1–4, we will thereby confirm that the surfaces $Y^{p}$ are
I${}_{n_{p}}$-components with $n_{p}$ subject to the constraints (A.53) and
(A.54).

#### Class 1: $(4,6,12)$

We first recall the expansion (A.17) of the Weierstrass section $f$ and the
expansion coefficients $\mathcal{F}_{i}$ therein, where the leading monomials
in $e_{p}$ define the exponents $\mu_{p,i}$ as (A.23). It then follows,
essentially from the relations (A.26), that these exponents obey

  * •

for $p=0$:

$\mu_{0,i}\left\\{\begin{array}[]{ l l }>4-a-i>0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{-}\\\ =0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{0}\\\ \geq 0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{+}\,,\end{array}\right.$ (A.62)

  * •

for $0<p<P$:

$\mu_{p,i}\left\\{\begin{array}[]{ l l }>0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{-}\\\ =0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{0}\\\ >0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{+}\,,\end{array}\right.$ (A.63)

  * •

for $p=P$:

$\mu_{P,i}\left\\{\begin{array}[]{ l l }\geq 0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{-}\\\ =0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{0}\\\ >0&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{+}\,.\end{array}\right.$ (A.64)

Similarly, the exponents $\nu_{p,j}$ defined through the leading monomials of
$\mathcal{G}_{j}$ in (A.24) satisfy the following set of analogous conditions:

  * •

for $p=0$:

$\nu_{0,j}\left\\{\begin{array}[]{ l l }>6-b-j>0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{-}\\\ =0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{0}\\\ \geq 0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{+}\,,\end{array}\right.$ (A.65)

  * •

for $0<p<P$:

$\nu_{p,j}\left\\{\begin{array}[]{ l l }>0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{-}\\\ =0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{0}\\\ >0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{+}\,,\end{array}\right.$ (A.66)

  * •

for $p=P$:

$\nu_{P,j}\left\\{\begin{array}[]{ l l }\geq 0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{-}\\\ =0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{0}\\\ >0&\quad\textrm{if
}\leavevmode\nobreak\ j\in\mathcal{J}_{+}\,.\end{array}\right.$ (A.67)

Note that the constraints (A.62) for $\mu_{0,i}$ imply that

$a+i+\mu_{0,i}\left\\{\begin{array}[]{ l l }>4&\quad\textrm{if
}\leavevmode\nobreak\ i\in\mathcal{I}_{-}\cup\mathcal{I}_{+}\\\
=4&\quad\textrm{if }\leavevmode\nobreak\
i\in\mathcal{I}_{0}\,,\end{array}\right.$ (A.68)

which, together with the expansion (A.17) and the leading behaviour (A.23),
tells us that the 3-fold-vanishing order of $f$ at $s=0$,

${\rm ord}_{\widehat{\mathcal{Y}}}(f)|_{(u,s)=0}=4+\alpha=4\,,$ (A.69)

is achieved by the bi-degree $(4,4)$ sector,

$\mathcal{F}_{4-a}s^{4}t^{4}=A_{4-a}s^{4}t^{4}+\cdots\,,$ (A.70)

while all other terms of $f$ have a higher 3-fold-vanishing order.
Analogously, we learn from the constraints (A.65) for $\nu_{0,j}$ that the
3-fold-vanishing order of $g$,

${\rm ord}_{\widehat{\mathcal{Y}}}(g)|_{(u,s)=0}=6+\beta=6\,,$ (A.71)

is achieved only by the bi-degree $(6,6)$ sector,

$\mathcal{G}_{6-b}s^{6}t^{6}=B_{6-b}s^{6}t^{6}+\cdots\,.$ (A.72)

Here, the coefficients $A_{4-a}$ and $B_{6-b}$ in (A.70) and (A.72) are non-
zero as addressed in (A.48) and (A.49), respectively. Therefore, the
expansions of $f$ and $g$ in each $e_{p}$ have a non-trivial constant sector,
i.e.,

$\displaystyle f_{p}:=f|_{e_{p}=0}\not\equiv 0\,,\quad
g_{p}:=g|_{e_{p}=0}\not\equiv 0\,,\qquad\text{for}\leavevmode\nobreak\
\leavevmode\nobreak\ p=0,\cdots,P\,.$ (A.73)

Focusing now on the discriminant $\Delta$, note that since its vanishing order
is

${\rm ord}_{\widehat{\mathcal{Y}}}(\Delta)|_{(u,s)=0}=12+\gamma=12\,,$ (A.74)

the order $12$ term arising from the cube of (A.70) and the one from the
square of (A.72) cannot cancel in $\Delta$, i.e.,

$4A_{4-a}^{3}+27B_{6-b}^{2}\neq 0\,.$ (A.75)

We then observe that the expansion of $\Delta$ in each $e_{p}$ is also
equipped with a non-vanishing constant sector, i.e.,

$\Delta_{p}:=\Delta|_{e_{p}=0}\not\equiv
0\,,\quad\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\ p=0,\cdots,P\,.$
(A.76)

This follows from

$\Delta_{p}=(4f^{3}+27g^{2})|_{e_{p}=0}=\left\\{\begin{array}[]{ l l
}(4A_{4-a}^{3}+27B_{6-b}^{2})s^{12}t^{12}+\mathcal{O}(s^{13})&\quad\textrm{for
}\leavevmode\nobreak\ p=0\\\
(4A_{4-a}^{3}+27B_{6-b}^{2})s^{12}t^{12}&\quad\textrm{for
}\leavevmode\nobreak\ 0<p<P\\\
(4A_{4-a}^{3}+27B_{6-b}^{2})s^{12}t^{12}+\mathcal{O}(t^{13})&\quad\textrm{for
}\leavevmode\nobreak\ p=P\,,\end{array}\right.$ (A.77)

which, when combined with (A.75), indicates that the coefficient of
$s^{12}t^{12}$ is non-vanishing in $\Delta_{p}$, guaranteeing the non-
triviality (A.76). We thus learn that $\Delta$ cannot acquire an overall
common factor of $e_{p}$ and hence that $Y^{p}$ is an I0-component, i.e.,

$n_{p}=0\quad\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\ 0\leq p\leq
P\,.$ (A.78)

#### Class 2: $(>4,6,12)$

The key ideas used in section A.2 for analyzing fibers of Class 1 apply also
to the analysis of Class 2 fibers. In particular, the conditions on the
exponents $\mu_{p,i}$ and $\nu_{p,j}$, as summarized in (A.62) through (A.67),
hold almost unaltered, with the only change being that the vanishing of
$\mu_{p,i}$ for $i\in\mathcal{I}_{0}$ should be modified to

$\mu_{p,i}>0\,,\quad\forall p\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ i\in\mathcal{I}_{0}\,.$
(A.79)

The reason is that the initial vanishing order of $f$ at $s=0$,

${\rm ord}_{\widehat{\mathcal{Y}}}(f)|_{(u,s)=0}=4+\alpha>4\,,$ (A.80)

is strictly bigger than $4$ for Class $2$ non-minimality. It thus follows that
the positivity (A.79) holds for $p=0$, which then propagates via (A.26) to all
$p>0$.

