To prove Claim 2.7, first we take finitely ramified base change from $K[[t]]$ to $K[[t^{1/d}]]$ for some $d\in \mathbb {Z}_{>0}$ , so that we can and do assume that the roots of $g_{8}, g_{12}, \Delta _{24}:=g_{8}^{3}-27g_{12}^{2}$ are Laurent (not only Puiseux), that is, there are $\xi _{i}\in K((t)) (i=1,\ldots ,8)$ , $\eta _{i}\in K((t)) (i=1,\ldots ,12)$ , and $\chi _{i}\in K((t)) (i=1,\ldots ,24)$ in the descending order of the valuations $v_{t}(-)$ along coordinates s with respect to t (or additive inverse of the valuation of $s^{-1}$ ).

We first set

(38) $$ \begin{align} e(0):=\min \{{\textrm{val}}_{t}(\xi_{1}),\ldots,{\textrm{val}}_{t}(\xi_{4}), {\textrm{val}}_{t}(\eta_{1}), \ldots, {\textrm{val}}_{t}(\eta_{6})\}, \end{align} $$

(39) $$ \begin{align} e(\infty):=\min \left\{{\textrm{val}}_{t}\biggl(\dfrac{1}{\xi_{5}}\biggr),\ldots,{\textrm{val}}_{t}\biggl( \dfrac{1}{\xi_{8}}\biggr), {\textrm{val}}_{t}\biggl(\dfrac{1}{\eta_{7}}\biggr), \ldots, {\textrm{val}}_{t}\biggl(\dfrac{1}{\eta_{12}}\biggr)\right\}, \end{align} $$

and after an appropriate elementary transform of the trivially extended $\mathbb {P}^{1}$ -bundle over $\mathbb {P}^{1}_{s}\times _{K}K[[t]]$ (we fix this ambiguity below soon), further blow it up to $\mathcal {B}_{1}\to \mathcal {B}_{\textrm {triv}}$ by the coherent ideal sheaf

(40) $$ \begin{align} \langle s,t^{e(0)} \rangle \cdot \langle s', t^{e(\infty)}\rangle \cdot \mathcal{O}_{\mathcal{B}_{\textrm{triv}}}. \end{align} $$

Here, $s' :=\frac {s_{2}}{s_{1}}$ and it is also a local uniformizer at $[s_{1}:s_{2}]=[1:0]$ ( $\infty $ -point) in the base $\mathbb {P}^{1}_{s}$ . Then, the special fiber of $\mathcal {B}_{1}$ over $t=0$ is

(41) $$ \begin{align} \mathbb{P}^{1}_{\frac{s}{t^{e(0)}}}\cup \mathbb{P}^{1}_{s} \cup \mathbb{P}^{1}_{\frac{s'}{t^{e(\infty)}}}, \end{align} $$

where the two ends are exceptional curves.

Accordingly, we can naturally degenerate the ambient space $\mathbb {P}_{\mathbb {P}_{s}^{1}}(\mathcal {O}_{\mathbb {P}^{1}}(4)\oplus \mathcal {O}_{\mathbb {P}^{1}}(6)\oplus \mathcal {O}_{\mathbb {P}^{1}})$ over $K((t))$ to over $K[[t]]$ , so that the special fiber over $t=0$ is a connected union of the following three irreducible components:

1. i. $\mathbb {P}_{\mathbb {P}_{\frac {s}{t^{e(0)}}}^{1}}(\mathcal {O}_{\mathbb {P}^{1}}(2)\oplus \mathcal {O}_{\mathbb {P}^{1}}(3)\oplus \mathcal {O}_{\mathbb {P}^{1}})$ over $\mathbb {P}^{1}_{\frac {s}{t^{e(0)}}}$ ,

2. ii. trivial $\mathbb {P}^{2}$ -bundle over $\mathbb {P}^{1}_{s}$ (i.e., $\mathbb {P}^{2}\times \mathbb {P}^{1}_{s}$ ), and

3. iii. $\mathbb {P}_{\mathbb {P}_{s}^{1}}(\mathcal {O}_{\mathbb {P}^{1}}(2)\oplus \mathcal {O}_{\mathbb {P}^{1}}(3)\oplus \mathcal {O}_{\mathbb {P}^{1}})$ over $\mathbb {P}^{1}_{\frac {s'}{t^{e(\infty )}}}$ .

