Proof. By a result of Mather and Yau [Reference Mather and Yau40] (see also [Reference Greuel, Lossen and Shustin25, Th. 2.26]), f and g are contact equivalent if and only if the Tjurina algebras $T_f$ and $T_g$ are isomorphic. A simple computation shows that $T_f \cong T_F$ and $T_g \cong T_G$ , which proves the proposition for contact equivalence.

The proof for right equivalence is similar. Namely, we use a statement analogous to [Reference Mather and Yau40]: two elements $h, k \in \mathbb C\{\boldsymbol {x}\}$ with zero constant term are right equivalent if and only if the Milnor algebras $M_h$ and $M_k$ are isomorphic as algebras over the ring $\mathbb C\{t\}$ , where t acts on $M_h$ (resp. $M_k$ ) by multiplying by h (resp. k) (see [Reference Greuel, Lossen and Shustin25, Th. 2.28]).

Proof. By [Reference Iskovskikh and Prokhorov30, Prop. 2.1.2], we have natural isomorphisms $\operatorname {Pic}(\mathbb P) \cong H^2(\mathbb P^{\mathrm {an}}_{\mathrm{top}}, {\mathbb Z})$ and $\operatorname {Pic}(X) \cong H^2(X^{\mathrm {an}}_{\mathrm{top}}, {\mathbb Z})$ , where $\mathbb{P}^{\mathrm {an}}_{\mathrm{top}}$ , respectively $X^{\mathrm {an}}_{\mathrm{top}}$ denotes the underlying topological space of the analytification of $\mathbb{P}$ , respectively $X$ . By [Reference Mavlyutov41, Proposition 1.4], the restriction map $H^{i}(\mathbb{P}^{\mathrm{an}}_{\mathrm{top}}, \mathbb{C}) \rightarrow H^{i}(X^{\mathrm{an}}_{\mathrm{top}}, \mathbb{C})$ is an isomorphism for is an isomorphism for $i < n$ . By [Reference Zhang53, Corollary 1], $X$ and $\mathbb{P}$ are simply connected. The proposition now follows from universal coefficienttheorems.

Proof. Replacing $D$ by a suitable multiple, it suffices to consider the case where $D$ isCartier. There are no non-zero effective principal divisors on a normal projective variety.Therefore, since $X$ has Picard number one, either $D$ or $-D$ is ample. Since $D$ intersectssome closed integral curve positively, $D$ is ample by Kleiman’s criterion. Again byKleiman’s criterion, $D$ intersects $C$ positively.

Proof. First, we show that Definition 2.17 is the $\boldsymbol {w}$ -blowup when X is the affine space $\mathbb A^n = \operatorname {Spec} \mathbb C[x_1, \ldots , x_n]$ . Let $S = \mathbb C[u, \boldsymbol {x}]$ be the $\mathbb Z$ -graded $\mathbb C$ -algebra with grading $(-1, \operatorname {wt} x_1, \ldots , \operatorname {wt} x_n)$ for $u, \boldsymbol {x}$ . Let $S_{\geq 0}$ be the nonnegatively graded part of S. By definition of the geometric quotient, the weighted blowup of $\mathbb A^n$ is given by $\operatorname {Proj} S_{\geq 0} \to \mathbb A^n$ . The $\mathbb Z_{\geq 0}$ -graded $\mathbb C$ -algebra isomorphism

$$\begin{align*}\begin{gathered} S_{\geq0} \to R_{\mathbb A^n}\\ u \mapsto t^{-1}, \quad x_i \mapsto t^{\operatorname{wt} x_i} x_i \end{gathered} \end{align*}$$

induces an isomorphism $\operatorname {Proj} R \to \operatorname {Proj} S_{\geq 0}$ over $\mathbb A^n$ .

We show that Definition 2.17 is the $\boldsymbol {w}$ -blowup for any X. Define $N = n \cdot \operatorname {lcm}(w_1, \ldots , w_n)$ . If $M = x_1^{a_1} \cdot \ldots \cdot x_n^{a_n}$ is any monomial such that $\sum a_i w_i> N$ , then M is divisible by $x_k^{N / (n w_k)}$ for some k. It follows that the Nth Veronese subring $R_X^{(N)}$ of $R_X$ is generated by its degree $1$ part $(R_X)_N$ . Therefore, $\operatorname {Proj} R_X$ is isomorphic over X to $\operatorname {Bl}_{(R_X)_N} X$ , where $\operatorname {Bl}_{(R_X)_N} X \to X$ is blowup of X along $(R_X)_N$ . Since the intersection of $\operatorname {Spec} (R_{\mathbb A^n})_N$ and X is $\operatorname {Spec} (R_X)_N$ , we find that $\operatorname {Bl}_{(R_X)_N} X$ is the strict transform of X under the blowup of $\mathbb A^n$ along $(R_{\mathbb A^n})_N$ , which coincides with the closure of the inverse image of $X \setminus \mathbb V(x_1, \ldots , x_n)$ in $\operatorname {Bl}_{(R_{\mathbb A^n})_N} \mathbb A^n$ .

We show that Definition 2.18 is the $\boldsymbol {w}$ -blowup. We similarly prove that $\operatorname {Projan} \mathcal B_X$ is the closure of the inverse image of $X \setminus \{\boldsymbol {0}\}$ in $\operatorname {Projan} \mathcal B_D$ . Now, it suffices to note that the analytification of $\operatorname {Proj} R_{\mathbb A^n} \to \mathbb A^n$ is $\operatorname {Projan} \mathcal B_{\mathbb C^n} \to \mathbb C^n$ .

Construction 2.25. Let X be a Fano variety embedded in a weighted projective space $\mathbb P$ , where X is a Mori fiber space, and let $Y_0 \to X$ be a divisorial contraction from a projective $\mathbb Q$ -factorial variety Y. By [Reference Ahmadinezhad and Kaloghiros2, Lem. 2.9], the divisorial contraction $Y_0 \to X$ can be part of a Sarkisov link only if $Y_0$ is a Mori dream space.

