Proof. See [Reference Hacking, Keel and Yu53, Propositions 7.2–7.3].

Proof. The statement follows from Proposition 2.1 and the toric description of minimal resolutions of cyclic quotient singularities [Reference Cox, Little and Schenck26, Chapter 10].

Proof. This follows from the proof of [Reference Engel and Friedman33, Theorem 5.4] applied to the smooth log Calabi–Yau pair $(Y',D')$ obtained as the minimal resolution $g' : (Y',D') \rightarrow (Y,D)$ as in Corollary 2.2, and to the big and nef line bundle $L':= (g')^{\star } L$ . Note that, while [Reference Engel and Friedman33, Theorem 5.4] is stated for an ample line bundle, its proof applies more generally to a big and nef line bundle as remarked in [Reference Engel and Friedman33, Remark 5.7].

Proof. By the existence of a toric model, Y is isomorphic to a projective toric surface $\overline {Y}$ , up to loci of complex dimension one, that is, of real dimension two. Since proper toric varieties have trivial first homology groups, we obtain $H_1(Y,\mathbb {Z})=H_1(\overline {Y},\mathbb {Z})=0$ . It follows that $H^2(Y,\mathbb {Z})$ is torsion free, and by the exponential exact sequence, $\mathrm {Pic}(Y)$ is also torsion free.

Construction 2.10. We describe the construction of a Symington polytope for a generic polarized log Calabi–Yau surface $(Y,D,L)$ endowed with a good toric model $(\overline {Y},\overline {D},\overline {L})$ as in Definition 2.7. We denote by $\overline {P}$ the toric momentum polytope associated to the ample line bundle $\overline {L}$ on the toric surface $(\overline {Y},\overline {D})$ . The edges $(\overline {P}_i)_{1\leq i\leq n}$ of the boundary $\partial \overline {P}$ of $\overline {P}$ are naturally in one-to-one correspondence with the toric irreducible components $(\overline {D}_i)_{1\leq i\leq n}$ of $\overline {D}$ . Moreover, for every $1\leq i\leq n$ , the integral length $\ell (\overline {P}_i)$ of the edge $\overline {P}_i$ is given by the intersection number $\overline {L} \cdot \overline {D}_i$ .

As in Definition 2.7, we denote the exceptional curves of the blow-up $h:\widetilde {Y} \to \overline {Y}$ contracted to $\overline {D}_i$ by $E_{ij}$ , for $1\leq j \leq N_i$ . By Definition 2.7, there exists a decomposition

(2.2) $$ \begin{align} \widetilde{L}:=g^\star L = \sum_{i=1}^n c_i \, h^\star \overline{D}_i -\sum_{i=1}^n \sum_{j=1}^{N_i} m_{ij} E_{ij} \,,\end{align} $$

such that $c_i \geq 0$ , $m_{ij} \geq 0$ , and $ m_{ij} \leq c_i$ for all $1\leq i \leq n$ and $1\leq j \leq N_i$ . Moreover, we have

(2.3) $$ \begin{align} \sum_j m_{ij} \leq \ell(\overline{P}_i) \end{align} $$

where $\ell (\overline {P}_i)$ is the lattice length of $\overline {P}_i$ , since $g^*L$ is nef and hence

$$\begin{align*}g^*L \cdot \tilde{D}_i = \overline{L} \cdot \overline{D}_i - \sum_{j=1}^{N_i}m_{ij} = \ell(\overline{P}_i) - \sum_{j=1}^{N_i}m_{ij} \geq 0 \, ,\end{align*}$$

using that $\overline {L}=h_\star g^\star L = \sum _{i=1}^n c_i \overline {D}_i$ .

As $\sum _{i=1}^n c_i \overline {D}_i$ is an effective torus-invariant divisor on $\overline {Y}$ representing the line bundle $\overline {L}$ , there exists a corresponding integral point $p \in \overline {P}$ at lattice distance $c_i$ from each edge $\overline {P}_i$ of $\overline {P}$ . The momentum polytope $\overline {P}$ admits a polyhedral decomposition,

$$\begin{align*}\overline{P} = \bigcup_{i=1}^n \overline{T}_i \, , \end{align*}$$

where $\overline {T}_i$ is the convex hull of $\overline {P}_i$ and the point p. Generally $\overline {T}_i$ ’s are triangles, unless $p \in \partial \overline {P}$ , in which case they may degenerate into line segments.