Inside the first component (i), the closure of X (limit component) appears as

(42) $$ \begin{align} X^{W}_{g_{4}^{\nu},g_{6}^{\nu}}:=[y^{2}z=4x^{3}-g_{4} ^{\nu}|_{t=0}xz^{2}+g_{6}^{\nu}|_{t=0}z^{3}], \end{align} $$

where $g_{4}^{\nu }=c_{4}\prod _{i=1}^{4}(s-\xi _{i})$ and $g_{6}^{\nu }=c_{6}\prod _{i=1}^{6}(s-\eta _{i}),$ with replaced roots $\xi $ s and $\eta $ s. Recall that construction of the model $\mathcal {B}_{1}$ above had an ambiguity modulo elementary transform with respect to $t=0$ , but we fix it by assuming $(c_{4},c_{6})\in K^{2}\setminus \vec {0}$ . From the construction, $g_{4}^{\nu }$ and $g_{6}^{\nu }$ are strictly degrees $4$ and $6$ , respectively, with coefficients $3$ and $1$ , respectively, and $\Delta ^{\nu }_{12}:=(g_{4}^{\nu })^{3}-27(g_{6}^{\nu })^{2}$ has degree at most $11$ . This means that the component $X^{W}_{g_{4}^{\nu },g_{6}^{\nu }}$ has singular fiber over $\infty $ , which corresponds to the fact that the degeneration is of type III.

Furthermore, from the definition of $e(0)$ , not all of $\xi _{i}$ s and $\eta _{i}$ s vanish. Similarly, in the last component (iii), the closure of X (limit component) appears as

(43) $$ \begin{align} X^{W}_{h_{4}^{\nu},h_{6}^{\nu}}:=[y^{2}z=4x^{3}-h_{4} ^{\nu}|_{t=0}xz^{2}+h_{6}^{\nu}|_{t=0}z^{3}], \end{align} $$

where $h_{4}^{\nu }=\prod _{i=5}^{8}(s-\xi _{i})$ and $h_{6}^{\nu }=\prod _{i=7}^{12}(s-\eta _{i}),$ again with newly replaced roots $\xi $ s and $\eta $ s. From the construction, due to [Reference KasKas77, Lem. 1], if Weierstrass surfaces are generically smooth, they automatically only have ADE singularities (at nonzero finite base coordinates).

When $K={\mathbb {C}}$ , in comparison with our asymptotic analysis of McLean’s real Monge–Ampére metrics in [Reference Odaka and OshimaOO21, §7.3.3], these end surfaces are where the term (denominator of the second term in [Reference Odaka and OshimaOO21, Lem. 7.16])

(44) $$ \begin{align} \log(|g_{8}|^{3}+27|g_{12}|^{2}) \end{align} $$

becomes dominant. On the other hand, the following step is relevant to expand the divergence of the $\log (|\Delta _{24}|)$ term.

Step 2. (Separating middle $\chi _{i}$ s)

In the next step, we consider toric model $\mathcal {B}$ with respect to some combinatorial data coming from the Newton polygon, as the method used classically by [Reference Alexeev and NakamuraAN99], [Reference DonaldsonDon02], and [Reference MumfordMum72b] as follows. We consider the Newton polygon ${\textrm {Newt}}(\Delta _{24})$ of $\Delta _{24}$ , that is, the convex hull of

(45) $$ \begin{align} \{(i,v_{t}(d_{i}))\mid 0\le i\le 24\}+{\mathbb{R}}_{\ge 0}(0,1). \end{align} $$

We regard it as a graph of PL convex function $\varphi _{\Delta }\colon [0,24]\to {\mathbb {R}}\cup \{\infty \}$ . Then, we modify this as follows (this process aims at including the previous step when we consider the toric models).

Set

(46) $$ \begin{align} i_{e(0)}:=\max\{i\mid \varphi_{\Delta}(i)-\varphi_{\Delta}(i+1)\ge e(0)\}, \end{align} $$

(47) $$ \begin{align} i_{e(\infty)}:=\min\{i\mid \varphi_{\Delta}(i+1)-\varphi_{\Delta}(i)\ge e(\infty)\}, \end{align} $$

where $e(0)$ and $e(\infty )$ as (38) and (39). We modify $\varphi _{\Delta }$ to $\overline {\varphi }_{\Delta }\colon [0, 24]\to {\mathbb {R}}\cup \{\infty \}$ defined as follows:

(48) $$ \begin{align} \bar{\varphi}_{\Delta}(i):= \begin{cases} \varphi_{\Delta}(i_{e(0)})-e(0)(i_{e(0)}-i) & (\text{if }0\le i\le i_{e(0)}), \\ \varphi_{\Delta}(i) & (\text{if }i_{e(0)}\le i\le i_{e(\infty)}), \\ \varphi_{\Delta}(i_{e(\infty)})+e(\infty)(i-i_{e(\infty)}) & (\text{if }i_{e(\infty)}\le i\le 24). \end{cases} \end{align} $$

Then, consider the toric model (test configuration of $\mathbb {P}^{1}$ ) $\mathcal {B}$ over $\mathbb {A}^{1}$ (hence also over $K[[t]]$ ), corresponding to $\bar {\varphi }_{\Delta }$ , that is, for

(49) $$ \begin{align} P_{\Delta,c}:=\{(x,y)\in {\mathbb{R}}^{2}\mid 0\le x\le 24, -c\le y\le -\bar{\varphi}_{\Delta}(x)\}, \end{align} $$

for some $c\ge 0$ , the moment polytope of (the natural compactification of) $\mathcal {B}$ becomes $P_{\Delta ,c}$ .