By [Reference Hu and Keel27, Prop. 2.11], we can embed a Mori dream space $Y_0$ into a projective toric variety $T_0$ with cyclic quotient singularities such that the Mori chambers of $Y_0$ are unions of finitely many Mori chambers of $T_0$ . Moreover, we can embed $Y_0$ in such a way that $Y_0$ is given by a homogeneous ideal $I_Y$ in $\operatorname {Cox} T_0$ , and the toric 2-ray link

restricts to a 2-ray link

where each $Y_i \subseteq T_i$ is given by the same ideal $I_Y \subseteq \operatorname {Cox} T_0 = \cdots = \operatorname {Cox} T_r$ , and $W_i \subseteq \mathcal W_i$ is given by the ideal $I_Y \cap \mathbb C[\nu _0, \ldots , \nu _s]$ , where $\mathcal W_i$ is given by $\operatorname {Proj} \mathbb C[\nu _0, \ldots , \nu _s]$ for some polynomials $\nu _j \in \operatorname {Cox} T_0$ that depend on i (see [Reference Ahmadinezhad and Zucconi4, Rem. 4]). In this case, $\operatorname {Cox}(T_0) / I_Y$ is a Cox ring for $Y_0$ and we say that $I_Y$ 2-ray follows $T_0$ . In contrast to [Reference Ahmadinezhad and Zucconi4, Def. 3.5], we emphasize the ideal $I_Y$ , since there could be other ideals I satisfying $\mathbb V(I_Y) = \mathbb V(I)$ such that the toric 2-ray link restricts to a 2-ray link for $I_Y$ but not for I.

Note that some of the small birational maps $T_i \dashrightarrow T_{i+1}$ may restrict to isomorphisms $Y_i \to Y_{i+1}$ . If all the varieties $Y_i$ are terminal and the anti-canonical divisor $-K_{Y_0}$ of $Y_0$ is inside the interior $\operatorname {int} (\operatorname {Mov} Y_0)$ of the movable cone, then the 2-ray link for $Y_0$ is a Sarkisov link (see [Reference Ahmadinezhad and Kaloghiros2, Lem. 2.9]), otherwise it is called a bad link.

Proof. It is proved in [Reference Greuel, Lossen and Shustin25, Th. 2.47] that there exist unique $g \in \mathbb C[[x, \boldsymbol {y}]]$ and $h \in \mathbb C[[\boldsymbol {y}]]$ , where the multiplicity of g is at least 2, such that $f = (x + g)^2 + h$ . Moreover, it is proved that the power series g and h are absolutely convergent around the origin, and the multiplicity of h is at least 2.

By the Weierstrass preparation theorem (see [Reference Greuel, Lossen and Shustin25, Th. 1.6]), there exist a unique unit $v \in \mathbb C\{x, \boldsymbol {y}\}$ and a unique $p \in \mathbb C\{\boldsymbol {y}\}$ such that $x + g = v(x + p)$ .

Proof. Taking the degree d part of the coefficient of $x^{i+1}$ in $f = (x + g)^2 + h$ where $i \geq 0$ , we find Equation (3.1). Taking all degree d terms of $f = (x + g)^2 + h$ that are not divisible by x, we find Equation (3.2). Taking the degree d part of the coefficient of $x^{i+1}$ in $x + g = v(x + p)$ where $i \geq 0$ , we find Equation (3.4), and taking all degree d terms not divisible by x, we find Equation (3.3).

Notation 3.4. Let X be the subscheme of $\mathbb P(1, 1, 1, 1, 3)$ , with variables $x, y, z, t, w$ , defined by f, where

(3.5) $$ \begin{align} \begin{aligned} f & = -w^2 + x^4 (t^2 + Q_2) \\ &\quad + x^3(4t^3a_0 + 4t^2a_1 + 2ta_2 + a_3) \\ &\quad + x^2(2t^4b_0 + 2t^3b_1 + 2t^2b_2 + 2tb_3 + b_4) \\ &\quad + x(2t^5c_0 + 2t^4c_1 + 2t^3c_2 + 2t^2c_3 + 2tc_4 + c_5) \\ &\quad + t^6d_0 + 2t^5d_1 + t^4d_2 + 2t^3d_3 + t^2d_4 + 2td_5 + d_6, \end{aligned} \end{align} $$

where the polynomials $a_j, b_j, c_j, d_j \in \mathbb C[y, z]$ and $Q_j \in \mathbb C[y, z, t]$ are homogeneous of degree j.

We define the following $11$ technical conditions, where $i \in \{1, 2, 3, 4\}$ :

1. (1) (This condition is always true).

2. (2) $Q_2 = 0$ .

3. (3) Condition (2) holds and $a_3 = 0$ .

4. (4) Condition (3) holds and $b_4 = a_2^2$ .

5. (5) Condition (4) holds and $c_5 = 2a_2b_3 - 4a_1a_2^2$ .

6. (6) Condition (5) holds and $d_6 = 2a_2c_4 + b_3^2 - 8a_1a_2b_3 - 2a_2^2b_2 + 4a_0a_2^3 + 16a_1^2a_2^2$ .