This in turn modifies (A.77) to

$\Delta_{p}=(4f^{3}+27g^{2})|_{e_{p}=0}=\left\\{\begin{array}[]{ l l
}27B_{6-b}^{2}s^{12}t^{12}+\mathcal{O}(s^{13})&\quad\textrm{for
}\leavevmode\nobreak\ p=0\\\ 27B_{6-b}^{2}s^{12}t^{12}&\quad\textrm{for
}\leavevmode\nobreak\ 0<p<P\\\
27B_{6-b}^{2}s^{12}t^{12}+\mathcal{O}(t^{13})&\quad\textrm{for
}\leavevmode\nobreak\ p=P\,,\end{array}\right.$ (A.81)

where the coefficient $B_{6-b}$ on the RHS is a non-zero constant as discussed
in (A.49). Therefore, we have

$\Delta_{p}\not\equiv 0\,,$ (A.82)

and hence $\Delta$ cannot acquire an overall factor of $e_{p}$ for any
$p=0,\cdots,P$, indicating that the surfaces $Y^{p}$ are I0-components.

#### Class 3: $(4,>6,12)$

In much the same way as how the expression (A.77) for the discriminant of a
Class 1 configuration had to change for Class 2 to (A.81), we now have

$\Delta_{p}=(4f^{3}+27g^{2})|_{e_{p}=0}=\left\\{\begin{array}[]{ l l
}4A_{4-a}^{3}s^{12}t^{12}+\mathcal{O}(s^{13})&\quad\textrm{for
}\leavevmode\nobreak\ p=0\\\ 4A_{4-a}^{3}s^{12}t^{12}&\quad\textrm{for
}\leavevmode\nobreak\ 0<p<P\\\
4A_{4-a}^{3}s^{12}t^{12}+\mathcal{O}(t^{13})&\quad\textrm{for
}\leavevmode\nobreak\ p=P\,.\end{array}\right.$ (A.83)

where $A_{4-a}$ is a non-zero constant in (A.48). We thus learn that $Y^{p}$
are I0-components.

#### Class 4: $(4,6,>12)$

With the vanishing orders of $f$ and $g$ given as

$\displaystyle{\rm ord}_{\widehat{\mathcal{Y}}}(f)|_{(u,s)=0}$
$\displaystyle=$ $\displaystyle 4+\alpha=4\,,$ (A.84) $\displaystyle{\rm
ord}_{\widehat{\mathcal{Y}}}(g)|_{(u,s)=0}$ $\displaystyle=$ $\displaystyle
6+\beta=6\,,$ (A.85)

the arguments in section A.2 for Class 1 persist up until (A.72). In
particular, the Weierstrass sections $f$ and $g$ are non-trivial on the
component curves $B^{p}$

$\displaystyle f_{p}=f|_{e_{p}=0}\not\equiv 0\,,\quad
g_{p}=g|_{e_{p}=0}\not\equiv 0\,,\qquad\text{for}\leavevmode\nobreak\
\leavevmode\nobreak\ p=0,\cdots,P\,,$ (A.86)

which already implies that the surfaces $Y^{p}$ are I${}_{n_{p}}$-components.

However, since $\Delta$ vanishes to a higher order as

${\rm ord}_{\widehat{\mathcal{Y}}}(\Delta)|_{(u,s)=0}=12+\gamma>12\,,$ (A.87)

the condition (A.75) changes to

$4A_{4-a}^{3}+27B_{6-b}^{2}=0\,,$ (A.88)

so that the order $12$ terms in $\Delta$ may cancel. Therefore, the
discriminant expression (A.77) for Class 1 simplifies to

$\Delta_{p}=(4f^{3}+27g^{2})|_{e_{p}=0}=\left\\{\begin{array}[]{ l l
}\mathcal{O}(s^{13})&\quad\textrm{for }\leavevmode\nobreak\ p=0\\\
0&\quad\textrm{for }\leavevmode\nobreak\ 0<p<P\\\
\mathcal{O}(t^{13})&\quad\textrm{for }\leavevmode\nobreak\
p=P\,,\end{array}\right.$ (A.89)

guaranteeing the positivity of the vanishing orders,

$v_{\mathcal{P}_{\rm gen}}(\Delta_{p})>0\quad\text{for}\leavevmode\nobreak\
\leavevmode\nobreak\ 0<p<P\,,$ (A.90)

where $\mathcal{P}_{\rm gen}$ is a generic point in $B^{p}$. We thus conclude
that

$n_{p}>0\quad\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\ 0<p<P\,,$
(A.91)

as claimed in (A.54).

#### Class 5: $(>4,>6,>12)$

Non-minimal fibers of Class 5 are distinguished from those of the previous
four classes in that the chain of $P$ blowups does not lead to a (blown-down)
Kulikov model. Specifically, as it turns out, we have

$f_{p}\equiv 0\,,\quad g_{p}\equiv 0\,,\quad\Delta_{p}\equiv
0\,,\qquad\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
p=1,\cdots,P-1\,,$ (A.92)

and therefore none of the surfaces $Y^{p}$ in the middle are In-components.

In order to show (A.92), we first recall from section A.2 that with the
vanishing order of $f$ given as

${\rm ord}_{\widehat{\mathcal{Y}}}(f)|_{(u,s)=0}=4+\alpha>4\,,$ (A.93)

the exponents $\mu_{p,i}$ should obey (A.62), (A.63), and (A.64), except that
the vanishing of $\mu_{p,i}$ for $i\in\mathcal{I}_{0}$ is modified to

$\mu_{p,i}>0\,,\quad\forall p\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ i\in\mathcal{I}_{0}\,.$
(A.94)

Similarly, since the vanishing order of $g$ is assumed to satisfy

${\rm ord}_{\widehat{\mathcal{Y}}}(g)|_{(u,s)=0}=6+\beta>6\,,$ (A.95)

the constraints (A.65), (A.66), and (A.67) still apply to $\nu_{p,j}$, with
the modification

$\nu_{p,j}>0\,,\quad\forall p\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ j\in\mathcal{J}_{0}\,.$
(A.96)

We thus deduce the following positivity properties,

$\mu_{p,i}>0\,,\quad\forall i\,,\quad\text{and}\quad\nu_{p,j}>0\,,\quad\forall
j\,,\qquad\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
p=1,\cdots,P-1\,,$ (A.97)

implying that $f$, $g$, and $\Delta$ identically vanish in each of the middle
components $B^{p}$ as claimed.

###  A.3 Proofs

In the following we will present the proofs of the propositions stated in
section A.1.

###### Proof 1

[Proposition 1].

Let us first note that for non-minimal degenerations of Class 1–4, either of
the following two equations holds:

${\rm
ord}_{\widehat{\mathcal{Y}}}(f)|_{(u,s)=0}=4+\alpha=4\,\quad\text{or}\quad{\rm
ord}_{\widehat{\mathcal{Y}}}(g)|_{(u,s)=0}=6+\beta=6\,.$ (A.98)

We suppose here that the former condition is obeyed (under the latter
condition the argument proceeds in an analogous manner). Since we may assume
without loss of generality that the number $P$ of the required blowups is

$P\geq 2\,,$ (A.99)

the first blow up by itself should not have removed the non-minimality at
$s=0$. As stated in section A.2, for every $i\in\mathcal{I}_{-}$ we thus have

$\mu_{1,i}\geq
4-(a+i)>0\quad\Longrightarrow\quad\mu_{0,i}+a+i=\mu_{1,i}+4>4\,,$ (A.100)

and hence the vanishing order (A.98) of $f$ is achieved by the sector with
$i\in\mathcal{I}_{0}$, i.e.,

$\mathcal{F}_{4-a}s^{4}t^{4}=A_{4-a}s^{4}t^{4}+\cdots\,,$ (A.101)

with $A_{4-a}$ a non-zero constant. Since the term (A.101) does not depend on
each $e_{p}$, the Weierstrass section $f$ has 3-fold-vanishing order $0$ at
$B^{p,p+1}$ for $p=0,\cdots,P-1$. Therefore, the elliptic fiber there is of
Type I${}_{n_{p,p+1}}$, where $n_{p,p+1}$ is subject to

${n_{p,p+1}\geq n_{p}+n_{p+1}}\,,$ (A.102)

since the fiber at $B^{p,p+1}$ is obtained by the collision of an
I${}_{n_{p}}$ and an I${}_{n_{p+1}}$ fiber.