In particular, the normal fan of the graph of $\bar {\varphi }_{\Delta }$ gives $\mathcal {B}$ by usual toric construction. We fix and take the natural c such that the obtained $\mathcal {B}$ has the same end components as $\mathcal {B}_{1}$ in Step 1, that is, the end components of $\mathcal {B}|_{t=0}$ are the bases $\mathbb {P}^{1}_{\frac {s}{t^{e(0)}}}$ and $\mathbb {P}^{1}_{\frac {s'}{t^{e(\infty )}}}$ of the ends at (63). Indeed, it is possible by our modification (73) of the PL function.

Furthermore, as desired, every other component of $\mathcal {B}|_{t=0}$ has at least one point of $D_{0}(=\bar {D}\cap (t=0))$ . Here, recall that D denotes the discriminant locus defined after Theorem 2.6 whose closure is denoted as $\bar {D}$ . This ensures the ampleness of the boundary R in the corresponding irreducible components of the Weierstrass (reducible) fibered surface.

If the end surface $X^{W}_{g_{4}^{\nu },g_{6}^{\nu }}\to \mathbb {P}^{1}_{s}$ is generically smooth, it is nothing but a rational elliptic surface, that is, type $\mathbb {E}_{k}$ in [Reference Alexeev, Brunyate and EngelABE20]. In that case, because of the construction, $\deg \Delta _{12}^{\nu }=12$ ( $\Delta _{12}^{\nu }$ does not vanish at $\infty $ ), so that the fiber over $\frac {s}{t^{e(0)}}=\infty $ cannot be singular.

On the other hand, if the end surface $X^{W}_{g_{4}^{\nu },g_{6}^{\nu }}$ has singular general fibers, it means that there is $P_{2}\in H^{0}(\mathcal {O}(2[\infty ]))$ such that

(50) $$ \begin{align} g_{4}^{\nu}=3P_{2}^{2}, g_{6}^{\nu}=P_{2}^{3}. \end{align} $$

${\textrm {deg}}(P_{2})$ cannot be less than $2$ from the construction. If this $P_{2}$ is square-free, then from our discussion in §2.2.2, we get the surface $\mathbb {D}$ type and end the step here. If $P_{2}$ is not square-free, we continue to next step.

Step 4. (Modifying almost $\mathbb {D}$ -type end)

Depending on formulation, this process may be included in Step 1, but, nevertheless, we separated it to make the steps clearer. From here, we treat the left end surfaces in the original sense of Step 1, that is, those map to $s=0$ , that is, defined by $g_{4}^{\nu }$ and $g_{6}^{\nu }$ . (For the right end surface which maps to $s=\infty $ , the completely similar arguments work by symmetry, so we avoid repetition of the details of the arguments.)

We continue from the previous step, so suppose $P_{2}$ is not square-free. Nevertheless, since our generic fiber at $t\neq 0$ , $X_{g_{8},g_{12}}^{W}$ , was originally at worst ADE, among those (a priori at total $10$ ) roots of $g_{4}^{\nu }$ or $g_{6}^{\nu }$ , that is, $\xi _{i} (1\le i\le 4), \eta _{j} (1\le j\le 6)$ , at least two of them do not coincide as elements of $K[[t]]$ (before substitution $t=0$ ). Suppose that they are $\{p,q\}\subset \{\xi _{i} (1\le i\le 4), \eta _{j} (1\le j\le 6)\}$ with respect to the new coordinates after Step 1. Write the local uniformizer at $p(0)=q(0)$ for the component $\mathbb {P}^{1}_{\frac {s}{t^{e(0)}}}$ , as $s_{p,q}$ .