Due to Equation (2.3), we can choose nonoverlapping line segments $\overline {P}_{ij} \subset \overline {P}_i$ of lattice length $m_{ij}$ . We denote by $\overline {T}_{ij}$ the triangle with base $\overline {P}_{ij}$ and third vertex p, and by $\overline {L}_{ij} \subset \overline {T}_{ij}$ the line segment of points in $ \overline {T}_{ij}$ at lattice distance $m_{ij}$ from $\overline {P}_i$ . Note that $\overline {L}_{ij}$ is nonempty since we have $m_{ij}\leq c_i$ by definition of a good toric model.

By the proof of [Reference Engel and Friedman33, Proposition 5.8], there exists an integral point $p_{ij}$ on $\overline {L}_{ij}$ . Let $\Delta _{ij}$ be the triangle with base $\overline {P}_{ij}$ and third vertex $p_{ij}$ . By construction, $\Delta _{ij}$ is an integral triangle, with its base $\overline {P}_{ij}$ of integral length $m_{ij}$ , and integral height also equal to $m_{ij}$ . We refer to $m_{ij}$ as the size of $\Delta _{ij}$ . We then define the Symington polytope P associated to $(Y,D,L)$ as the integral affine manifold with singularities P obtained from complement $\overline {P} \setminus \cup _{ij} \mathrm {Int}(\Delta _{ij})$ , by gluing the edges adjacent to $p_{ij}$ using a matrix conjugate to

$$\begin{align*}\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)\,.\end{align*}$$

This process introduces a specific type of integral affine singularity, known as a focus-focus singularity, at each point $p_{ij}$ . The corresponding monodromy invariant direction is the line $L_{ij}$ which is parallel to the edge $\overline {P}_i$ and passes through $p_{ij}$ . The discriminant locus of the affine structure is given by the union of points

$$\begin{align*}\Delta = \{p_{ij} ~ | ~ 1\leq i \leq n ~ \mathrm{and} ~ 1\leq j \leq N_i \} \,. \end{align*}$$

Proof. We use the notation introduced in Construction 2.10. First, since $g^\star L = \overline {L}-\sum _{i,j} m_{ij}E_{ij}$ , it follows that $L^2=\overline {L}^2 - \sum _{i,j} m_{ij}^2$ . On the other hand, since P is obtained from $\overline {P}$ by removing the nonoverlapping triangles $\Delta _{ij}$ of sizes $m_{ij}$ , we obtain $\mathrm {Area}(P)=\mathrm {Area}(\overline {P}) - \frac {1}{2}\sum _{i,j}m_{i,j}^2$ . Finally, by standard toric geometry [Reference Fulton39, §5.3, p111, Corollary], we have $\mathrm {Area}(\overline {P})=\frac {1}{2}\overline {L}^2$ , and so the result follows.

Proof. We first prove the result when $\Delta _{ij}$ is the only cut triangle. Let $\ell _i=0$ be the affine equation defining $\overline {P}_i$ , so that $L_{ij}$ is the line of equation $\ell _i=m_{ij}$ . Define $d_{ij}$ as the greatest integral distance between the line $L_{ij}$ and a point $q \in \overline {P} \cap \{ \ell _i \geq m_{ij}\}$ . To prove Lemma 2.12, it suffices to show that $d_{ij}>0$ . Indeed, in this situation, let q be a point in $\overline {P}$ with $\ell _i(q)>m_{ij}$ , and denote by $p_1$ and $p_2$ the two endpoints of the line segments $\overline {P}_{ij}$ . Since the points $p_1, p_2, p_{ij}$ are not collinear, at least one of the triangles with vertices $p_1, p_{ij}, q$ or $p_2, p_{ij}, q$ must be nonflat – see Figure 2. Such a nonflat triangle necessarily has a nontrivial intersection with $L_{ij}$ and is contained within $\overline {P}$ , as the polygon $\overline {P}$ is convex and contains the points $p_1$ , $p_2$ , $p_{ij}, q$ .