Let us now suppose that

$n_{p}=0=n_{p+1}\,,$ (A.103)

i.e., that we start with a non-minimal fiber of Class 1–3, not of Class 4.
Then, the middle degree sectors,

$\mathcal{F}_{4-a}s^{4}t^{4}\quad\text{and}\quad\mathcal{G}_{6-b}s^{6}t^{6}$
(A.104)

of $f$ and $g$, respectively, lead to the bi-degree $(12,12)$ sector in
$\Delta$ of the form,

$\mathcal{D}_{12-c}(e_{0},\cdots,e_{P})s^{12}t^{12}\,,$ (A.105)

where the coefficient $\mathcal{D}_{12-c}$ contains the following non-trivial
sector independent of each $e_{p}$,

$\mathcal{D}_{12-c}(0,\cdots,0)=\left\\{\begin{array}[]{ l l
}4A_{4-a}^{3}+27B_{6-b}^{2}&\quad\textrm{for Class 1}\\\
27B_{6-b}^{2}&\quad\textrm{for Class 2}\\\ 4A_{4-a}^{3}&\quad\textrm{for Class
3}\,.\end{array}\right.$ (A.106)

We therefore conclude that

$n_{p,p+1}=0\,,$ (A.107)

i.e. that the fiber at $B^{p,p+1}$ is of Type $I_{0}$ for $p=0,\cdots,P-1$.
$\Box$

###### Proof 2

[Proposition 2].

We start by noting that the leading exponents $\mu_{p,i}$ of $e_{p}$ in the
expansion coefficients $\mathcal{F}_{i}$ are subject to the positivity
condition

$\mu_{p,i}>0\quad\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
0<p<P\,,\leavevmode\nobreak\ i\not\in\mathcal{I}_{0}\,.$ (A.108)

This is manifest from the fact that $\\{\mu_{p,i}\\}_{p=0}^{P}$ is an
increasing (resp., decreasing) sequence for $i\in\mathcal{I}_{-}$ (resp.,
$i\in\mathcal{I}_{+}$). On the other hand, for $i\in\mathcal{I}_{0}$ we have

$\mu_{p,4-a}=0\quad\forall p\,,$ (A.109)

since the non-minimality is of Class 4. Similarly, the exponents $\nu_{p,j}$
obey

$\nu_{p,j}>0\quad\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
0<p<P\,,\leavevmode\nobreak\ j\not\in\mathcal{J}_{0}\,,$ (A.110)

and

$\nu_{p,6-b}=0\quad\forall p\,.$ (A.111)

Combining the constraints (A.108)–(A.109), it follows for $0<p<P$ that

$\displaystyle f_{p}$ $\displaystyle=$ $\displaystyle A_{4-a}s^{4}t^{4}\,,$
(A.112) $\displaystyle g_{p}$ $\displaystyle=$ $\displaystyle
B_{6-b}s^{6}t^{6}\,,$ (A.113)

where $A_{4-a}$ and $B_{6-b}$ are non-vanishing constants, and hence, that

$\left(\frac{g}{f}\right)\bigg{|}_{e_{p}=0}=\frac{B_{6-b}}{A_{4-a}}s^{2}t^{2}\,.$
(A.114)

Since this ratio is a perfect square, each middle component $B^{p}$ supports a
codimension-zero I${}_{n_{p}}$ fibers of split form. However, the expression
(A.114) for the ratio does not hold in general along the end components, for
which both split and non-split forms are allowed in principle. $\Box$

###### Proof 3

[Proposition 3].

Let us recall that we have either (A.108) and (A.109) for $\mu_{p,i}$ (if
$\alpha=0$) or (A.110) and (A.111) for $\nu_{p,j}$ (if $\beta=0$). As the
argument proceeds in a symmetric fashion, we will assume that the former holds
with $\alpha=0$, i.e.,

${\rm ord}_{\widehat{\mathcal{Y}}}(f)|_{(u,s)=0}=4+\alpha=4\,.$ (A.115)

Then, for $0<p<P$, we have

$f_{p}=A_{4-a}s^{4}t^{4}\,,$ (A.116)

which vanishes nowhere in $B^{p}$. Therefore, at every point $\mathcal{P}$ in
the middle component $B^{p}$ the K3-vanishing order of $f$ vanishes, i.e.,

${\rm ord}_{Y_{0}}(f)|_{\mathcal{P}}=0\,,\quad\forall\mathcal{P}\in
B^{p}\,,\leavevmode\nobreak\ 0<p<P\,,$ (A.117)

and hence, only In enhancements can occur for the $8$-dimensional gauge
theory. $\Box$

###### Proof 4

[Proposition 4].

The discriminant $\Delta$ factorizes as

$\Delta=\prod_{q=0}^{P}e_{q}^{n_{q}}\Delta^{\prime}\,,$ (A.118)

with $\Delta^{\prime}$ admitting no overall factors of $e_{q}$ for any $q$.
Here, the exponents $n_{q}$ appear since $Y^{q}$ are I${}_{n_{q}}$-components
for Class 1–4 models, but we can define $n_{q}$ even for Class 5
degenerations.

In studying the branes localized in a fixed base component $B^{p}$, it
suffices to analyze the terms in $\Delta$ that come precisely with the minimal
power of $e_{p}$, i.e., with $e_{p}^{n_{p}}$. Each such term originates either
from a cubic product of the terms in $f$ or a quadratic product of the terms
in $g$. Let us focus first on terms of the former type, given by

$4A_{i_{i}}A_{i_{2}}A_{i_{3}}\Big{(}\prod_{q=0}^{P}e_{q}^{\mu_{q,i_{1}}+\mu_{q,i_{2}}+\mu_{q,i_{3}}}\Big{)}s^{3a+i_{1}+i_{2}+i_{3}}t^{24-3a-i_{1}-i_{2}-i_{3}}\,,$
(A.119)

for a triple of indices $(i_{1},i_{2},i_{3})$ subject to

$\mu_{p,i_{1}}+\mu_{p,i_{2}}+\mu_{p,i_{3}}=n_{p}\,.$ (A.120)

First, for the left-end component with $p=0$, the points on the rational curve
$B^{0}$ are parametrized by the homogeneous coordinates $[t:e_{1}]$, and the
terms of the form (A.119) in $\Delta$ are of homogeneous degree

$\displaystyle(24-3a-i_{1}-i_{2}-i_{3})+(\mu_{1,i_{1}}+\mu_{1,i_{2}}+\mu_{1,i_{3}})$
$\displaystyle=$ $\displaystyle(\mu_{0,i_{1}}+\mu_{0,i_{2}}+\mu_{0,i_{3}})+12$
(A.121) $\displaystyle=$ $\displaystyle n_{0}+12\,,$ (A.122)

where we have used the recursion relation (A.26) in the first step and the
constraint (A.120) in the second.272727Note that we are implicitly assuming
that the triple products that involve subleading terms in the expansion (A.23)
remain subleading in $\Delta$. This may not be the case if cancellations occur
amongst products of the form (A.119) from $f^{3}$ that only involve the
leading terms and analogous products from $g^{2}$. However, all that we have
used in deriving the homogeneous degree (A.122) is the recursion relation
(A.26) and the constraint (A.120), which continue to hold for the subleading
terms. The argument therefore goes through even if cancellations occur in
$\Delta$. With this understanding, and to be explicit, we will henceforth only
consider cross products of leading terms whenever we need to consider such
products. One can also see that, for a similar reason, the same homogeneous
degree (A.122) arises for the terms that originate from $27g^{2}$. Since the
discriminant $\Delta$ has an overall factor of $e_{1}^{n_{1}}$ as in (A.118),
we thus conclude that the zeroes of $\Delta^{\prime}$ in $B^{0}$ are counted
as