We make Puiseux expansions of $p, q$ , and set $e_{p,q}:=v_{t}(p-q)$ , where $v_{t}$ denotes the t-adic (additive) valuation. Then, do blowup of $\mathcal {B}$ (which was the outcome of processes until the previous step) along $\langle s_{p,q}, t^{e_{p,q}}\rangle \mathcal {O}_{\mathcal {B}}$ whose cosupport is in $\mathbb {P}^{1}_{\frac {s}{t^{e(0)}}} \times \{t=0\}$ , and blow down the surface without $\bar {D}\cap (t=0)$ if necessary, so that we obtain the situation with square-free $P_{2}$ . Note that by this last step, the resulting model $\mathcal {B}$ may not be toric, while toroidal, with respect to the original coordinates (since $p(0)=q(0)$ may not be zero).

Subcase. (to $\widetilde {\mathbb {D}_{16}}$ )

This case essentially follows from the GIT picture in [Reference Odaka and OshimaOO21, §7] by applying the GIT stable reduction. Recall that the Satake–Baily–Borel compactification $\overline {M_{W}}^{\textrm {SBB}}$ coincides with the GIT compactification with respect to the Weierstrass expression [Reference Odaka and OshimaOO21, §7.2.1]. As [Reference Odaka and OshimaOO21, §7.1] shows, the locus $M_{W}^{\textrm {nn}}$ is in the strictly stable locus, which parametrizes the semi-log-canonical surface of the form (24), which is nothing but $\tilde {\mathbb {D}}_{16}$ -type in [Reference Alexeev, Brunyate and EngelABE20].

If we have $(g_{8},g_{12})\in H^{0}(\mathcal {O}(8))\times H^{0}(\mathcal {O}(12))$ over the base $K[[t]]$ , with reduction sits in the stable locus mapping down to $M_{W}^{\textrm {nn}}$ , then the GIT stable reduction proves that after finite base change if necessary, if we apply an element of $SL(2)$ in the coefficient $K((t))$ , we get reduction with special fiber of the surface of type (24). This completes the required process.

Subcase. (to $\widetilde {\mathbb {E}_{8}}\widetilde {\mathbb {E}_{8}}$ )

We treat the case of degenerating from $M_{W}$ to $M_{W}^{\textrm {seg}} \subset M_{W}^{\textrm {SBB}}$ , which we recall to be the $\tilde {E}_{8}^{\oplus 2}$ -type $1$ -cusp (see also its GIT interpretation in [Reference Odaka and OshimaOO21, §7]).

Take $(X,R)\twoheadrightarrow B$ in $\mathcal {M}_{W}(K((t)))$ which degenerates to $M_{W}^{\textrm {seg}}$ at the closed point. From [Reference Odaka and OshimaOO21, §7], it follows that we can lift these data to $(g_{8}, g_{12})\in H^{0}(\mathcal {O}(8))\times H^{0}(\mathcal {O}(12))$ with coefficients in $K[[t]]$ , so that its reduction is $(cs^{4},s^{6})$ for $c\neq 3$ .

Then, we can exploit the same procedure as Step 1 of Case 1, to replace the reduction as the reducible fibered surface

(51) $$ \begin{align} (X_{1}\cup X_{2})\to \mathbb{P}^{1}\cup \mathbb{P}^{1}, \end{align} $$

where $X_{1}$ (resp. $X_{2}$ ) is a hypersurface of the $\mathbb {P}^{2}$ -bundle $\mathbb {P}_{\mathbb {P}^{1}}(\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3))$ over the first $\mathbb {P}^{1}$ (resp. the second $\mathbb {P}^{1}$ ), defined by

(52) $$ \begin{align} [y^{2}z=4x^{3}-g_{4}^{\nu}xz^{2} +g_{6}^{\nu}z^{3}], \end{align} $$

(53) $$ \begin{align} [y^{2}z=4x^{3}-h_{4}^{\nu}xz^{2} +h_{6}^{\nu}z^{3}], \end{align} $$

respectively. Then, from our assumption that $c\neq 3$ , it follows that the double locus $X_{1}\cap X_{2}$ is smooth elliptic curve fiber, hence this is of $\tilde {E}_{8}\tilde {E}_{8}$ -type surface as desired. We have $12$ real discriminant points in each base.

Subcase. ( $\mathbb {A}_{l-1}$ to $\mathbb {A}_{l_{1}-1}\mathbb {A}_{l_{2}-1}\cdots \mathbb {A}_{l_{m}-1}$ )

We now concentrate on the base of the component of A-type in the degenerated

(54) $$ \begin{align} [(X,R)\to \mathbb{P}^{1} \cup\cdots \cup \mathbb{P}^{1}]\in \overline{M_{W}}^{\textrm{ABE}} (K((t))), \end{align} $$

which we denote as $X_{A}\to \mathbb {P}^{1}$ here, with coordinate $s_{A}$ . The real discriminant points $\chi _{a+1},\ldots ,\chi _{a+l}$ can be seen as formal Puiseux series, that is, elements of $\overline {K((t))}$ . Note that any of $\chi _{a+i}$ is not $0$ nor $\infty $ (as element of $\mathbb {P}^{1}(K((t)))$ . Hence, after finite base change, we can suppose that they all lie in $K((t))$ and we write $\Delta _{A}(s_{A}):=\prod _{1\le i\le l} (s_{A}-\chi _{a+i})$ .