Figure 2 Illustration of the first part of the proof of Lemma 2.12.

We now prove that $d_{ij}>0$ . Consider the toric variety $(\overline {Y}',\overline {D}')$ obtained from $(\overline {Y},\overline {D})$ by adding to its fan the ray opposite to the ray corresponding to $\overline {P}_i$ . These two opposite rays define a generically $\mathbb {P}^1$ -fibration $\overline {Y}' \rightarrow \mathbb {P}^1$ . Denote by $(\tilde {Y}', \tilde {D}')$ the corresponding blow-up of $(Y,D)$ , with a generically $\mathbb {P}^1$ -fibration $\tilde {Y}' \rightarrow \mathbb {P}^1$ . Then, the preimage of $E_{ij}$ is an irreducible component $E_{ij}'$ of a fiber of this map. Let $F_{ij}$ be the other irreducible component of this fiber and let $g': \widetilde {Y}' \rightarrow Y$ be the contraction induced by g. Define $G_{ij}$ as the curve class of a fiber, so that $G_{ij}=E_{ij}'+F_{ij}$ . By standard toric geometry [Reference Fulton39, §5.3, p112, Corollary], the intersection number $L \cdot g^{\prime }_\star G_{ij}$ equals the maximal integral distance between the line $\ell _i=0$ and a point in $\overline {P}$ . On the other hand, we have $L \cdot g^{\prime }_\star E_{ij}=m_{ij}$ , and so the intersection number $L \cdot g^{\prime }_\star F_{ij}$ is equal to the maximal integral distance between the line $\ell _i=m_{ij}$ and a point in $\overline {P}$ , that is, we have $d_{ij}=L \cdot g^{\prime }_\star F_{ij}$ . Since $g'$ is a corner blow-up, we have $g^{\prime }_\star F_{ij}\neq 0$ , and so we conclude that $d_{ij}=L \cdot g^{\prime }_\star F_{ij}>0$ since L is ample. This ends the proof of Lemma 2.12 when $\Delta _{ij}$ is the only cut triangle.

Finally, we prove Lemma 2.12 in the general case where $\Delta _{ij}$ is not the only cut triangle. If $\Delta _{ij}$ has an interior edge that is not entirely contained within another cut triangle, then by the first part of the proof, the corresponding segment of $L_{ij}$ has a nontrivial intersection with P. From Construction 2.10, the only way both interior edges of $\Delta _{ij}$ are fully contained within other cut triangles is when $p_{ij}$ coincides with the central point p. In this case, one can attempt to propagate $L_{ij}$ through the neighboring cut triangles to obtain a nontrivial intersection with P. This will always work unless the cut triangles completely cover $\overline {P}$ . However, this does not occur, as the Symington polytope P has positive area by Lemma 2.11 and $L^2>0$ due to the ampleness of L. This concludes the proof of Lemma 2.12.

Proof. Without loss of generality, we can assume that the vertex $p_{ij}$ has coordinates $(0,0)$ , and that the other two vertices of $\Delta _{ij}$ on $\overline {P}_{ij}$ have coordinates $(a,b)$ and $(a,c)$ respectively, with $b>c$ . Since the edge of $\Delta _{ij}$ on the side $\overline {P}_{ij}$ has the same integral length as the height of $\Delta _{ij}$ by Construction 2.10, we have $a=b-c$ . Then, we obtain $d:=\gcd (a,b)=\gcd (a,c)$ , and the sets of integral points on the edges connecting $p_{ij} = (0,0)$ to $(a,b)$ and to $(a,c)$ respectively have the same cardinality, equal to $d+1$ :

$$\begin{align*}\left| \left\{ \left( \frac{ka}{d} , \frac{kb}{d} \right) \,\ | \,\ 0 \leq k \leq d \right\} \right| = \left| \left\{ \left( \frac{ka}{d} , \frac{kc}{d} \right) \,\ | \,\ 0 \leq k \leq d \right\} \right| = d+1.\\[-45pt] \end{align*}$$

Proof. We will construct regular polyhedral decompositions of $\overline {P}$ by successively pulling off integral vertices. This process involves iteratively modifying a piecewise linear convex function by lowering its value at integral vertices by a small enough positive amount – see for instance [Reference De Loera, Rambau and Santos28, §4.3].