$x_{0}=(n_{0}+12)-n_{1}\,,$ (A.123)

as claimed by (A.56). Similarly, we can count the zeroes of $\Delta^{\prime}$
on the right-end component $B^{P}$ as

$x_{P}=(n_{P}+12)-n_{P-1}\,,$ (A.124)

as in (A.57). Finally, if $0<p<P-1$, the homogenous degree of
$\Delta^{\prime}$ in the coordinates $e_{p\pm 1}$ of $B^{p}$ is read off,
e.g., from the cubic products (A.119), as

$\displaystyle\sum_{\delta\in\\{-1,1\\}}\left(\sum_{k=1}^{3}\mu_{p+\delta,i_{k}}-n_{p+\delta}\right)$
$\displaystyle=$ $\displaystyle
2\sum_{k=1}^{3}\mu_{p,i_{k}}-(n_{p-1}+n_{p+1})$ (A.125) $\displaystyle=$
$\displaystyle 2n_{p}-n_{p-1}-n_{p+1}\,,$ (A.126)

where, once again, the relation (A.26) and the constraint (A.120) have been
used in turn. We thus learn that $\Delta^{\prime}$ has

$x_{p}=2n_{p}-n_{p-1}-n_{p+1}\,,$ (A.127)

zeroes in the middle components $B^{p}$, thereby confirming (A.58). $\Box$

###### Proof 5

[Proposition 5].

We prove the proposition in the following two steps.

#####  Step 1 :

Let us start by performing the base change

$u\to u^{(4-a)!(6-b)!}$ (A.128)

where $a$ and $b$ are the K3-vanishing orders (A.6) of the Weierstrass
sections $f$ and $g$ for a generic $u\neq 0$.282828To be precise, we are
assuming $a\leq 4$ and $b\leq 6$ in (A.128). For $a>4$ the base change should
be replaced by $u\to u^{(6-b)!}$, and similarly for $b>6$, by $u\to
u^{(4-a)!}$. Since the leading exponents $\mu_{p,i}$ and $\nu_{p,j}$ obey the
relations (A.26) and (A.27), after the base change (A.128), there must exist
either $i_{0}\in\mathcal{I}_{-}$ with $\mu_{P,i_{0}}=0$ or
$j_{0}\in\mathcal{J}_{-}$ with $\nu_{P,j_{0}}=0$. In the rest of this proof we
will assume the former (a similar argument can be applied in the latter case).

The Weierstrass section $f$ restricted to the right-end component thus behaves
as

$f_{P}:=f|_{e_{P}=0}=\sum_{i\in\mathcal{I}_{-}^{(0)}}\mathcal{F}_{i}s^{a+i}t^{8-a-i}+\mathcal{F}_{4-a}|_{e_{P}=0}s^{4}t^{4}\,,$
(A.129)

where $\mathcal{I}_{-}^{(0)}\subset\mathcal{I}_{-}$ is the index subset with
the defining property $\mu_{P,i}=0$. Importantly, this subset
$\mathcal{I}_{-}^{(0)}$ is non-empty since $i_{0}\in\mathcal{I}_{-}^{(0)}$ by
assumption. Therefore, the K3-vanishing order of $f_{P}$ at $s=0$ is strictly
less than $4$, and hence, the fiber there is no longer physically non-minimal.

#####  Step 2 :

While the absence of potential physical non-minimality is guaranteed at $s=0$
thanks to the base change (A.128), it may still be present away from $s=0$ on
the right-end $B^{P}$. We thus suppose that the fiber is physically non-
minimal at a point $\mathcal{P}_{*}\in B^{P}$,

$\mathcal{P}_{*}:\\{s=s_{*}e_{P-1}\\}\,,\quad\text{for}\leavevmode\nobreak\
\leavevmode\nobreak\ s_{*}\neq 0\,,$ (A.130)

with $[s:e_{P-1}]$ being the homogeneous coordinates of the rational component
$B^{P}$.

Let us now recall that $f_{P}$ and $g_{P}$ are sections of degree $4$ and $6$
line bundles $L_{P}^{\otimes 4}$ and $L_{P}^{\otimes 6}$, respectively, where
$L_{P}$ is of degree $1$ on $B^{P}$. We then note that $f_{P}$ is non-trivial
since it acquires at least one non-trivial term from the summand with
$i_{0}\in\mathcal{I}_{-}^{(0)}$ in (A.129). As the maximal power of $s$
appearing in (A.129) is $4$, the following necessary condition for physical
non-minimality,

${\rm ord}_{Y_{0}}(f)|_{\mathcal{P}_{*}}\geq 4\,,$ (A.131)

must be saturated and $f_{P}$ must take the form,

$f_{P}=A(s-\sigma_{*}e_{\downarrow}t)^{4}t^{4}\,.$ (A.132)

Here, $A$ and $\sigma_{*}$ are non-zero constant and the latter is determined
in terms of $s_{*}$ via

$\sigma_{*}e_{\downarrow}t=s_{*}e_{P-1}\,,$ (A.133)

where $e_{\downarrow}$ denotes the product of $e_{p}$,

$e_{\downarrow}=\prod_{p=0}^{P}e_{p}^{P-p}\,,$ (A.134)

with the exponents decreasing in $p$. We also observe from (A.132) that the
physical non-minimality assumption at $\mathcal{P}_{*}$ implies

$a=0$ (A.135)

for the initial K3-vanishing order of $f$ and

$\alpha=0$ (A.136)

for the class distinction, i.e. Class 2 is ruled out. In fact the global
behavior of $f$ can be specified as

$f=A(s-\sigma_{*}e_{\downarrow}t)^{4}t^{4}+\sum_{i=5}^{8}A_{i}(e_{\rightarrow})^{\mu_{0,i}}(e_{\uparrow})^{i-4}s^{i}t^{8-i}\,,$
(A.137)

where $e_{\uparrow}$ and $e_{\rightarrow}$ are defined as

$\displaystyle e_{\uparrow}$ $\displaystyle=$
$\displaystyle\prod_{p=0}^{P}e_{p}^{p}\,,$ (A.138) $\displaystyle
e_{\rightarrow}$ $\displaystyle=$ $\displaystyle\prod_{p=0}^{P}e_{p}\,.$
(A.139)

Note that, for simplicity, any contributions to $f$ from the higher-order
terms in (A.23) have been neglected in (A.137) and the constant $A$ can been
further specified as $A_{4}$.