7. (7.i) Condition (6) holds and there exist polynomials $q, r, s, e \in \mathbb C[y, z]$ that are, respectively, homogeneous of degrees $i-1$ , $3-i$ , $4-i$ , $i+1$ , where $0$ is considered to be the only polynomial homogeneous of degree $-1$ , such that

   $$\begin{align*}\begin{aligned} a_2 & = qr,\\ b_3 & = qs + 4a_1qr,\\ c_4 & = 2a_1qs - 6a_0q^2r^2 + 8a_1^2qr + er,\\ d_5 & = 2b_2qs - 8a_1^2qs - es - b_1q^2r^2 + c_3qr. \end{aligned} \end{align*}$$

8. (8) Condition (7.1) holds and there exist a constant $A_0 \in \mathbb C$ and a polynomial $B_1 \in \mathbb C[y, z]$ homogeneous of degree $1$ such that

   $$\begin{align*}\begin{aligned} e_2 & = 4A_0r_2 + b_2 - 6a_1^2,\\ c_3 & = 6a_0s_3 - 4A_0s_3 + 4a_0a_1r_2 - 8A_0a_1r_2 + B_1r_2 + 2a_1b_2 - 4a_1^3,\\ d_4 & = -2s_3B_1 + 16r_2^2A_0^2 - 8b_2r_2A_0 + 16a_1^2r_2A_0 + 4b_1s_3\\ &\quad - 8a_0a_1s_3 - 2b_0r_2^2 + 2c_2r_2 + b_2^2 - 4a_1^2b_2 + 4a_1^4. \end{aligned} \end{align*}$$

Note that zero is homogeneous of every nonnegative degree, so, for example, in Condition (7.1), the term e can be zero.

Next, define the set of $11$ rational indices

$$\begin{align*}\mathrm{Inds} := \{1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7.1,\, 7.2,\, 7.3,\, 7.4,\, 8\}. \end{align*}$$

Let $\lfloor k\rfloor $ denote the greatest integer not greater than k. For every $k \in \mathrm {Inds}$ , let $R_k$ denote the $\mathbb C$ -algebra freely generated by the coefficients of the polynomials

• • $Q_2, a_i, b_i, c_i, d_i$ if $k \leq 6$ ,

• • $a_i, b_i, c_i, d_i, q, r, s, e$ if $k \in \{7.1,\, 7.2,\, 7.3,\, 7.4\}$ , and

• • $a_i, b_i, c_i, d_i, q, r, s, e, A_0, B_1$ if $k = 8$ ,

where we consider the coefficients to be variables satisfying Condition $(k)$ . Define

$$\begin{align*}F_k = \operatorname{Spec} \left( R_k[x, y, z, t, w] / (f) \right), \end{align*}$$

where $f \in R_k[x, y, z, t, w]$ is the polynomial in Equation (3.5). Let family k denote the set of fibers of $F_k \to \operatorname {Spec} R_k$ over closed points. We say that a general sextic double solid in family k satisfies a property if the property is satisfied by all the fibers of $F_k \to \operatorname {Spec} R_k$ over the closed points of some Zariski open dense set in $\operatorname {Spec} R_k$ . We say that an analytically very general sextic double solid in family k satisfies a property if there is a Zariski open dense subset U of $\operatorname {Spec} R_k$ such that the property is satisfied by all the fibers of $F_k \to \operatorname {Spec} R_k$ over the closed points of U that are in the complement of some countable union of closed analytic proper subsets.

Table 2

Dimension of the space of sextic double solids with an isolated $cA_{\lfloor k\rfloor }$

Proof. If Condition $(6)$ holds and $a_2 = b_3 = 0$ , then $a_3 = b_4 = c_5 = d_6 = 0$ . Let C be the curve defined by the ideal $(t, w, 2xc_4 + 2d_5)$ . Note that C contains $P_x$ . Taking partial derivatives, we see that every point of C is a singular point of X.

Proof. Let $D \in \mathbb C[y, z]$ be a common prime divisor of r and s or a common prime divisor of q and e. Then D divides $a_2, b_3, c_4, d_5$ , and $D^2$ divides $a_3, b_4, c_5, d_6$ . Let C be the curve defined by the ideal $(D, t, w)$ . Note that C contains $P_x$ . Taking partial derivatives, we see that X is singular at every point of C

Proof. Obvious.

Proof of Theorem A(b)

First, we prove that every sextic double solid $Y \subseteq \mathbb P(1, 1, 1, 1, 3)$ with a singular point P (not necessarily of type $cA_n$ ) is isomorphic to some X in Notation 3.4, with the isomorphism sending P to $P_x = [1, 0, 0, 0, 0]$ . For this, it suffices to note that Notation 3.4 describes all sextic double solids with a singular point at $P_x$ , and that we can move any point of Y to $P_x$ using an automorphism of $\mathbb P(1, 1, 1, 1, 3)$ . This proves the case $n = 1$ . For the rest of the proof, X is given by some f in Notation 3.4 with a (not necessarily isolated) $cA_n$ singularity at $P_x$ and n is at least $2$ .

Let $X^{\mathrm {an}}$ denote the analytification of X. By Propositions 2.4 and 2.5 and Corollary 2.6, after applying a suitable linear invertible coordinate change on $y, z, t$ , Condition (2) holds. This proves the case $n = 2$ . For the rest of the proof, Condition (2) holds and n is at least $3$ .

Let

$$\begin{align*}X_x = \operatorname{Spec} \left(\mathbb C[y, z, t, w] / (f(1, y, z, t, w))\right) \end{align*}$$

denote the affine open of X given by inverting x. Let $g \in \mathbb C\{y, z, t\}$ and $h \in \mathbb C\{y, z\}$ be the unique convergent power series of multiplicity at least 2 such that

$$\begin{align*}f(1, y, z, t, w) = -w^2 + (t + g)^2 + h. \end{align*}$$

Since by assumption $(X^{\mathrm {an}}, P_x)$ is a $cA_n$ singularity, Propositions 2.4 and 2.5 and Corollary 2.6 imply that $h_2 = \cdots = h_n = 0$ , where $h_j \in \mathbb C[x_3, x_4]$ is the homogeneous degree j part of h.

Using the explicit splitting lemma (Proposition 3.2), it is straightforward to compute that $h_2 = \cdots = h_n = 0$ is equivalent to satisfying Condition $(n)$ when $n \leq 6$ , even if $P_x$ is not an isolated singularity. This proves the cases $n \in \{3, \ldots , 6\}$ . For the rest of the proof, Condition (6) holds, $(X^{\mathrm {an}}, P_x)$ is an isolated $cA_n$ singularity, and n is at least $7$ .