Let $\overline {\mathscr {P}}_1$ be the regular polyhedral decomposition of $\overline {P}$ obtained from the trivial decomposition by first pulling the central point p, and then pulling the vertices of the triangles $\Delta _{ij}$ – see Figure 3. Denote by $\varphi _1$ a convex piecewise linear function $\varphi _1: \overline {P} \to \mathbb {R}$ inducing $\overline {\mathscr {P}}_1$ .

Figure 3 The left-hand figure illustrates $\overline {\mathscr {P}}_2$ . To obtain the final decomposition $\overline {\mathscr {P}}$ the red edges are inserted, as in the right-hand figure.

By Proposition 2.13, there are the same number of integral points on each of the two interior edges of $\Delta _{ij}$ ’s. Let $\{ a^1_{ij}, \ldots , a^d_{ij} \}$ and $\{ b^1_{ij}, \ldots , b^d_{ij} \}$ respectively denote the set of integral points on the interior edges of $\Delta _{ij}$ , except for the interior vertex $p_{ij}$ . Let ${ \epsilon _{ij}^1, \ldots , \epsilon _{ij}^d }$ be a set of positive real numbers such that the points

(2.6) $$ \begin{align} (p_{ij}, \varphi_0(p_{ij}))\,, (a_{ij}^1, \varphi_0(a_{ij}^1) - \epsilon_{ij}^1)\,, \cdots, \,,(a_{ij}^d, \varphi_0(a_{ij}^d) - \epsilon_{ij}^d)\end{align} $$

are exactly the bending points of a piecewise linear function on the interior edge of $\Delta _{ij}$ containing the points $a_{ij}^k$ ’s. Since $\varphi _1$ is affine on $\Delta _{ij}$ , the points

(2.7) $$ \begin{align} (p_{ij}, \varphi_0(p_{ij}))\,, (b_{ij}^1, \varphi_0(b_{ij}^1) - \epsilon_{ij}^1)\,, \cdots, \,,(b_{ij}^d, \varphi_0(b_{ij}^d) - \epsilon_{ij}^d) \end{align} $$

are also exactly the bending points of a piecewise linear function on the interior edge of $\Delta _{ij}$ containing the points $b_{ij}^k$ ’s. Let $\varphi _2: \overline {P} \rightarrow \mathbb {R}$ be the convex piecewise linear function whose graph is the lower convex hull of the points in (2.6), the points in (2.7), and the points $(v,\varphi (v))$ , where v are vertices of $\overline {\mathscr {P}}_1$ not contained in triangles $\Delta _{ij}$ . Denote by $\overline {\mathscr {P}}_2$ the regular decomposition of $\overline {P}$ induced by $\varphi _2$ . If $\epsilon _{ij}^1, \cdots , \epsilon _{ij}^d$ are small enough for all $1\leq i\leq n$ , $1\leq j\leq N_i$ , then $\overline {\mathscr {P}}_2$ satisfies condition i) of Definition 2.15.

To obtain a good decomposition that also satisfies condition ii) of Definition 2.15, we start with $\overline {\mathscr {P}}_2$ and then successively pull generically all the integral points v of $\overline {P}$ which are not vertices of $\overline {\mathscr {P}}_2$ . The resulting decomposition $\overline {\mathscr {P}}$ , illustrated in Figure 3, coincides with $\overline {\mathscr {P}}_2$ on each triangle $\Delta _{ij}$ , and so also satisfies condition i) of Definition 2.15. Moreover, on the complement in $\overline {P}$ of the interior of the triangles $\Delta _{ij}$ , $\overline {\mathscr {P}}$ is a triangulation containing all integral points, and so $\overline {\mathscr {P}}$ satisfies condition ii) of Definition 2.15, since by Pick’s formula an integral triangle containing no integral point apart from its vertices is necessarily a triangle of size 1.