Similarly to Step 2 of Case 1, we take Newton polygon ${\textrm {Newt}}(\Delta _{A})$ , its supporting function $\varphi _{A}$ , and the toric degeneration model $\mathcal {B}_{A}$ over $\mathbb {A}^{1}$ (hence also over $K[[t]]$ ) whose corresponding fan is the normal fan of the graph of $\varphi _{A}$ . Or in other words, the natural compactification has moment polytope

(55) $$ \begin{align} \{(x,y)\in {\mathbb{R}}^{2}\mid 0\le x\le l, -c\le y\le - \varphi_{A}(x)\} \end{align} $$

for a constant $c\gg 0$ . This is one component of our desired $\mathcal {B}$ , that is, the closure of $\mathbb {P}^{1}_{s_{A}}$ . Then, accordingly, we degenerate the ambient space $\mathbb {P}^{2}\times \mathbb {P}_{s_{A}}^{1}\simeq \mathbb {P}_{\mathbb {P}_{s_{A}}^{1}}(\mathcal {O}^{\oplus 3})$ to still trivial $\mathbb {P}^{2}_{s_{A}}$ -bundle over $\mathcal {B}$ , so that we obtain the (semi-log-canonical) union of $\mathbb {A}$ -type log surfaces as the closure of $X_{A}$ inside the ambient model $\mathcal {B}\times \mathbb {P}^{2}$ .

Subcase. ( $\mathbb {D}_{k+l}$ to $\mathbb {D}_{k}\mathbb {A}_{l_{1}-1}\cdots \mathbb {A}_{l_{m}-1}$ )

Next, we consider the base of the component of D-type in the degenerated

(56) $$ \begin{align} [(X,R)\to \mathbb{P}^{1} \cup\cdots \cup \mathbb{P}^{1}]\in \overline{M_{W}}^{\textrm{ABE}} (K((t))), \end{align} $$

which we denote as $X_{D}\to \mathbb {P}^{1}$ here, with coordinate $s_{D}$ . We can and do suppose that the only double curve in $X_{D}$ , which is the intersection of next surface component, has coordinates $s_{D}=\infty $ .

Recall from §2.2.2.4 that we have an explicit Weierstrass-type equation for the $\mathbb {D}$ -type surface (20) in terms of a quadratic polynomial $P_{2}(s_{D})$ whose coefficients live in $K((t))$ . By quadratic base change, if necessary, we can further suppose that its two roots are also in $K((t))$ . Then, by multiplying appropriate powers $t^{2c}$ , $t^{3c}$ of t to $g_{4}^{\nu }$ and $g_{6}^{\nu }$ which do not change the isomorphism class of original $X_{D}\to \mathbb {P}^{1}$ (over $t\neq 0$ ), we can and do assume that coefficients of both lie in $K[[t]]$ and do not vanish at $t=0$ generically (with respect to $s_{D}$ ).

If some of real discriminants $\chi _{i}$ in the base of $X_{D}$ (including two roots of $P_{2}$ ) converges to $\infty $ , whose fiber is in the double locus of the surface, then we do weighted blowup of the model finite times, so that all $\chi _{i}$ in the base of $X_{D}$ never diverge to $ \infty $ when $t=0$ . Furthermore, in a similar manner, if two distinct roots of $P_{2}$ converge to the same point for $t\to 0$ , then we do further weighted blowup at the point, so that the two roots converge to different points. After these compositions of weighted blowups of the base surface, we contract all irreducible components of $t=0$ which do not contain any real discriminant.

Then, we degenerate the bundle $\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3)$ on $\mathcal {B}_{t\neq 0}$ to the whole model obtained above, so that it restricts to $\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3)$ on the component where the roots of $P_{2}$ converge, to $\mathcal {O}_{\mathbb {P}^{1}}^{3}$ otherwise. We consider its projectivization as the ambient model and take closure of X to get desired model of type $\mathbb {D}\mathbb {A}\cdots \mathbb {A}$ .