By Lemma 3.7, $a_2 \neq 0$ or $b_3 \neq 0$ . Define q to be a homogeneous greatest common divisor of $a_2$ and $b_3$ . Define r and $s \in \mathbb C[y, z]$ to be the unique homogeneous polynomials such that

$$\begin{align*}\begin{aligned} a_2 & = qr,\\ b_3 & = qs + 4a_1qr. \end{aligned} \end{align*}$$

Then r and s are coprime. Using the explicit splitting lemma (Proposition 3.2), we compute that

$$\begin{align*}h_7 = q(r(-12a_0q^2rs+4b_2qs-2b_1q^2r^2+2c_3qr-2d_5) - s(2c_4-4a_1qs)). \end{align*}$$

Using Lemma 3.9, the equations $h_2 = \cdots = h_7 = 0$ imply the existence of a polynomial $e \in \mathbb C[y, z]$ such that

$$\begin{align*}\begin{aligned} c_4 & = 2a_1qs - 6a_0q^2r^2 + 8a_1^2qr + er,\\ d_5 & = 2b_2qs - 8a_1^2qs - es - b_1q^2r^2 + c_3qr. \end{aligned} \end{align*}$$

Therefore, $h_2 = \cdots = h_7 = 0$ implies Condition $(7.i)$ , where i is defined by

$$\begin{align*}i := \deg \gcd(a_2, b_3) + 1, \end{align*}$$

where $\deg \gcd (a_2, b_3)$ is the degree of a greatest common divisor of $a_2 \in \mathbb C[y, z]$ and $b_3 \in \mathbb C[y, z]$ . This proves $n = 7$ .

Next, we show that if $h_2 = \cdots = h_8 = 0$ and one of Conditions $(7.2)$ – $(7.4)$ holds, then r and s have a common prime divisor or q and e have a common prime divisor, which contradicts Proposition 3.8. In Condition $(7.2)$ , we calculate that $h_8 + e^2r^2$ is divisible by q, giving $r = Cq$ for some $C \in \mathbb C$ . Substituting into $h_8$ , we compute that $h_8 - 2qes^2$ is divisible by $q^2$ . Therefore, q and s have a common prime divisor, giving that r and s have a common prime divisor, a contradiction. Conditions $(7.3)$ and $(7.4)$ are similar.

Hence, if $h_2 = \cdots = h_8 = 0$ , then Condition $(7.1)$ holds. Using the explicit splitting lemma, we calculate $h_8$ , and using Lemma 3.9, we can show that $h_2 = \cdots = h_8 = 0$ implies Condition $(8)$ .

Proof of Theorem A(a)

Assume that X is a sextic double solid with an isolated $cA_n$ singularity where $n \geq 9$ . Using the notation in the proof of Theorem A(b), we find that Condition (8) holds and $h_2 = \cdots = h_9 = 0$ . Using the explicit splitting lemma, we compute $h_9$ , and using Lemma 3.9, we find that there exists $B_0 \in \mathbb C$ such that

$$\begin{align*}\begin{aligned} A_0 & = a_0,\\ B_1 & = b_1,\\ d_3 & = -s_3B_0 + 2b_0s_3 - 2a_0^2s_3 + c_1r_2 - 4a_0b_1r_2 \\ &\quad + 16a_0^2a_1r_2 + b_1b_2 - 4a_0a_1b_2 - 2a_1^2b_1 + 8a_0a_1^3,\\ c_2 & = r_2B_0 - 6a_0^2r_2 + 2a_0b_2 + 2a_1b_1 - 12a_0a_1^2. \end{aligned} \end{align*}$$

Substituting into f gives

$$\begin{align*}\begin{aligned} x^3a_3 + x^2b_4 + xc_5 + d_6 & = (s_3 + 2a_1r_2 + xr_2)^2,\\ x^3a_2 + x^2b_3 + xc_4 + d_5 & = (s_3 + 2a_1r_2 + xr_2)(-2a_0r_2 + b_2 - 2a_1^2 + 2xa_1 + x^2). \end{aligned} \end{align*}$$

Define the curve C by the ideal $(w, t, s_3 + 2a_1r_2 + xr_2)$ . Taking partial derivatives, we find that X is singular at every point of C, a contradiction.

Proof of Theorem A(c)

Let $X^{\mathrm {an}}$ denote the analytification of X. Using the explicit splitting lemma (Proposition 3.2), we can compute that the complex space germ $(X^{\mathrm {an}}, P_x)$ is isomorphic to $(\mathbb V(-w^2 + t^2 + h))$ , where $h \in \mathbb C[y, z]$ is zero or has multiplicity at least $\lfloor k\rfloor + 1$ . By Propositions 2.4 and 2.5 and Corollary 2.6, X has either a (possibly non-isolated) $cA_m$ singularity or the singularity $(\mathbb V(x_1^2 + x_2^2), \boldsymbol {0}) \subseteq (\mathbb C^4, \boldsymbol {0})$ at $P_x$ , where $m \geq \lfloor k\rfloor $ and $\mathbb C^4$ has variables $x_1, x_2, x_3, x_4$ .