Proof. We denote by $P_{\mathbb {Z}}(0)$ (resp. $P_{\mathbb {Z}}(1)$ and $P_{\mathbb {Z}}(2)$ ) the set of integral points $v \in P_{\mathbb {Z}}$ such that $X^v$ is a 0-surface (resp. a 1-surface and a 2-surface). By Definition 3.7 ii), we have $P_{\mathbb {Z}}(0)\neq \emptyset $ , and, for every boundary integral point $v \in (\partial P)_{\mathbb {Z}}$ , we have either $v \in P_{\mathbb {Z}}(0)$ or $v \in P_{\mathbb {Z}}(1)$ . By Definition 3.7 i), P is topologically a disk, and so the boundary $\partial P$ is topologically a circle. If one fixes an orientation of $\partial P$ , we have a corresponding cyclic orientation of the boundary vertices $v \in (\partial P)_{\mathbb {Z}} =P_{\mathbb {Z}}(0) \cup P_{\mathbb {Z}}(1)$ . We denote by $(v_i)_{i \in \mathbb {Z}/N \mathbb {Z}}$ the cyclically ordered vertices in $P_{\mathbb {Z}}(0)$ . Moreover, $P_{\mathbb {Z}}(2)$ is exactly the set of vertices of $\mathscr {P}$ contained in the interior of P.

For every integral point $v \in P_{\mathbb {Z}}$ , we denote by $(\overline {X}^v ,\partial \overline {X}^v)$ the affinization of $(X^v, \partial X^v)$ , that is,

$$\begin{align*}(\overline{X}^v ,\partial \overline{X}^v):=(\mathrm{Spec}\, H^0(X^v, \mathcal{O}_{X^v}), \mathrm{Spec}\, H^0(\partial X^v, \mathcal{O}_{\partial X^v})) \,. \end{align*}$$

If $v \in P_{\mathbb {Z}}(2)$ , then $X^v$ is compact, and so $(\overline {X}^v, \partial \overline {X}^v)$ is just a point. If $v \in P_{\mathbb {Z}}(1)$ , then $(\overline {X}^v, \partial \overline {X}^v)=(\mathbb {A}^1,\mathbb {A}^1)$ by Definition 3.4. For $i \in \mathbb {Z}/N \mathbb {Z}$ , corresponding to an integral point $v_i \in P_{\mathbb {Z}}(0)$ , the affinization $(\overline {X}^{v_i}, \partial \overline {X}^{v_i})$ is an affine toric surface by Definition 3.6. Moreover, $\partial \overline {X}^{v_i}$ is the corresponding toric divisor, that is, the union of two irreducible toric divisors $(\partial \overline {X}^{v_i})_{i-1}$ and $(\partial \overline {X}^{v_i})_{i+1}$ corresponding to the two boundary edges of $\mathscr {P}$ adjacent to $v_i$ and contained in the intervals $[v_{i-1},v_i]$ and $[v_i, v_{i+1}]$ of $\partial P$ respectively. Both $(\partial \overline {X}^{v_i})_{i-1}$ and $(\partial \overline {X}^{v_i})_{i+1}$ are isomorphic to $\mathbb {A}^1$ , with $0$ corresponding to the torus fixed point of $\overline {X}^{v_i}$ .

Figure 5 Affinization $f_0: \mathcal {X}_0 \rightarrow \mathcal {X}^{\mathrm {can}}_0$ of an open Kulikov surface $\mathcal {X}_0$ .