Subcase. ( $\mathbb {E}_{k+l}$ to $\mathbb {E}_{k}\mathbb {A}_{l_{1}-1}\cdots \mathbb {A}_{l_{m}-1}$ )

Next, we consider the base of the component of $\mathbb {E}$ -type in the degenerated

(57) $$ \begin{align} [(X,R)\to \mathbb{P}^{1} \cup\cdots \cup \mathbb{P}^{1}]\in \overline{M_{W}}^{\textrm{ABE}} (K((t))), \end{align} $$

which we denote as $\pi _{E}\colon X_{E}\to \mathbb {P}^{1}_{s_{E}}$ here, with coordinate $s_{E}$ . We can and do suppose that the only double curve in $X_{E}$ , which is the intersection of next surface component, has coordinates $s_{E}=\infty $ . We consider stable reduction of generic fiber, thus, over $K(s_{E})$ , which is from elliptic curve to either elliptic curve or (irreducible) nodal rational curve over whole $K(s_{E})[[t]])$ . Correspondingly, we realize this model by multiplying $t^{2c}$ (resp. $t^{3c}$ )

$$ \begin{align*}g_{4} \in H^{0}(\mathbb{P}^{1}_{s_{E}},\mathcal{O}(4)) (\text{resp.} g_{6} \in H^{0}(\mathbb{P}^{1}_{s_{E}},\mathcal{O}(6)))\end{align*} $$

with appropriate c (we fix this normalization from now on), so that $g_{4}, g_{6}$ both become nonzero at $t=0$ .

In this subcase, we focus when the generic fiber at $t=0$ is smooth, that is, elliptic curve, which we suppose from now on, and leave the nodal reduction case to the next subcase.

Suppose that the real discriminant points $\chi _{e+1},\ldots ,\chi _{e+k+l+3}$ below $X_{E}$ also all sit in $K((t))$ after finite base change if necessary. Then, in a similar manner as before, with respect to the variable $s^{\prime }_{E}:=s_{E}^{-1}$ , we set

(58) $$ \begin{align} P_{E}(s^{\prime}_{E}):=\prod_{i=1}^{3+k+l} (s^{\prime}_{E}-\chi_{e+i}^{-1}), \end{align} $$

consider its Newton polygon ${\textrm {Newt}}(P_{E})$ , then corresponding toric blow-up model $\mathcal {B}_{E}\to \mathbb {P}_{s_{E}}^{1}\times {\textrm {Spec}}(K[[t]])$ with cosupport at $t=0, s_{E}=\infty $ . Then, generalizing the stable reduction over $K(s_{E})[[t]]$ , we extend ambient space $\mathbb {P}_{\mathbb {P}_{s_{E}}^{1}}(\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3))$ of $X_{E}$ to that of $\mathcal {B}$ , so that its restriction to $\mathbb {P}_{s_{E}}^{1}$ is $\mathbb {P}(\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3))$ which includes the t-direction stable reduction of the generic fiber of $X_{E}$ , and trivial $\mathbb {P}^{1}$ -bundle over the rest of the components of $\mathcal {B}|_{t=0}$ . Then, it is easy to see that the closure inside the ambient model over $K[[t]]$ gives reduction to the surface of type $\mathbb {E}\mathbb {A}\cdots \mathbb {A}$ in [Reference Alexeev, Brunyate and EngelABE20].

Subcase. ( $\mathbb {E}_{k+l}$ to $\mathbb {D}_{k-1}\mathbb {A}_{l_{1}-1}\cdots \mathbb {A}_{l_{m}-1}$ )

Similarly to the previous subcase, we next treat the case when t-direction stable reduction of the generic fiber of $X_{E}$ becomes nodal (i.e., $j=\infty $ ). This assumption means

(59) $$ \begin{align} \Delta_{12}=(g_{4}^{\nu})^{3}-27(g_{6}^{\nu})^{2}=0, \end{align} $$

hence we can write $g_{4}^{\nu }=3P_{2}^{2}, g_{6}^{\nu }=P_{2}^{3}$ . Since we normalized our $g_{i}^{\nu }$ to give the t-direction stable reduction of the generic fiber, $P_{2}|_{t=0}\neq 0$ as a polynomial.

If the roots of $P_{2}|_{t=0}$ remain finite and distinct, then we only need to do toric modifications of the base model $\mathbb {P}^{1}_{s_{E}}\times {\textrm {Spec}}K[[t]]$ at cosupport $\infty \times 0(\text {closed point})$ . As it is completely similar to the just previous subcase, using Newton polygon of the polynomial of $s^{\prime }_{E}$ with roots $\chi _{i}^{-1}$ converging to $0$ , we omit details.