Proof. Let $\widehat {\mathcal A}_k$ denote the set of closed points Q of $\operatorname {Spec} R_k$ such that the fiber of $F_k \to \operatorname {Spec} R_k$ over Q has a singular point at $P_t = [0, 0, 0, 1, 0]$ . We find

$$\begin{align*}f(P_t) = d_0, \quad \frac{\partial f}{\partial x}(P_t) = 2c_0, \quad \frac{\partial f}{\partial y}(P_t) = 2\frac{\partial d_1}{\partial y}, \quad \frac{\partial f}{\partial z}(P_t) = 2\frac{\partial d_1}{\partial z}, \quad \frac{\partial f}{\partial t}(P_t) = 6d_0. \end{align*}$$

By the Jacobian criterion ([Reference Liu39, Exer. 4.2.10]), $\widehat {\mathcal A}_k$ is the set of closed points of

$$\begin{align*}\mathcal A_k = \mathbb V_{\operatorname{Spec} R_k}\left(d_0, c_0, \frac{\partial d_1}{\partial y}, \frac{\partial d_1}{\partial z}\right). \end{align*}$$

We see that $\dim \mathcal A_k = \dim \operatorname {Spec} R_k - 4$ .

The $\mathbb C$ -algebra automorphism $x \mapsto x + \alpha _x t$ , $y \mapsto y + \alpha _y t$ , $z \mapsto z + \alpha _z t$ of $\mathbb C[x, y, z, t, w]$ defines a morphism

$$\begin{align*}\pi_{\mathcal A_k}\colon \operatorname{Spec} \mathcal A_k \times \operatorname{Spec} \mathbb C[\alpha_x, \alpha_y, \alpha_z] \to \operatorname{Spec} R_k \end{align*}$$

with closed image. The set of closed points Q of $\operatorname {Spec} R_k$ , where the fiber of $F_k \to \operatorname {Spec} R_k$ over Q has a singular point with t-coordinate nonzero, is precisely the set of closed points of the image of $\pi _{\mathcal A_k}$ . The image of $\pi _{\mathcal A_k}$ has codimension at least $1$ .

Proof. Let $P = [0, \beta , \gamma , 0, 0] \in \mathbb P(1, 1, 1, 1, 3)$ , where $(\beta , \gamma ) \in \mathbb C^2 \setminus \{(0, 0)\}$ . We find

$$\begin{align*}f(P) = d_6(P),\ \frac{\partial f}{\partial x}(P) = c_5(P),\ \frac{\partial f}{\partial y}(P) = \frac{\partial d_6}{\partial y}(P),\ \frac{\partial f}{\partial z}(P) = \frac{\partial d_6}{\partial z}(P),\ \frac{\partial f}{\partial t}(P) = 2d_5(P). \end{align*}$$

Define the linear polynomial $l = \gamma y - \beta z$ . By the Jacobian criterion ([Reference Liu39, Exer. 4.2.10]), P is a singular point of X if and only if the following divisibility constraint is satisfied:

(3.6) $$ \begin{align} l\text{ divides }c_5\text{ and }d_5\text{ and }l^2\text{ divides }d_6. \end{align} $$

The set of closed points $Q \in \operatorname {Spec} R_k$ , where the fiber of $F_k \to \operatorname {Spec} R_k$ over Q is singular at P for some $(\beta , \gamma ) \in \mathbb C^2 \setminus \{(0, 0)\}$ , coincides with the set of closed points of a closed subset $\mathcal B_k$ of $\operatorname {Spec} R_k$ . We show that $\dim \mathcal B_k$ is at most $\dim \operatorname {Spec} R_k - 2$ .

• • If $k \leq 4$ , then the $19$ coefficients of $c_5, d_5$ , and $d_6$ are algebraically independent in $R_k$ . By the divisibility constraint (3.6), $\dim \mathcal B_k = \dim \operatorname {Spec} R_k - 3$ .

• • If $k = 5$ , then the $20$ coefficients of $a_2, b_3, d_5$ , and $d_6$ are algebraically independent in $R_k$ . We have $c_5 = a_2(2b_3 - 4a_1a_2)$ . If l divides $c_5$ , then l divides $a_2$ or l divides $b_3 - 2a_1a_2$ . By the divisibility constraint (3.6), in both cases we have three less degrees of freedom. More formally, $\mathcal B_k$ is the union of the images of two morphisms, both having codimension exactly $3$ in $\operatorname {Spec} R_k$ . Therefore, $\dim \mathcal B_k = \dim \operatorname {Spec} R_k - 3$ .

• • If $k = 6$ , then the $23$ coefficients of $a_2, b_3, c_4, d_5$ are algebraically independent in $R_k$ . We have $c_5 = a_2(2b_3 - 4a_1a_2)$ and $d_6 = a_2 \cdot (2 c_4 + G) + b_3^2$ for a polynomial $G \in \mathbb C[y, z]$ homogeneous of degree $4$ which does not contain $c_4$ .

  If l divides $a_2$ , then using the divisibility constraint (3.6), we find that l divides $b_3$ . Now, $l^2$ divides $a_2$ or l divides $2 c_4 + G$ . So, there are three less degrees of freedom in choosing $a_2$ , $b_3$ , $c_4$ , and $d_5$ .

  If l does not divide $a_2$ , then l divides $b_3 - 2a_1a_2$ , so $b_3 = 2a_1a_2 + Ql$ for some homogeneous quadratic form $Q \in \mathbb C[y, z]$ . From $l \mid d_6$ , we find that l divides $c_4 - a_2b_2 + 2a_0a_2^2 + 2a_1^2a_2$ , so $c_4 = Cl + a_2b_2 - 2a_0a_2^2 - 2a_1^2a_2$ for some homogeneous cubic form $C \in \mathbb C[y, z]$ . From $l^2 \mid d_6$ , we find that l divides $C - 4Qa_1$ . Therefore, after fixing $a_0, a_1, a_2$ , and $b_2$ , there are at least two less degrees of freedom in choosing $b_3$ , $c_4$ , and $d_5$ .

  In both cases, we see that $\dim \mathcal B_k \leq \dim \operatorname {Spec} R_k - 2$ .