The irreducible components of $\mathcal {X}^{\mathrm {can}}_0$ are the affine toric surfaces $\overline {X}^{v_i}$ for $i \in \mathbb {Z}/N\mathbb {Z}$ . Moreover, the double locus of $\mathcal {X}^{\mathrm {can}}_0$ is a union of copies of affine lines $(\partial X^{v_i})_{i+1} \simeq (\partial X^{v_{i+1}})_{i} \simeq \mathbb {A}^1$ , and all intersecting at a single common point $x_0 \in \mathcal {X}^{\mathrm {can}}_0$ , referred to below as the central point of $\mathcal {X}^{\mathrm {can}}_0$ , see Figure 5.

For every $v \in (\partial P)_{\mathbb {Z}}$ , the affinization map $X^v \rightarrow \overline {X}^v$ is proper and surjective, and so $f_0$ is also proper and surjective. To compute $R^i (f_0)_{\star } \mathcal {O}_{\mathcal {X}_0}$ , one uses the resolution of $\mathcal {O}_{\mathcal {X}_0}$ given by the complex

(3.1) $$ \begin{align} 0 \rightarrow \bigoplus_{v \in \mathscr{P}^{[0]}} \mathcal{O}_{X^v} \rightarrow \bigoplus_{e \in \mathscr{P}^{[1]}} \mathcal{O}_{X^e} \rightarrow \bigoplus_{f \in \mathscr{P}^{[2]}} \mathcal{O}_{X^f} \rightarrow 0 \,,\end{align} $$

as in the proof of Lemma 3.10. We have the following vanishing results:

1. i) If $v \in P_{\mathbb {Z}}(0)$ , then, $\overline {X}^v$ is an affine toric surface, so with cyclic quotient, and so rational, singularities, and the affinization map $X^v \rightarrow \overline {X}^v$ is a resolution of singularities. It follows that $R^i (f_0)_\star \mathcal {O}_{X^v}=0$ for $i>0$ .

2. ii) If $v \in P_{\mathbb {Z}}(1)$ , then the affinization map $X^v \rightarrow \overline {X}^v$ is generically a $\mathbb {P}^1$ -fibration, and we have $R^i (f_0)_\star \mathcal {O}_{X^v}=0$ for $i>0$ .

3. iii) If $v \in P_{\mathbb {Z}}(2)$ is an interior vertex or if e is an interior edge of $\mathscr {P}$ , then $f_0$ is constant on $X^v$ or $X^e$ , and so $R^i (f_0)_{\star } \mathcal {O}_{\mathcal {X}^v} = R^i (f_0)_{\star } \mathcal {O}_{\mathcal {X}^e}=0$ for $i>0$ follows from the vanishings $H^i(\mathcal {X}^v, \mathcal {O}_{\mathcal {X}^v})= H^i(\mathcal {X}^e, \mathcal {O}_{\mathcal {X}^e})=0$ , which hold since $\mathcal {X}^v$ is a rational surface and we have $\mathcal {X}^e \simeq \mathbb {P}^1$ .

4. iv) If e is a boundary edge of $\mathscr {P}$ , then the induced map $X^e \simeq \mathbb {A}^1 \rightarrow \overline {X}^e \simeq \mathbb {A}^1$ is an isomorphism. In particular, we have $R^i (f_0)_\star \mathcal {O}_{X^e}=0$ for $i>0$ .

Hence, the first page of the spectral sequence obtained by applying $R (f_0)_\star $ to the complex (3.1) reduces to the complex

$$\begin{align*}0 \rightarrow \bigoplus_{v \in \mathscr{P}^{[0]}} (f_0)_\star\mathcal{O}_{X^v} \rightarrow \bigoplus_{e \in \mathscr{P}^{[1]}} (f_0)_\star\mathcal{O}_{X^e} \rightarrow \bigoplus_{f \in \mathscr{P}^{[2]}} (f_0)_\star\mathcal{O}_{X^f} \rightarrow 0 \,.\end{align*}$$

The stalk of this complex at the point $x_0$ is isomorphic to the tensor product of the local ring $\mathcal {O}_{\mathcal {X}_0^{\mathrm {can}}, x_0}$ with the simplicial chain complex defined by the polyhedral decomposition $\mathscr {P}$ of the tropicalization P. Since P is topologically a disk, we have $H^0(P,\mathbb {Z})=\mathbb {Z}$ and $H^i(P,\mathbb {Z})=0$ for $i>0$ , and so the stalk of $R^i (f_0)_{\star } \mathcal {O}_{\mathcal {X}_0}=0$ at $x_0$ is equal to zero for $i>0$ . Similarly, at a point of the double locus (resp. a smooth point) of $\mathcal {X}_0^{\mathrm {can}}$ , the disk is replaced by an interval (resp. a point), and so Lemma 3.13 follows.