If at least one of the roots of $P_{2}|_{t=0}$ diverges, then we do toric blowup at $\infty \times 0\in \mathbb {P}^{1}_{s_{E}}\times {\textrm {Spec}}K[[t]]$ , so that $\mathcal {B}|_{t=0}$ becomes union of $\mathbb {P}^{1}_{s}$ with one or two exceptional divisors to each of which the diverging real discriminant converges. Furthermore, if the roots of $P_{2}$ converge to same points q in $\mathbb {P}^{1}_{s_{E}}$ , we do weighted blowup of the base model surface at the point q, so that the roots converge to different points in the same component which we (still) denote as $\mathbb {P}^{1}_{s}$ . After that, we contract all irreducible components (curves) of $t=0$ which do not contain any real discriminant. Then, again similarly, we take ambient space whose restriction to $\mathbb {P}^{1}_{s}\times \{t=0\}$ (resp. other components) is $\mathbb {P}_{\mathbb {P}^{1}} (\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3))$ (resp. trivial $\mathbb {P}^{2}$ -bundle).

After all these procedures, we obtain the model of reduction type $\mathbb {D}\mathbb {A}\cdots \mathbb {A}$ .

Subcase. ( $\widetilde {\mathbb {E}_{8}}\widetilde {\mathbb {E}_{8}}$ to $\mathbb {E}_{9-l}\mathbb {A}_{l_{1}-1}\cdots \mathbb {A}_{l_{m}-1}$ )

First, we treat the case when the generic point of ${\textrm {Spec}}K[[t]]$ maps to $M_{W}^{\textrm {seg}}$ . (Other case when the generic point of ${\textrm {Spec}}K[[t]]$ maps to $M_{W}^{\textrm {nn}}$ is treated in the Subcase after next.) We write the component of $\mathbb {E}_{9}$ -surface [Reference Alexeev, Brunyate and EngelABE20], that is, rational elliptic surface with double locus a single smooth fiber, as $X_{E}\to B_{E}\simeq \mathbb {P}^{1}$ as local notation. We suppose the double locus fibers over $\infty $ .

In case the reduction $t=0$ gives divergence of some real discriminants in the base $B_{E}$ to $\infty $ , then we again do the toric blowups of the model completely similarly as in previous steps via Newton polygon technique, so that the real discriminant points only converge finite in the strict transform of $B_{E}$ and smooth points in $\mathcal {B}|_{t=0}$ in general. Then, again in the similar manner, we obtain model of polarization whose restriction to $B_{E}$ is $\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3)$ while trivial $\mathcal {O}^{\oplus 3}$ otherwise, projectify it, and take closure of $X_{E}$ inside.

If such model is generically smooth over the strict transform of $B_{E}$ (otherwise, proceed to the next subcase). Then, by the assumption $(*_{II,III})$ , it follows that the fiber over $\infty $ becomes nodal at $t=0$ (otherwise, it remains to be in type II locus, that is, $1$ -cusps of $\overline {M_{W}}^{\textrm {SBB}}$ . Hence, the reduction for $t=0$ is the desired fibred surface of type $\mathbb {E}\mathbb {A}\cdots \mathbb {E}$ .

Subcase. ( $\widetilde {\mathbb {E}_{8}}\widetilde {\mathbb {E}_{8}}$ to $\mathbb {D}_{8-l}\mathbb {A}_{l_{1}-1}\cdots \mathbb {A}_{l_{m}-1}$ )

If the obtained model of $(X_{E},R)\to B_{E}$ in the last step is not generically smooth over the strict transform of $B_{E}$ , then the corresponding elements of $H^{0}(\mathcal {O}_{\mathbb {P}^{1}}(4))$ (resp. $H^{0}(\mathcal {O}_{\mathbb {P}^{1}}(6))$ ) which we still prefer to write $g_{4}^{\nu }$ , $g_{6}^{\nu }$ are of the form $(3P_{2}^{2},P_{2}^{3})$ with some $P_{2}\in H^{0}(\mathcal {O}_{\mathbb {P}^{1}}(2))$ . If $P_{2}$ vanishes at $\infty $ , that is, degree at most $1$ as a polynomial, then it means that one of the roots of $P_{2}$ , which is also a real discriminant point, diverges (or converges) to $\infty $ . We do toric blowup of the model of $B_{E}$ at this stage by the Newton polygon of the polynomial whose roots are diverging real discriminants, as in the previous steps. The process avoids the divergence of real discriminants $\infty $ while procuring further rational components in the reduction of base $\mathcal {B}|_{t=0}$ . If $P_{2}$ is not square-free, we do the same process as Step 4 of Case 1. Then, we contract all irreducible components of $t=0$ which do not contain any real discriminants.

