{P}(1,9,1,6,3,2,2,1) \\ (t , s, \xi, x_1, y_3, x_2, y_2, y_1, y_4) & \longmapsto (t y_4^3,s y_4^{10},\xi y_4, x_1 y_4^6, y_3 y_4^2, x_2 y_4, y_2 y_4, y_1) \;. \end{align*} $$

This is the map given by the sections multiples of $H^0(T^{y_4},\mathcal {O}_{T^{y_4}}(0,1))$ and it corresponds to a divisorial contraction to a point in $\mathbb {P}(1,9,1,6,3,2,2,1)$ . Since $\Phi ^{y_4}$ is an isomorphism away from the locus $(y_4=0)$ , the action on $T^{y_4}$ is carried through to $\mathbb {P}(1,9,1,6,3,2,2,1)$ .

The action at each point of this weighted projective space is then given by

$$ \begin{align*} \epsilon \cdot (t: s: \xi: x_1: y_3: x_2: y_2: y_1: y_4) &= (\epsilon t: s: \epsilon \xi : x_1 : y_3 : \epsilon x_2 : \epsilon y_2 : y_1) \\ &=( t:\epsilon s: \xi : x_1 :\epsilon y_3 : \epsilon x_2 : \epsilon y_2 : \epsilon y_1) \;. \end{align*} $$

Then, $\Phi ^{y_4}$ descends to a divisorial contraction of quotient spaces

$$ \begin{align*} \Phi' \colon T'=T^{y_4}/\boldsymbol{\mu}_2 \rightarrow \mathbb{P}(1,9,1,6,3,2,2,1)/\boldsymbol{\mu}_2 \;. \end{align*} $$

Let $Y'$ be the image of $\tau $ restricted to Y. The map $\Phi '$ restricts to a divisorial contraction $\varphi ' \colon Y' \rightarrow X'$ to the point $\mathbf {p_{y_1}} \sim \frac {1}{2}(1,1,1) \in X'$ in the complete intersection of a quartic and a quintic $X' \subset \mathbb {P}(1,1,3,2,2,1)/\boldsymbol {\mu }_2$ with homogeneous variables $t,\xi , y_3, x_2, y_2, y_1$ which is given by the equations

$$ \begin{align*} \begin{cases} t^4 + t \xi^3 - y_3 y_1 - y_2^2 = 0 \\ t x_2^2 + t y_1^4 - \xi^5 - \xi^2 x_2 y_1 + y_3 y_2 = 0 \;. \end{cases} \end{align*} $$

Note that $X'$ is invariant under the action of $\boldsymbol {\mu }_2$ . By Lemma 4.8, the map $\varphi '$ has discrepancy $\frac {1}{2}$ and its exceptional divisor is isomorphic to $\mathbb {P}^2$ . Hence, $\varphi '$ is the Kawamata blowup centred at $\mathbf {p_{y_1}}\in X'$ .

Cases II.b. Divisorial contractions to a rational curve.

By assumption, $\kappa _6 = 0$ in the grading of $T'$ in equation (4.1). The map $\Phi '$ is given by the sections that are multiples of the divisor $(z_4 =0)$ , and is a divisorial contraction to the rational curve $\Gamma ' := \operatorname {Proj} \mathbb {C}[\lambda _6,d]$ . By Lemma 3.5, $\Phi '$ restricts to a divisorial contraction to a curve at the level of 3-folds. By terminality, it follows that $\gcd (\lambda _6,d)=1$ since, otherwise, the curve $\Gamma '$ would be a line of singularities. In fact, by looking at each case we can see that $d=1$ . The map $\Phi '$ is then

$$ \begin{align*} \Phi' \colon T' &\longrightarrow \mathcal{F}' \\ (t , s, \xi, x_1, z_1, z_2, z_3, z_4, z_5) & \longmapsto (tz_5^{\kappa_0},sz_5^{\kappa_1},\xi z_5^{\kappa_2},x_1 z_5^{\kappa_3}, z_1z_5^{\kappa_4}, z_2 z_5^{\kappa_5}, z_3,z_4) \;. \end{align*} $$

Hence, the divisor $(z_5=0) \subset T'$ which is isomorphic to $\mathbb {P}(\kappa _0,\ldots ,\kappa _5)$ is contracted to $\Gamma ' \subset \mathbb {P}(\lambda _0,\ldots ,\lambda _6,1)$ . In Table 4, we summarise the details regarding all cases falling into II.b.

Table 4 We list the restriction $\Phi '|_{Y'}=\varphi '$ and the model to which $\varphi '$ contracts to. In each case, $\varphi '$ is a contraction to a curve $\Gamma '$ inside a Fano 3-fold $X'$ .

#40672.

Let with homogeneous coordinates $y_1,y_2,x_2,\xi ,y_4,x_1,y_3,s$ be a quasi-smooth member of the family #40672 as in Section 3 and [Reference Campo13].