If the initial non-minimality is of Class 1 or 4 with $\beta=0$ on top of
(A.136), $g_{P}$ is also non-trivial in presence of the non-trivial
contribution,

$\mathcal{G}_{6-b}s^{6}t^{6}\,,$ (A.140)

and hence, a similar conclusion can be drawn for $g_{P}$,

$g_{P}=B(s-\sigma_{*}e_{\downarrow}t)^{6}t^{6}\,,$ (A.141)

where $B$ is a non-zero constant. This implies that

$b=0\,,$ (A.142)

and also leads to the global behavior of $g$,

$g=B(s-\sigma_{*}e_{\downarrow}t)^{6}t^{6}+\sum_{j=7}^{12}B_{j}(e_{\rightarrow})^{\nu_{0,j}}(e_{\uparrow})^{j-6}s^{j}t^{12-j}\,,$
(A.143)

again modulo higher-order terms in (A.24). In case the initial non-minimality
was of Class 3 with $\beta>0$ instead, the global form (A.141) would still be
valid, except that $B$ is not a constant but rather

$B=B_{6}(e_{\rightarrow})^{\nu_{0}}\,,$ (A.144)

where $\nu_{0}$ is the common exponent of $e_{p}$ in $\mathcal{G}_{6}$.
However, this modification does not affect the argument that will follow, and
hence, we will assume for simplicity of the notation that $\beta=0$.

All in all, we have so far seen that the Weierstrass sections, modulo higher-
order terms, should take the form

$\displaystyle f$ $\displaystyle=$ $\displaystyle
A_{4}(s-\sigma_{0}e_{\downarrow}t)^{4}t^{4}+\sum_{i=5}^{8}A_{i}(e_{\rightarrow})^{\mu_{0,i}}(e_{\uparrow})^{i-4}s^{i}t^{8-i}\,,$
(A.145) $\displaystyle g$ $\displaystyle=$ $\displaystyle
B_{6}(s-\sigma_{0}e_{\downarrow}t)^{6}t^{6}+\sum_{j=7}^{12}B_{j}(e_{\rightarrow})^{\nu_{0,j}}(e_{\uparrow})^{j-6}s^{j}t^{12-j}\,.$
(A.146)

We will now prove that the physical non-minimality at $\mathcal{P}_{*}$,

$[s:e_{P-1}]=[s_{*}:1]\,,$ (A.147)

can eventually be removed by an additional chain of blowups. To be explicit,
we will start by introducing a new pair of homogeneous coordinates for
$B^{P}$:

$[s:e_{P-1}]\to[s:v]\,,$ (A.148)

where the coordinate $v$ is defined as

$v:=s-s_{*}e_{P-1}\,.$ (A.149)

Then, the Weierstrass section $f$ can be rewritten as

$\displaystyle f$ $\displaystyle=$ $\displaystyle
A_{4}v^{4}t^{4}+\sum_{i=5}^{8}A_{i}\prod_{p=0}^{P}e_{p}^{\mu_{0,i}+(i-4)p}s^{i}t^{8-i}\,$
(A.150) $\displaystyle=$ $\displaystyle
A_{4}v^{4}t^{4}+\sum_{i=5}^{8}A_{i}\prod_{p=0}^{P-2}e_{p}^{\mu_{0,i}+(i-4)p}s_{*}^{-K_{i}}(s-v)^{K_{i}}e_{P}^{\mu_{0,i}+(i-4){P}}s^{i}t^{8-i}\,$
(A.151)

with $K_{i}:=\mu_{0,i}+(i-4)(P-1)$, and similarly, $g$ as

$\displaystyle g$ $\displaystyle=$ $\displaystyle
B_{6}v^{6}t^{6}+\sum_{j=7}^{12}B_{j}\prod_{p=0}^{P-2}e_{p}^{\nu_{0,j}+(j-6)p}s_{*}^{-L_{j}}(s-v)^{L_{j}}e_{P}^{\nu_{0,j}+(j-6){P}}s^{j}t^{12-j}\,,$
(A.152)

with $L_{j}:=\nu_{0,j}+(j-6)(P-1)$.292929Strictly speaking, the expressions
(A.151) and (A.152) are not the most appropriate ones for a global algebraic-
geometric description since there appear negative powers of the coordinates,
$t,e_{0},\cdots,e_{P-2}\,,$ (A.153) via $s_{*}^{-K_{i}}$ and $s_{*}^{-L_{j}}$,
respectively, where  $s_{*}=\sigma_{*}te_{0}^{P}e_{1}^{P-1}\cdots
e_{P-2}^{1}\,.$ (A.154) For the purpose of analyzing the behavior of $f$ and
$g$ restricted to $B^{P+q}$ for $q=0,\cdots,Q$, however, one may set all the
coordinates (A.153) to unity.

Clearly, we have to blow up at $v=e_{P}=0$ since $P$ has been made large
enough via the initial base change (A.128). To make a precise statement, let
us first define

$Q:={\rm min}\\{Q_{f},Q_{g}\\}\,,$ (A.155)

where $Q_{f}$ and $Q_{g}$ are given as

$\displaystyle Q_{f}$ $\displaystyle=$ $\displaystyle{\rm
min}_{i=5}^{8}\\{\lfloor{\frac{\mu_{0,i}+(i-4)P}{4}}\rfloor\\}\,,$ (A.156)
$\displaystyle Q_{g}$ $\displaystyle=$ $\displaystyle{\rm
min}_{j=7}^{12}\\{\lfloor{\frac{\nu_{0,j}+(j-6)P}{6}}\rfloor\\}\,.$ (A.157)

Our claim is that the non-minimality is removed by performing additional $Q$
blowups, where the $q^{\rm th}$ additional blowup for $1\leq q\leq Q$ is the
operation that transforms $f$ and $g$ as

$\displaystyle f(s,t;e_{0},\ldots,e_{P+q-1})$ $\displaystyle\longrightarrow$
$\displaystyle f(s,t;e_{0},\ldots,e_{P+q})\,,$ (A.158) $\displaystyle
g(s,t;e_{0},\ldots,e_{P+q-1})$ $\displaystyle\longrightarrow$ $\displaystyle
g(s,t;e_{0},\ldots,e_{P+q})\,,$ (A.159)

by applying the coordinate replacement,

$\displaystyle e_{P+q-1}$ $\displaystyle\longrightarrow$ $\displaystyle
e_{P+q-1}\,e_{P+q}\,,$ (A.160) $\displaystyle v$
$\displaystyle\longrightarrow$ $\displaystyle v\,e_{P+q}\,,$ (A.161)

and dividing the resulting $f$ and $g$ by $e_{P+q}^{4}$ and $e_{P+q}^{6}$,
respectively. At the end of these additional $Q$ blowups, we thus obtain

$\displaystyle f$ $\displaystyle=$ $\displaystyle
A_{4}v^{4}+\sum_{i=5}^{8}A_{i}\prod_{p=0}^{P-2}e_{p}^{\mu_{0,i}+(i-4)p}s_{*}^{-K_{i}}(s-v\prod_{q=1}^{Q}e_{P+q})^{K_{i}}\prod_{q=0}^{Q}e_{P+q}^{\mu_{0,i}+(i-4){P-4q}}s^{i}t^{8-i}\,,$
(A.162) $\displaystyle g$ $\displaystyle=$ $\displaystyle
B_{6}v^{6}+\sum_{j=7}^{12}B_{j}\prod_{p=0}^{P-2}e_{p}^{\nu_{0,j}+(j-6)p}s_{*}^{-L_{j}}(s-v\prod_{q=1}^{Q}e_{P+q})^{L_{j}}\prod_{q=0}^{Q}e_{P+q}^{\nu_{0,j}+(j-6){P-6q}}s^{j}t^{12-j}\,,$

and the chain (A.51) of base curves further expands to

$B^{0}-B^{1}-\cdots-B^{P+Q-1}-B^{P+Q}\,.$ (A.163)

The self-intersection numbers are $-2$ for the middle components and $-1$ for
the end components $B^{0}$ and $B^{P+Q}$. Obviously, $f$ and $g$ are non-
trivial on each of the additional components $B^{P+q}$. Therefore, the special
fibers on the components $B^{P+q}$ with $q=0,\cdots,Q-1$ can only be of
Kodaira Type In.