Proof. By Lemma 3.13, the surface $\mathcal {X}_0^{\mathrm {can}}$ has semirational singularities in the sense of [Reference Berquist23, Definition 4.1]. Hence, $\mathcal {X}_0^{\mathrm {can}}$ is Cohen-Macaulay by [Reference Berquist23, Corollary 4.4]. Moreover, it follows from the proof of Lemma 3.13 that the seminormalization of $\mathcal {X}_0^{\mathrm {can}}$ is the disjoint union of the affine toric surfaces $\overline {X}^{v_i}$ for $i \in \mathbb {Z}/N\mathbb {Z}$ , that is, of cyclic quotient singularities. In addition, the irreducible components $\overline {X}^{v_i}$ of $\mathcal {X}_0^{\mathrm {can}}$ are glued together cyclically along their toric boundary divisors. Hence, $\mathcal {X}_0^{\mathrm {can}}$ is a degenerate cusp singularity as in [Reference Shepherd-Barron90, §Reference Abramovich, Chen, Gross and Siebert1]. It then follows from [Reference Shepherd-Barron90, Lemma 1.1] that $\mathcal {X}_0^{\mathrm {can}}$ is Gorenstein with $\omega _{\mathcal {X}_0^{\mathrm {can}}}=\mathcal {O}_{\mathcal {X}_0^{\mathrm {can}}}$ , and from [Reference Kollár and Shepherd-Barron74, Theorem 4.21] that $\mathcal {X}_0^{\mathrm {can}}$ is semilog-canonical.

Proof. We first prove Theorem 3.22 i) and iii). By Lemma 3.10, we have $H^1(\mathcal {X}_0,\mathcal {O}_{\mathcal {X}_0})=0$ , and so it follows from [Reference Wahl100, Theorem 1.4] (see also [Reference Cynk and van Straten27, Theorem 3.1]) that f is a flat deformation of $f_0$ . To prove that $\mathcal {X}^{\mathrm {can}}$ has Gorenstein canonical singularities, first note that $\mathcal {X}$ has trivial canonical line bundle – this can be shown analogously as in [Reference Friedman35, Theorem 5.10, 3)], where a similar result is proved in the context of degenerations of K3 surfaces. Hence, since $\mathcal {X}$ is a smooth threefold with trivial canonical class, it follows by [Reference Kawamata66, Corollary 1.5] that $\mathcal {X}^{\mathrm {can}}$ has Gorenstein canonical singularities. Note that [Reference Kawamata66, Corollary 1.5] is stated in the context of projective morphism between normal algebraic varieties. In this context, the result follows from the base point free theorem [Reference Kawamata66, Theorem 1.3]. The same argument shows that the result for projective morphisms of formal schemes follows from the corresponding version of the base point free theorem in [Reference Lyu and Murayama80, §Reference Altmann11].

To prove Theorem 3.22 ii), denote by $\mathcal {X}_\eta $ the rigid-analytic generic fiber of $\mathcal {X}$ , which is smooth since $\mathcal {X} \rightarrow \Delta $ is smoothing. Standard results on birational geometry of surfaces generalize to rigid-analytic surfaces by [Reference Ueno99, §Reference Alexeev5]. Hence, it follows from the triviality of the canonical class of $\mathcal {X}_\eta $ that the birational map $\mathcal {X}_\eta \rightarrow \mathcal {X}^{\mathrm {can}}_\eta $ can only contract $(-2)$ -curves, and so that $\mathcal {X}^{\mathrm {can}}_\eta $ has at worst Du Val (ADE) singularities.