After the Kawamata blowup of the point $\mathbf {p_s} \sim \frac {1}{3}(1,1,2)$ and the isomorphism $\tau _0$ in the birational link, we have

The small $\mathbb {Q}$ -factorial modification $\tau _1$ contracts $\mathbb {P}(3,3,1,1) \subset T_1$ to a point and extracts $\mathbb {P}(1,1,1,2) \subset T_2$ , where $T_2$ has the same Cox ring as $T_1$ but irrelevant ideal $(t,s,\xi ,x_1,y_3)\cap (y_1,y_2,x_2,y_4)$ . This restricts to the flip

where $C_1 \colon (t+s+2 x_1^3=\xi -x_1=0) \subset \mathbb {P}(3,3,1,1)$ , that is, $C_1 \simeq \mathbb {P}(3_t,1_{x_1})$ and $C_2 \colon (y_1+x_2=y_2+x_2=0) \subset \mathbb {P}(1,1,1,2)$ , that is $C_2 \simeq \mathbb {P}(1_{x_2},2_{y_4})$ . Also, $-K_{Y_1} \sim (\xi =0) \subset Y_1$ and $E \sim (t=0) \subset Y_1$ .

It is clear that $-K_{Y_1}\cdot C_1=\frac {1}{3}$ . On the other hand, we have $-K_{Y_2}\sim D_{y_3} - D_{x_2}$ . Hence, $-K_{Y_2} \cdot C_2 = -\frac {1}{2}$ . Consider the map $\Phi '$

$$ \begin{align*} \Phi' \colon T_2 &\rightarrow \mathbb{P}(1,1,1,2) \\ (t,s,\xi,x_1,y_3,y_1,y_2,x_2,y_4) &\mapsto (y_1,y_2,x_2,y_4) \;. \end{align*} $$

Then $\Phi '$ is a fibration whose fibres are isomorphic to $\mathbb {P}(1,2,1,1,1)$ . Consider the two consecutive projections $X \dashrightarrow X' \dashrightarrow X"$ , where $X \dashrightarrow X'$ is the projection away from $\mathbf {p}_s \in X$ and $X' \dashrightarrow X"$ is the projection away from $\mathbf {p}_{y_3} \in X'$ . The equations of $X"$ are given explicitly by

Let $\Gamma \subset \mathbb {P}(1,1,1,2)$ be defined by the three $2\times 2$ minors of the matrix above. The curve $\Gamma $ has generically two irreducible and reduced components, which can be easily checked on Magma: One is rational, and the other has genus 4. We have that $\Phi '|_{Y_2}$ is a divisorial contraction to $\Gamma \subset \mathbb {P}(1,1,1,2)$ . Since $\Gamma $ has two irreducible components, $\rho _X =2$ .

#40671.

This case is completely analogous to the previous one. We give the $2\times 3$ matrix whose $2\times 2$ minors define $\Gamma \subset \mathbb {P}(1,1,1,2)$ . This is

The curve $\Gamma $ has one rational irreducible component, and another irreducible component with genus 5. As before, $\rho _X =2$ .

4.2.1 Wrapping up: conclusion of nonsolidity proof

We finalise the proof of nonsolidity for the families whose link terminates with a divisorial contraction.

We prove that all models $X'$ in Tables 3 and 4 admit a structure of a strict Mori fibre space, with the possible exception of two.

Lemma 4.11. Let X be a family in Table 4. Then X is nonsolid.

Proof. We consider the projection $\pi \colon X' \dashrightarrow \mathbb {P}^1$ in each case. Then, the generic fibre $S \subset X'$ is a surface for which $-K_S \sim \mathcal {O}_S(1)$ by adjunction since $X'$ has Fano index 2. The conclusion follows from Corollary 2.3.

We show next that the families #39890 and #39928 in Table 3 admit singular birational models in a family whose general member is a quasi-smooth complete intersection of a cubic and a quartic $X^{\prime }_{3,4} \subset \mathbb {P}(1,1,1,1,2,2)$ . It was shown by Corti and Mella [Reference Corti and Mella18] that its general member is birigid. In this case, we were not able to show that these are nonsolid: These are the nonbirationally rigid families mentioned in Theorem 1.2.

Lemma 4.12. Let X be a quasi-smooth member of either family #39890 or #39928 in Table 3. Then X is birational to a nonquasismooth complete intersection of a cubic and a quartic $X^{\prime }_{3,4} \subset \mathbb {P}(1,1,1,1,2,2)$ .

Example 4.13. Consider the singular complete intersection

$$ \begin{align*} \begin{cases} -t \xi z_2 + t z_2^2 + t z_1 + \xi u + z_1 y + u y =0\\ t^3 \xi - \xi^4 + t y^3 - t^3 z_2 - t \xi u - \xi z_1 y - t u y + t u z_2 - z_1^2 =0 \end{cases} \end{align*} $$

inside $\mathbb {P}(1,1,1,1,2,2)$ with homogeneous variables $t,\,\xi ,\,y,\,z_2,\,z_1,\,u$ . Then, there is a weighted blowup from the point $\mathbf {p_{z_2}}$ that initiates a birational link to a quasismooth family in Tom format with ID #39928 (see Table 3).

Lemma 4.14. Let X be a quasi-smooth member of a family in Table 3 not treated in Lemma 4.12. Then, the model $X'$ is birational to a del Pezzo fibration.

Proof. Notice that each model $X'$ in Table 3 has Fano index 1. General (quasi-smooth) members in each of these families of $X'$ have been treated in [Reference Brown and Zucconi10], [Reference Okada29] and [Reference Campo12]. In particular, it was shown that these are nonsolid. The same results extend to terminal families provided that the key monomials are still present in the equations of $X'$ , which is the case for each model.