Let us finally turn to the right-end component, $B^{P+Q}$, with homogeneous
coordinates $[v:e_{P+Q-1}]$. Note that, upon restricting to this end
component, the following coordinates,

$s,t,e_{0},e_{1},\cdots,e_{P+Q-2}\,,$ (A.164)

may all be set to unity, and hence, the Weierstrass sections (A.162) and (A.3)
simplify as

$\displaystyle f_{P+Q}$ $\displaystyle=$ $\displaystyle
A_{4}v^{4}+\sum_{i=5}^{8}A_{i}\sigma_{*}^{-K_{i}}e_{P+Q-1}^{\mu_{0,i}+(i-4){P-4(Q-1)}}\big{(}e_{P+Q}^{\mu_{0,i}+(i-4){P-4Q}}\big{)}|_{e_{P+Q}=0}\,,$
(A.165) $\displaystyle g_{P+Q}$ $\displaystyle=$ $\displaystyle
B_{6}v^{6}+\sum_{j=7}^{12}B_{j}\sigma_{*}^{-L_{j}}e_{P+Q-1}^{\nu_{0,j}+(j-6){P-6(Q-1)}}\big{(}e_{P+Q}^{\nu_{0,j}+(j-6){P-6Q}}\big{)}|_{e_{P+Q}=0}\,.$
(A.166)

Here, the relation (A.133) has been used to replace $s_{*}$ by $\sigma_{*}$
and the summands are non-trivial only for $i\in\mathcal{I}_{+}$ with

$\mu_{0,i}+(i-4)P-4Q=0\,,$ (A.167)

and for $j\in\mathcal{J}_{+}$ with

$\nu_{0,j}+(j-6)P-6Q=0\,.$ (A.168)

Let us define the subsets $\mathcal{I}_{+}^{(0)}\subset\mathcal{I}_{+}$ and
$\mathcal{J}_{+}^{(0)}\subset\mathcal{J}_{+}$ by collecting those $i$ and $j$,
respectively. Now, with the base change (A.128) assumed in the very beginning,
it is clear that $Q_{f}$ and $Q_{g}$ in (A.156) and (A.157) could as well have
been defined without the floor symbols. Therefore, we learn that either
$\mathcal{I}_{+}^{(0)}$ or $\mathcal{J}_{+}^{(0)}$ should be non-empty, and
hence, that the expressions (A.165) and (A.166) can be schematically written
as

$\displaystyle f_{P+Q}$ $\displaystyle=$ $\displaystyle
A_{4}(v^{4}+Ke_{P+Q-1}^{4})\,,$ (A.169) $\displaystyle g_{P+Q}$
$\displaystyle=$ $\displaystyle B_{6}(v^{6}+Le_{P+Q-1}^{6})\,,$ (A.170)

where $K$ and $L$ are constants that cannot both be $0$. This in turn
guarantees that no physical non-minimality may arise in $B^{P+Q}$. $\Box$

##  Appendix B From Kulikov Models to Non-minimal Degenerations

In Appendix A we have shown that to each non-minimal Weierstrass degeneration
$\widehat{\cal Y}$ of Class 1, 2, 3, or 4 one can associate a Kulikov
Weierstrass model $\cal Y$, which is the fibral blowdown of a Kulikov model
${\cal X}$ of Type II or III. We now prove the converse, in the sense that
every elliptic Type II or Type III Kulikov model ${\cal X}$, and therefore
every Kulikov model associated with a degeneration at infinite distance, is
the blowup of a Weierstrass degeneration of Class 1, 2, 3 or 4, or
alternatively of a Weierstrass model with degenerate fibers in codimension
zero, but without non-minimal fibers in codimension one.

The idea is a straightforward reversal of the arguments of Appendix A: We
start by fixing a Kulikov model $\mathcal{X}$ for a degeneration of Kulikov
Type II or III. Let us denote by $\mathcal{Y}$ the Weierstrass model
associated with $\mathcal{X}$ and by $Y_{0}$ the central element of the family
$\mathcal{Y}$, which decomposes as

$Y_{0}=\cup_{p=0}^{P}Y^{p}\,.$ (B.1)

As explained in Appendix A, the base of $Y_{0}$ is guaranteed to decompose
accordingly and in fact takes the form of a chain,

$B^{0}-B^{1}-\cdots-B^{P-1}-B^{P}\,.$ (B.2)

The components $B^{p}$ are rational curves and, under the assumption that
$P>0$, their self-intersection numbers on the base ${\cal B}$ of $\mathcal{Y}$
are $-2$ for $0<p<P$ and $-1$ for $p=0,P$. Therefore, we can perform a chain
of blowdowns starting from the right-end to contract the $P$ base components,
$B^{P},\ldots,B^{1}$, which leads to a single-component configuration. The aim
of this Appendix is then to prove that this end configuration contains a non-
minimal fiber of Class 1, 2, 3 or 4. More precisely, we put forward the
following claim:

###### Proposition 6

Let $\mathcal{X}$ be a Type II or III Kulikov model of elliptic K3 surfaces
and $\mathcal{Y}$ be the associated Weierstrass model, whose central fiber
$Y_{0}$ decomposes into $P+1$ components as in (B.1), elliptically fibered
itself over the chain of base curves (B.2). Let us define
$\widehat{\mathcal{Y}}$ as the blowdown of $\mathcal{Y}$ along the base
$\mathcal{B}$ of the elliptic threefold $\mathcal{V}$, obtained by contracting
the $P$ base curves $B^{P},\ldots,B^{1}$ in turn, and $\hat{Y}_{0}$ as the
central fiber of $\widehat{\mathcal{Y}}$. Then, the surface $\hat{Y}_{0}$ can
never have a non-minimal fiber of Class 5. Specifically, there exists a non-
minimal fiber of Class 1, 2, 3 or 4 in $\hat{Y}_{0}$ unless $P=0$ in (B.1).
For $P=0$ the Weierstrass model (A.1) of $\widehat{\mathcal{Y}}$ is given by

$f_{u}=-3h^{2}+u\eta\,,\qquad g_{u}=-2h^{3}+u\rho\,,$ (B.3)

where $h$, $\eta$ and $\rho$ are sections of degree 4, 8 and 12, respectively,
and $h$ has four distinct zeroes.

Proof. We employ the same notation as in Appendix A.1, although we start here
from the $(P+1)$-component configuration $Y_{0}$ rather than its blown-down
version $\hat{Y}_{0}$. For instance, the deformation parameter of the model
$\widehat{\mathcal{Y}}$ will be denoted by $u$ and the homogeneous coordinates
for the rational base of $\hat{Y}_{0}$ will be denoted by $[s:t]$.
Furthermore, we use the symbols $e_{p}$ with $p=0,\ldots,P$ for the
coordinates whose vanishing loci on the base $\mathcal{B}$ of $\mathcal{Y}$
describe the curve components $B^{p}$.