• • If $\lfloor k\rfloor = 7$ , then

  $$\begin{align*}\begin{aligned} c_5 & = 4q^2r(2s + a_1r),\\ d_5 & = -es + q(2b_2s - a_1^2s - 4b_1qr^2 + c_3r),\\ d_6 & = 4q(er^2 + q(s^2 + a_1rs - 8a_0qr^3 - b_2r^2 + a_1^2r^2)). \end{aligned} \end{align*}$$
  Let us consider f for a closed point in $\mathcal B_k$ . If $l \mid q$ , then since q and e are coprime, we have $l \mid r$ and $l \mid s$ , a contradiction. If $l \mid r$ , then since $l \mid d_6$ , we find $l \mid s$ , a contradiction. Therefore, l divides neither q nor r.

  So, l divides $2s + a_1r$ . Using $l^2 \mid d_6$ , we see that $l^2$ divides $-32a_0q^2r - 4b_2q + 3a_1^2q + 4e$ . After fixing $a_0, a_1, b_2, q$ , and r, we see that there are at least two less degrees of freedom in choosing s and e. So, we have $\dim \mathcal B_k \leq \dim \operatorname {Spec} R_k - 2$ .

• • If $k = 8$ , then

  $$\begin{align*}\begin{aligned} c_5 & = 2r_2(s_3 + 2a_1r_2),\\ d_5 & = r_2(r_2B_1 - 8s_3A_0 - 8a_1r_2A_0 + 6a_0s_3 - b_1r_2 + 4a_0a_1r_2 + 2a_1b_2 - 4a_1^3) \\ &\quad + s_3(b_2 - 2a_1^2),\\ d_6 & = r_2(8r_2^2A_0 + 4a_1s_3 - 8a_0r_2^2 + 4a_1^2r_2) + s_3^2. \end{aligned} \end{align*}$$
  We consider f for a closed point in $\mathcal B_k$ . If $l \mid r_2$ , then $l \mid s_3$ , a contradiction. So, l divides $s_3 + 2a_1r_2$ . Since l divides $d_6$ , we have $l \mid r_2^3(A_0 - a_0)$ . So, $A_0 = a_0$ . Since l divides $d_5$ , we see that $l \mid r_2^2(B_1 - b_1)$ . We find that the coefficients of f have at least two less degrees of freedom, namely $A_0 = a_0$ , and the polynomials $B_1 - b_1$ and $s_3 + 2a_1r_2$ have a common prime divisor. So, we have $\dim \mathcal B_k \leq \dim \operatorname {Spec} R_k - 2$ .

The $\mathbb C$ -algebra automorphism $x \mapsto x + \alpha y$ of $\mathbb C[x, y, z, t, w]$ defines a morphism

$$\begin{align*}\pi_{\mathcal B_k, 1}\colon \operatorname{Spec} \mathcal B_k \times \operatorname{Spec} \mathbb C[\alpha] \to \operatorname{Spec} R_k, \end{align*}$$

and the $\mathbb C$ -algebra automorphism $x \mapsto x + \alpha z$ of $\mathbb C[x, y, z, t, w]$ defines a morphism

$$\begin{align*}\pi_{\mathcal B_k, 2}\colon \operatorname{Spec} \mathcal B_k \times \operatorname{Spec} \mathbb C[\alpha] \to \operatorname{Spec} R_k. \end{align*}$$

Every closed point Q of $\operatorname {Spec} R_k$ , where the fiber of $F_k \to \operatorname {Spec} R_k$ over Q has a singular point different from $P_x$ with t-coordinate zero, belongs to the image of $\pi _{\mathcal B_k, 1}$ or $\pi _{\mathcal B_k, 2}$ . The union of the images of $\pi _{\mathcal B_k, 1}$ and $\pi _{\mathcal B_k, 2}$ has codimension at least $1$ .

Proof of Theorem A(d)

It follows from Lemmas 3.10 and 3.11 that a general sextic double solid in family k has exactly one singular point, namely the point $P_x$ . The singularity of X at $P_x$ is of type $cA_{\lfloor k\rfloor }$ if the homogeneous part $h_{\lfloor k\rfloor + 1} \in \mathbb C[y, z]$ of h is nonzero, where h is as in the proof of Theorem A(b). Since this is an open condition, a general sextic double solid in family k has a $cA_{\lfloor k\rfloor }$ singularity at $P_x$ .

Proof. Let X be a log terminal variety in family 7.4. The Cartier divisor $\mathbb V_X(t)$ is the sum of the two prime divisors $D_1 = \mathbb V(t, q - w)$ and $D_2 = \mathbb V(t, q + w)$ . Let $l \varepsilon \mathbb{C}[y,z]$ be a non-zero linear form that does not divide $q$ . Define the curve $C = V(q+w, x, l)$ . If $D_1$ is $\mathbb{Q}$ -Cartier, then $D_1\cdot C=0$ , which contradicts Proposition 2.12 and lemma 2.13. Therefore, neither $D_1$ nor $D_2$ is $\mathbb{Q}$ -Cartier.

Proof. Let U be any Zariski open set in the smooth locus of X that contains D. By Remark 2.11(1), we have an isomorphism of class groups $\operatorname {Cl}(X) \cong \operatorname {Cl}(U)$ . Since U is smooth, we have an isomorphism $\operatorname {Cl}(U) \cong \operatorname {Pic}(U)$ . It follows from the proof of [Reference Namikawa42, Prop. 2] that we can choose a small enough U such that $\operatorname {Pic}(U)$ injects into $\operatorname {Pic}(D)$ .

Proof. By adjunction, every smooth anti-canonical divisor of a Fano variety is a K3 surface. A very general projective K3 surface has Picard number $1$ . Therefore, X is smooth along an analytically very general anti-canonical divisor with Picard number $1$ . By Corollary 3.14, X is $\mathbb Q$ -factorial. By Lemma 3.12, X is factorial.