Proof. By Lemma 3.10, we have $H^1(\mathcal {X}_0, \mathcal {O}_{\mathcal {X}_0})=H^2(\mathcal {X}_0, \mathcal {O}_{\mathcal {X}_0})=0$ , and so, by formal deformation theory of line bundles, we have $\mathrm {Pic}(\mathcal {X}) \simeq \mathrm {Pic}(\mathcal {X}_0)$ . On the other hand, it follows from the explicit description of $\mathcal {X}_0^{\mathrm {can}}$ given by Lemma 3.14 that $\mathrm {Pic}(\mathcal {X}_0^{\mathrm {can}})=0$ , $H^1(\mathcal {X}_0^{\mathrm {can}}, \mathcal {O}_{\mathcal {X}_0^{\mathrm {can}}})=H^2(\mathcal {X}_0^{\mathrm {can}},\mathcal {O}_{\mathcal {X}_0^{\mathrm {can}}})=0$ , and so that $\mathrm {Pic}(\mathcal {X}_0^{\mathrm {can}})=0$ .

Proof. The only nonimmediate implication is i) implies ii). By Lemma 3.23, so there are no obstructions to formal deformations over $\Delta $ of an ample line bundle on $\mathcal {X}_0$ , and so the result follows.

Proof. The equivalence between i), ii), iii), and iv) is the equivalence between (1.1), (1.2), (1.3), and (1.4) in [Reference Gross, Hacking and Keel44, Lemma 6.9].

If $W=-(K_Y+\sum _i c_i D_i)$ is big and nef, then $-K_Y=W+\sum _i c_i D_i$ is the sum of a big divisor with an effective divisor, and so $-K_Y$ is big. This shows that iv) implies v). Finally, we show that v) implies iii). If $-K_Y$ is big, then $\sum _i D_i=-K_Y$ is big, and so the result follows from the proof that (1.1) implies (1.2) in the proof of [Reference Gross, Hacking and Keel44, Lemma 6.9].

Proof. By assumption, the open Kulikov degeneration $\pi : \mathcal {X}\rightarrow \Delta $ is quasi-projective. Hence, by Lemma 3.24, the contraction $f: \mathcal {X} \rightarrow \mathcal {X}^{\mathrm {can}}$ is projective. Thus, by the relative Kodaira Lemma applied to $f: \mathcal {X} \rightarrow \mathcal {X}^{\mathrm {can}}$ , there exists an effective divisor F on $\mathcal {X}$ such that $-F$ is f-ample. Write $F=a_i X^i+G$ , where $a_i$ is an integer and G is a divisor not containing $X^i$ . Since $X^i$ is a compact irreducible component of $\mathcal {X}_0$ , $X^i$ is contracted by f to a point and so it follows from the Negativity Lemma [Reference Kollár and Mori73, Lemma 3.39] that $a_i>0$ . Restricting F to $X^i$ , we obtain:

$$\begin{align*}F|_{X^i}=a_i X^i|_{X^i}+G|_{X^i} \,.\end{align*}$$

Since $K_{\mathcal {X}}=0$ , it follows by adjunction that the normal bundle to $X^i$ in $\mathcal {X}$ is $K_{X^i}$ , and so $X^i|_{X^i}=K_{X^i}$ . Hence, we have

(3.8) $$ \begin{align} -K_{X^i}=\frac{1}{a_i}(-F|_{X^i}+G|_{X^i})\,. \end{align} $$

As $X^i$ is contracted by f and $-F$ is f-ample, it follows that $-F|_{X^i}$ is ample on $X^i$ . On the other hand, as G does not contain $X^i$ , $G|_{X^i}$ is effective. Therefore, $-F|_{X^i}+G|_{X^i}$ is the sum of an ample divisor with an effective divisor, and so is big. Hence, $-K_{X^i}$ is big by the Equation (3.8) above and using the fact that $a_i>0$ . Finally, we conclude that $(X^i,\partial X^i)$ is positive using that v) implies i) in Lemma 3.26