Then finally, similarly, we create the model of $\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(2)\oplus \mathcal {O}_{\mathbb {P}^1}(3)$ at $t\neq 0$ as before, its projectivization, and take the closure of $X_{E}$ inside, which is our desired model. In this manner, we obtain further degeneration to surface of type $\mathbb {D}\mathbb {A}\cdots \mathbb {A}$ .

Subcase. ( $\widetilde {\mathbb {D}_{16}}$ to $\mathbb {D}_{a}\mathbb {A}_{l_{1}-1}\cdots \mathbb {A}_{l_{m}-1}\mathbb {D}_{b} $ with $a+b+l=16$ )

Now, we treat the case when the generic point of ${\textrm {Spec}}K[[t]]$ maps to $M_{W}^{\textrm {nn}}$ , while the closed point maps to $M_{W}^{\textrm {seg}}\cap M_{W}^{\textrm {nn}}$ , that is, degenerations of $\widetilde {\mathbb {D}}_{16}$ -type surfaces to type III surfaces.

We lift the $K((t))$ -rational point at $\mathcal {M}_{W}^{\textrm {nn}}$ to $(g_{8}=3G_{4}^{2},g_{12}=G_{4}^{3})$ with $G_{4}\in H^{0}(\mathcal {O}(4))$ with coefficient $K((t))$ . By multiplying appropriate integer power of t, we can first assume that $G_{4}$ has all coefficients in $K[[t]]$ . We also set the solutions of $G_{4}$ as $\sigma _{1},\sigma _{2},\tau _{1}, \tau _{2}$ , which we can and do assume to be in $K((t))$ after finite base change of $K[[t]]$ if necessary. We suppose $\sigma _{i}|_{t=0}=0, \tau _{i}|_{t=0}=\infty $ .

Similarly to Step 1 of Case 1, we set

(60) $$ \begin{align} f(0):=\min \{{\textrm{val}}_{t}(\sigma_{1}),{\textrm{val}}_{t}(\sigma_{2})\}, \end{align} $$

(61) $$ \begin{align} f(\infty):=\min \{{\textrm{val}}_{t}(\tau_{1}^{-1}),{\textrm{val}}_{t}(\tau_{2}^{-1})\}, \end{align} $$

and consider the blowup $\mathcal {B}_{1}\to \mathcal {B}_{\textrm {triv}}$ by

(62) $$ \begin{align} \langle s,t^{f(0)} \rangle \cdot \langle s^{-1}, t^{f(\infty)}\rangle. \end{align} $$

Then, the special fiber of $\mathcal {B}_{1}$ over $t=0$ is

(63) $$ \begin{align} \mathbb{P}^{1}_{\frac{s}{t^{f(0)}}}\cup \mathbb{P}^{1}_{s} \cup \mathbb{P}^{1}_{\frac{1}{s\cdot t^{f(\infty)}}}, \end{align} $$

where the two ends are exceptional curves.

Then, as in Step 1 of Case 1, the first component contains the limit of $\sigma _{i}|_{t\neq 0}$ , and the last component contains the limit of $\tau _{i}|_{t\neq 0}$ both different from the nodal points.

Then, similarly to the above,

we degenerate $\mathcal {O}_{\mathbb {P}^{1}}(2)\oplus \mathcal {O}_{\mathbb {P}^{1}}(3)$ on the original base $\mathbb {P}^{1}$ to the whole model, so that its reduction restricts to:

1. i. $\mathbb {P}_{\mathbb {P}_{\frac {s}{t^{f(0)}}}^{1}}(\mathcal {O}_{\mathbb {P}^{1}}(2)\oplus \mathcal {O}_{\mathbb {P}^{1}}(3)\oplus \mathcal {O}_{\mathbb {P}^{1}})$ over $\mathbb {P}^{1}_{\frac {s}{t^{f(0)}}}$ ,

2. ii. trivial $\mathbb {P}^{2}$ -bundle over $\mathbb {P}^{1}_{s}$ (i.e., $\mathbb {P}^{2}\times \mathbb {P}^{1}_{s}$ ), and

3. iii. $\mathbb {P}_{\mathbb {P}_{s}^{1}}(\mathcal {O}_{\mathbb {P}^{1}}(2)\oplus \mathcal {O}_{\mathbb {P}^{1}}(3)\oplus \mathcal {O}_{\mathbb {P}^{1}})$ over $\mathbb {P}^{1}_{\frac {s'}{t^{f(\infty )}}}$ .

Then, our first step is to take closure of original X inside the projectivization of the above $\mathbb {P}^{2}$ -bundle on the rational chain.

After this, we do the same procedures as Step 2, Step 3, and then Step 4 of Case 1. Then, we obtain the desired reduction to $\mathbb {D}\mathbb {A}\cdots \mathbb {A}\mathbb {D}$ -type surface.