Proof. Let $S_k$ be the $\mathbb C$ -algebra freely generated by the $28$ coefficients, considered as variables, of polynomials $g \in \mathbb C[y, z, t]$ homogeneous of degree $6$ . By Remark 3.5(b), closed points P of $\operatorname {Spec} R_k$ bijectively correspond to polynomials $f_P \in \mathbb C[x, y, z, t, w]$ in Notation 3.4. Let $\theta \colon \mathbb C[x, y, z, t, w] \to \mathbb C[y, z, t]$ be the homomorphism $x \mapsto 0$ , $w \mapsto 0$ . Let $\pi _k\colon \operatorname {Spec} R_k \to \operatorname {Spec} S_k$ be the morphism of affine spaces given on closed points by $f_P \mapsto \theta (f_P)$ . The $\mathbb C$ -algebra automorphisms $t \mapsto \alpha y + \beta z + t$ of $\mathbb C[y, z, t]$ induce a morphism . Define $\rho _k$ to be the composition

We can compute that the rank of the Jacobian matrix of $\rho _k$ at some specified point is $28$ for all $k \in \mathrm {Inds} \setminus \{7.4\}$ . It follows that $\rho _k$ is a dominant morphism of affine spaces for all $k \in \mathrm {Inds} \setminus \{7.4\}$ .

The closed points Q of $\operatorname {Spec} S_k$ bijectively correspond to polynomials $g_Q \in \mathbb C[y, z, t]$ homogeneous of degree $6$ , and therefore also to subschemes $Z_Q$ of $\mathbb P(1, 1, 1, 3)$ with variables $y, z, t, w$ given by $-w^2 + g_Q$ . Smooth schemes $Z_Q \subseteq \mathbb P(1, 1, 1, 3)$ are K3 surfaces that are called sextic double planes. It is known that a very general projective K3 surface has Picard number $1$ . It follows that an analytically very general sextic double solid X in family $k \in \mathrm {Inds} \setminus \{7.4\}$ satisfies that $\mathbb V_X(x)$ has Picard number $1$ .

Proof of Theorem A(e)

By Theorem A(d), a general sextic double solid in family k is terminal and is smooth along the anti-canonical divisor $\mathbb V(x)$ . By Corollary 3.15 and Lemma 3.16, an analytically very general sextic double solid in family $k \neq 7.4$ is factorial.

Proof. Use the explicit splitting lemma (Proposition 3.2) and repeatedly apply Lemma 3.9 similarly to the proof of Theorem A(b).

Proof. Let $\psi \colon X \to X'$ be any local biholomorphism taking the origin to the origin. Composing with a suitable weight-respecting biholomorphic map and using Corollary 4.4, it suffices to consider the case where $\psi $ is a linear biholomorphism. Since the elliptic curve defined by f in $\mathbb P^2$ with variables $x_1, x_2, x_3$ has only two automorphisms, there are only four possibilities for a linear biholomorphism $X \to X'$ , namely $(x_1, x_2, x_3) \mapsto (x_1, \pm x_2, \pm x_2 + x_3)$ .

Let $Y \to X$ and $Y' \to X'$ be the $(1, 1, 2)$ -blowups of X and $X'$ , respectively. Then Y is given by $\mathbb V(g)$ where

$$\begin{align*}g(u, x_1, x_2, x_3) = u x_2^2 x_3 + x_1^3 + a u^2 x_1 x_3^2 + b u^3 x_3^3. \end{align*}$$

Denoting the points of Y and $Y'$ by $[u, x_1, x_2, x_3]$ , the lifted map $\psi _Y\colon Y \to Y'$ is given by $[u, x_1, x_2, x_3] \mapsto [u, x_1, \pm x_2, \pm x_2/u + x_3]$ , which is not holomorphic on the exceptional locus $\mathbb V(u)$ .

Proof. “ $\Longrightarrow $ .” The induced morphism $\mathcal B_{X'} \to \psi _* \mathcal B_X$ is given by

$$\begin{align*}\begin{aligned} \mathcal B_{X'}(U) & \to \mathcal B_X(\psi^{-1} U)\\ t^d \bar y_j & \mapsto t^d \bar{\psi}_j, \end{aligned} \end{align*}$$

where $U \subseteq X'$ is open. Since $\operatorname {wt} \psi _j \geq \operatorname {wt} y_j$ , the morphism $\mathcal B_{X'} \to \psi _* \mathcal B_X$ is well-defined. Similarly, we define a morphism $\mathcal B_{X'} \gets \psi _* \mathcal B_X$ , which is its inverse.

“ $\Longleftarrow $ .” Let $t^{\operatorname {wt} y_j} \Psi _j$ be the image of $t^{\operatorname {wt} y_j} y_j$ under $\mathcal B_{X'}(X') \to \psi _* \mathcal B_X(X)$ . There exists $\psi _j \in \mathbb C\{\boldsymbol {x}\}$ such that $\operatorname {wt} \psi _j \geq \operatorname {wt} y_j$ and $\bar {\psi }_j = \Psi _j$ . Similarly, we can find $\theta _i$ , showing that $\psi $ is weight-respecting.

Notation 4.5. We choose positive integer weights $\boldsymbol {w} = (r_1, r_2, a, 1)$ for variables $\boldsymbol {x} = (x_1, x_2, x_3, x_4)$ on $\mathbb C^4$ and define $n = (r_1 + r_2)/a - 1$ such that

• • a divides $r_1 + r_2$ and is coprime to both $r_1$ and $r_2$ ,

• • $r_1 \geq r_2$ , and

• • $n \geq 2$ .

Proof. First, we remind that the terms homogeneous, degree, and multiplicity are with respect to the standard weights $(1, \ldots , 1)$ . Let the quadratic part of f denote the homogeneous part of f of degree $2$ . After a suitable invertible linear weight-respecting coordinate change, the quadratic part of f is $x_1 x_2$ .