Let us first consider the special cases where the decomposition (B.1) is
trivial in the sense that $P=0$, which indicates that $n_{0}>0$; otherwise,
the Kulikov model $\mathcal{X}$ would be of Type I, contradicting the starting
assumption. The unique I${}_{n_{0}}$-component $Y^{0}$ of the central fiber
$Y_{0}$ is thus described by a Weierstrass model of the form (B.3). Here, it
follows that the degree-four section $h$ has four distinct zeroes since
otherwise $Y^{0}$ would develop a non-minimal fiber from the perspective of
the K3 surface, which would, in turn, contradict the fact that $Y^{0}$ is
obtained as a fibral blowdown of the central K3 fiber of the Kulikov model
$\mathcal{X}$.

Having characterized the blowndown configuration of a model with $P=0$, let us
now focus on the generic cases for which $P\geq 1$. Since the $P$ blowdowns
transform $\mathcal{Y}$ to $\widehat{\mathcal{Y}}$, there must exist a well-
defined sequence of reverse operations, i.e., $P$ blowups in the base, which
turn $\widehat{\mathcal{Y}}$ back to $\mathcal{Y}$. However, these blowups are
not necessarily performed at the “same” point in the following sense: We first
perform $P_{1}$ blowups, labelled by $p=1,\ldots,P_{1}$, at the points

$v^{(1)}:=s-s_{*}^{(1)}e_{p-2}=0\,,$ (B.4)

where $[s:v^{(1)}]$ are homogeneous coordinates of the right-end base curve
$B^{p-1}$ after the $(p-1)^{\rm th}$ blowup.303030We formally define
$e_{-1}:=t$ for the coordinate definition (B.4) to be valid also for $p=1$. We
next perform another set of blowups labelled by $p=P_{1}+1,\cdots,P_{2}$ at

$v^{(2)}:=s-s_{*}^{(2)}e_{p-2}=0\,,$ (B.5)

where $[s:v^{(2)}]$ are coordinates of $B^{p-1}$ with $s_{*}^{(2)}\neq
s_{*}^{(1)}$, and so on. In this way, we perform a total of
$P=\sum_{k=1}^{m}P_{k}$ blowups, subdivided into $P_{k}$ blowups at
$v^{(k)}=0$ for $k=1,\cdots,m$, where the coordinates $v^{(k)}$ of $B^{p-1}$
after the $(p-1)^{\rm th}$ blowup are defined as

$v^{(k)}:=s-s_{*}^{(k)}e_{p-2}\,.$ (B.6)

Here the total of $P$ blowups are performed at $m$ “different” points in that

$s_{*}^{(k)}\neq s_{*}^{(k+1)}\,,\quad\text{for}\quad k=1,\cdots,m-1\,.$ (B.7)

Notably, this general class of blowup chains reproduces the subclass described
in section (A.1) if $m=1$, in which case we may assume without loss of
generality that $s_{*}^{(1)}=0$.

Let us first suppose that the blowups are indeed to be repeatedly performed
only at the point with $m=1$. If the initial non-minimality at $s=0$ had been
of Class 5, according to (A.92), none of the middle components would be an In-
component, which contradicts the fact that $\mathcal{Y}$ is the fibral
blowdown of a Kulikov model.313131In principle, we may imagine a certain
degenerate surface $\hat{Y}_{0}$ containing a Class 5 fiber that leads to
(B.1) with $P=1$, so that $Y_{0}$ may have no middle components. However, in
such a model, the right-end component $Y^{P=1}$ can never be an In-component
for the following reason: Since the initial 3-fold-vanishing orders of $f$ and
$g$ at $s=u=0$ are strictly bigger than 4 and 6, respectively, the
coefficients $\mathcal{F}_{i}$ for $i\in\mathcal{I}_{-}\cup\mathcal{I}_{0}$
have the leading powers $\mu_{0,i}$ strictly bigger than $4-a-i$, and
similarly, $\nu_{0,j}$ are strictly bigger than $6-b-j$ for
$j\in\mathcal{J}_{-}\cup\mathcal{J}_{0}$. This means that $\mu_{1,i}$ and
$\nu_{1,j}$ are strictly positive for all index values of $i$ and $j$, i.e.,
that $f$ and $g$ are both divisible by $e_{1}$. Therefore, the codimension-
zero fibers over $B^{1}$ cannot be of Type In.

An analogous argument works even when $m>1$, in which case we blow the
configuration down in $m$ steps. In other words, we first perform $P_{m}$
blowdowns, by which the base chain (B.2) reduces to

$B^{0}-B^{1}-\cdots-B^{P-P_{m}}\,.$ (B.8)

Here, the right-end component $B^{P-P_{m}}$ supports a non-minimal fiber at
$v^{(m)}=0$, where $v^{(m-1)}\neq 0$. However, this does not keep us from
blowing the geometry further down, which is precisely the second situation of
Proposition 5. We thus proceed further to perform $P_{m-1}$ blowdowns even
though non-minimal fibers arise now at a different point, i.e., at
$v^{(m-1)}=0$. The base chain then further reduces to

$B^{0}-B^{1}-\cdots-B^{P-P_{m}-P_{m-1}}\,.$ (B.9)

By repeating the operations of this kind, we end up with a base of the form

$B^{0}-B^{1}-\cdots-B^{P_{1}}\,,$ (B.10)

where each component curve $B^{p}$ supports fibers of Type I${}_{n_{p}}$ at
codimension-zero. Therefore, for the same reason as in the $m=1$ case, we
conclude that after performing the remaining $P_{1}$ blowdowns, the non-
minimal fiber at $v^{(1)}=u=0$ cannot be of Class 5. $\Box$

As an immediate consequence of Proposition 6, we learn that it suffices to
consider Class 1–4 fibers or Weierstrass models of the form (B.3) in
classifying the effective physics of non-minimal fibers at infinite distance
in the complex structure moduli space.

However, Proposition 6 does not imply that every Class 5 model must
necessarily be equivalent to a Type I Kulikov model. Rather, starting with a
Type II or Type III Kulikov model ${\cal X}$ and its Weierstrass model ${\cal
Y}$, we can first blow down the latter along the base to a degenerate
Weierstrass model $\widehat{\cal Y}$, with a non-minimal degeneration of Class
1 – 4; but depending on the details, it might be possible to apply to this
$\widehat{\cal Y}$ a rescaling of the Weierstrass sections together with a
base change in such a way as to artificially enhance the fiber over generic
points of the base to a non-In fiber. Then performing suitable blowups to
remove the non-minimality may give rise to a degenerate Weierstrass model
${\cal Y}^{\prime}$ (whose central fiber $Y_{0}^{\prime}$ contains components
with generic fibers not of Kodaira Type Ik) which in turn admit a different
blowdown to a model $\widehat{\cal Y}^{\prime}$ with a Class 5 non-minimal
degeneration.

In Section 4.5 we present two examples of Class 5 degenerations which give
rise to Type II Kulikov Weierstrass models, by the reverse of this process.
However, in all such cases it remains true that the Kulikov Type II or Type
III Weierstrass model is associated with a more economical degeneration
$\widehat{\cal Y}$ with non-minimal fibers of Class 1, 2, 3, or 4, as claimed
by Proposition 6.

##  Appendix C Special Tunings for Type III Kulikov Models

According to the general classification of Section 3.2, Kulikov Weierstrass
model of Type III come in three versions: Type III.a models with two rational
elliptic end components, those with only one rational elliptic end component
and finally Type III.b models, where none of the end components is a rational
elliptic surface. These three models are therefore distinguished by the value
of $n_{p}$ in the generic I${}_{n_{p}}$ fiber of the components $Y^{p}$ for
$p=0$ and $p=P$. The criterion for a Type III.b model is that for both end
components

$\text{Type III.b:}\qquad n_{0}>0\quad\text{and}\quad n_{P}>0\,,$ (C.1)

while for Type III.a models with only one rational component only $n_{P}>0$
and $n_{0}=0$.