We find that $f = x_1 x_2 + x_1 G + H$ , where $G \in \mathbb C\{x_1, \ldots , x_4\}$ has weight at least $r_2$ and multiplicity $m \geq 2$ , and $H \in \mathbb C\{x_2, x_3, x_4\}$ . The coordinate change $x_2 \mapsto x_2 - G_m$ , where $G_m$ is the homogeneous degree m part of G, takes f to $x_1 x_2 + x_1 G' + H'$ , where $G'$ has multiplicity at least $m + 1$ . By induction, this defines the unique formal power series $K \in \mathbb C[[x_1, \ldots , x_4]]$ of multiplicity at least $2$ and weight at least $r_2$ such that the transformation $x_2 \mapsto x_2 + K$ takes f to the form $x_1 x_2 + H"$ where $H" \in \mathbb C[[x_2, x_3, x_4]]$ . Similarly, we transform f into $x_1 x_2 + h$ where $h \in \mathbb C[[x_3, x_4]]$ , using $x_1 \mapsto x_1 + L$ where $L \in \mathbb C[[x_2, x_3, x_4]]$ .

We show how to find a convergent weight-respecting coordinate change which changes f to $x_1 x_2 + h$ . Instead of the coordinate changes $x_2 \mapsto x_2 + K$ , $x_1 \mapsto x_1 + L$ , which might not be convergent, we do a coordinate change $\Theta _N$ with truncated power series $K_{\leq N}$ and $L_{\leq N}$ of homogeneous parts of K and L of degree at most N. The coordinate change $\Psi \colon x_1 \mapsto x_1 + i x_2$ , $x_2 \mapsto x_1 - i x_2$ takes $x_1 x_2$ into $x_1^2 + x_2^2$ . Now we use the splitting lemma, which gives a convergent coordinate change $\Phi _N$ which respects the weighting when N is large enough, to give f the form $x_1^2 + x_2^2 + h(x_3, x_4)$ where h converges. Applying $\Psi ^{-1}$ , we get $x_1 x_2 + h$ . Note that the coordinate changes $\Psi $ and $\Psi ^{-1}$ might not respect the weighting $\boldsymbol {w}$ , but the total coordinate change $\Psi ^{-1} \circ \Phi _N \circ \Psi \circ \Theta _N$ is weight-respecting if N is large enough.

Since the singularity is $cA_n$ where $n = (r_1 + r_2)/a - 1$ , h must contain a monomial of degree $(r_1 + r_2)/a$ . Since $x_1 x_2 + h$ has weight $r_1 + r_2$ , if $a> 1$ , then the coefficient of $x_3^{(r_1 + r_2)/a}$ in h is nonzero. If $a = 1$ , then after a suitable invertible linear coordinate change on $\mathbb C\{x_3, x_4\}$ , the coefficient of $x_3^{(r_1 + r_2)/a}$ in h is nonzero.

We found that we can transform f into the form $x_1 x_2 + h$ where the coefficient of $x_3^{(r_1 + r_2)/a}$ in h is nonzero, by using only weight-respecting coordinate changes. By Corollary 4.4, the weighted blowup of f is locally analytically equivalent to the weighted blowup of $x_1 x_2 + h$ , which is precisely a Kawakita blowup.

Proof. Let $f = x_1 x_2 + g(x_3, x_4)$ and $f' = x_1 x_2 + g'(x_3, x_4)$ be contact equivalent, where $g, g' \in \mathbb C\{x_3, x_4\}$ have weight $r_1 + r_2$ and $x_3^{(r_1 + r_2)/a}$ appears in both g and in $g'$ with nonzero coefficient. It suffices to show that there exist a weight-respecting map from $\mathbb V(f)$ to $\mathbb V(f')$ .

Since f and $f'$ are contact equivalent, there exist a unit $u \in \mathbb C\{\boldsymbol {x}\}$ and a local biholomorphism $\psi \colon (\mathbb C^4, \boldsymbol {0}) \to (\mathbb C^4, \boldsymbol {0})$ such that $f' = u(f \circ \psi )$ . Note that $f'$ and $f \circ \psi $ have the same weight $r_1 + r_2$ , and $x_3^{(r_1 + r_2)/a}$ appears in $f \circ \psi $ with nonzero coefficient. It suffices to show that there exist a weight-respecting map from $\mathbb V(f)$ to $\mathbb V(f \circ \psi )$ .

Using arguments similar to the proof of Proposition 4.6, we can find a weight-respecting biholomorphic map germ $\theta \colon (\mathbb C^4, \boldsymbol {0}) \to (\mathbb C^4, \boldsymbol {0})$ such that $f \circ \psi \circ \theta $ is of the form $x_1 x_2 + g"$ where $g" \in \mathbb C\{x_3, x_4\}$ contains $x^{(r_1 + r_2)/a}$ and has weight $r_1 + r_2$ . It suffices to show that there exist a weight-respecting map from $\mathbb V(f)$ to $\mathbb V(f \circ \psi \circ \theta )$ .

By Proposition 2.5, g and $g"$ are right equivalent, meaning there exists an automorphism $\Phi $ of $\mathbb C\{x_3, x_4\}$ such that $\Phi (g) = g"$ . Since $x_3^{(r_1 + r_2)/a}$ has nonzero coefficient in both g and $g"$ , and both g and $g"$ have weight $r_1 + r_2$ , the image of $x_3$ has weight a under both $\Phi $ and $\Phi ^{-1}$ . Define the biholomorphic map germ $\varphi \colon (\mathbb V(f \circ \psi \circ \theta ), \boldsymbol {0}) \to (\mathbb V(f), \boldsymbol {0})$ by $\boldsymbol {x} \mapsto (x_1, x_2, \Phi (x_3), \Phi (x_4))$ . By Corollary 4.4, the $\boldsymbol {w}$ -blowups of $\mathbb V(f \circ \psi \circ \theta ) \subseteq \mathbb C^4$ and $\mathbb V(f) \subseteq \mathbb C^4$ are locally analytically equivalent.