In the following we rewrite the criterion (C.1) in terms of the explicit
tuning of the Weierstrass sections $(f,g)$. To this end, we first define the
index subsets

$\displaystyle\mathcal{I}_{+}^{(0)}$ $\displaystyle:=$
$\displaystyle\\{\,i\in\mathcal{I}_{+}\;|\;\mu_{0,i}=0\,\\}\,,$ (C.2)
$\displaystyle\mathcal{J}_{+}^{(0)}$ $\displaystyle:=$
$\displaystyle\\{\,j\in\mathcal{J}_{+}\;|\;\nu_{0,j}=0\,\\}\,,$ (C.3)

in terms of which the restrictions of the Weierstrass sections $f$ and $g$ to
the end component $B^{0}$ are given respectively as

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle\Big{(}\mathcal{F}_{4-a}s^{4}t^{4}+\sum_{i\in\mathcal{I}_{+}^{(0)}}\mathcal{F}_{i}s^{a+i}t^{8-a-i}\Big{)}\,,$
(C.4) $\displaystyle g_{0}$ $\displaystyle=$
$\displaystyle\Big{(}\mathcal{G}_{6-b}s^{6}t^{6}+\sum_{j\in\mathcal{J}_{+}^{(0)}}\mathcal{G}_{j}s^{b+j}t^{12-b-j}\Big{)}\,.$
(C.5)

By abuse of notation, we here denote by $\mathcal{F}_{i}$ and
$\mathcal{G}_{j}$ only the leading monomials in the expansions (A.23) and
(A.24). Similarly, with the index subsets

$\displaystyle\mathcal{I}_{-}^{(0)}$ $\displaystyle:=$
$\displaystyle\\{\,i\in\mathcal{I}_{-}\;|\;\mu_{P,i}=0\,\\}\,,$ (C.6)
$\displaystyle\mathcal{J}_{-}^{(0)}$ $\displaystyle:=$
$\displaystyle\\{\,j\in\mathcal{J}_{-}\;|\;\nu_{P,j}=0\,\\}\,,$ (C.7)

we find on the end component $B^{P}$ the restrictions

$\displaystyle f_{P}$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{i\in\mathcal{I}_{-}^{(0)}}\mathcal{F}_{i}s^{a+i}t^{8-a-i}+\mathcal{F}_{4-a}s^{4}t^{4}\Big{)}\,,$
(C.8) $\displaystyle g_{P}$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{j\in\mathcal{J}_{-}^{(0)}}\mathcal{G}_{j}s^{b+j}t^{12-b-j}+\mathcal{G}_{6-b}s^{6}t^{6}\Big{)}\,,$
(C.9)

where, once again, $\mathcal{F}_{i}$ and $\mathcal{G}_{j}$ represent only the
leading monomials therein.

Given these expressions for the Weierstrass sections, the criterion (C.1) can
be tested via the equivalent criterion,

$\Delta_{0}\equiv 0\quad\text{and}\quad\Delta_{P}\equiv 0\,,$ (C.10)

which, in turn, requires

$\displaystyle f_{0}$ $\displaystyle=$ $\displaystyle-3h^{2}\,,$ (C.11)
$\displaystyle g_{0}$ $\displaystyle=$ $\displaystyle-2h^{3}\,,$ (C.12)

and

$\displaystyle f_{P}$ $\displaystyle=$ $\displaystyle-3l^{2}\,,$ (C.13)
$\displaystyle g_{P}$ $\displaystyle=$ $\displaystyle-2l^{3}\,.$ (C.14)

Here, $h=h(s,t;e_{1},\cdots,e_{P})$ and $l=l(s,t;e_{0},\cdots,e_{P-1})$ are
polynomials of the form,

$h=s^{2}(h_{0}t^{2}+h_{1}s\,t+h_{2}s^{2})\quad\text{and}\quad
l=t^{2}(l_{0}t^{2}+l_{1}s\,t+l_{2}s^{2})\,,$ (C.15)

with the coefficient functions given as

$h_{0}=H_{0}\,;\qquad h_{1}=H_{1}\prod_{p=1}^{P}e_{p}^{p}\,;\qquad
h_{2}=H_{2}\prod_{p=1}^{P}e_{p}^{2p}\,,$ (C.16)

and

$l_{0}=L_{0}\prod_{p=0}^{P-1}e_{p}^{2(P-p)}\,;\qquad
l_{1}=L_{1}\prod_{p=0}^{P-1}e_{p}^{P-p}\,;\qquad l_{2}=L_{2}\,,$ (C.17)

for some complex constants $H_{0,1,2}$ and $L_{0,1,2}$.

To test if a degenerate $\hat{Y}_{0}$ with a Class 4 fiber has this weak-
coupling property, let us simply rewrite the above set of requirements in
terms of the initial Weierstrass data as the following factorizability
criteria:

$\displaystyle
f_{+}:=\mathcal{F}_{4-a}s^{4}t^{4}+\sum_{i\in\mathcal{I}_{+}^{(0)}}\mathcal{F}_{i}s^{a+i}t^{8-a-i}$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}$
$\displaystyle-3s^{4}(H_{0}t^{2}+H_{1}s\,t+H_{2}s^{2})^{2}\,,$ (C.18)
$\displaystyle
g_{+}:=\mathcal{G}_{6-b}s^{6}t^{6}+\sum_{j\in\mathcal{J}_{+}^{(0)}}\mathcal{G}_{j}s^{b+j}t^{12-b-j}$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}$
$\displaystyle-2s^{6}(H_{0}t^{2}+H_{1}s\,t+H_{2}s^{2})^{3}\,,$ (C.19)

and

$\displaystyle
f_{-}:=\sum_{i\in\mathcal{I}_{-}^{(0)}}\mathcal{F}_{i}s^{a+i}t^{8-a-i}+\mathcal{F}_{4-a}s^{4}t^{4}\
$ $\displaystyle\stackrel{{\scriptstyle!}}{{=}}$
$\displaystyle-3t^{4}(L_{0}u^{2P}+L_{1}u^{P}s\,t+L_{2}s^{2})^{2}\,,$ (C.20)
$\displaystyle
g_{-}:=\sum_{j\in\mathcal{J}_{-}^{(0)}}\mathcal{G}_{j}s^{b+j}t^{12-b-j}+\mathcal{G}_{6-b}s^{6}t^{6}$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}$
$\displaystyle-2t^{6}(L_{0}u^{2P}+L_{1}u^{P}s\,t+L_{2}s^{2})^{3}\,.$ (C.21)

Here, $f_{+}$ and $g_{+}$ have been defined to only collect the
$u$-independent terms in $f$ and $g$ of $s$-degree not less than $4$ and $6$,
respectively. Similarly, $f_{-}$ and $g_{-}$ collect terms of $s$-degree not
bigger than $4$ and $6$, whose coefficient functions $\mathcal{F}_{i}$ and
$\mathcal{G}_{i}$ have the leading monomials $u^{(4-a-i)P}$ and
$u^{(6-b-j)P}$, respectively. Here $P$ is the number of required blowups that
can simply be read off as

$P={\rm
min}\left[\\{\lfloor\frac{\mu_{0,i}}{4-a-i}\rfloor\;|\;{i\in\mathcal{I}_{-}}\\}\cup\\{\lfloor\frac{\nu_{0,j}}{6-b-j}\rfloor\;|\;{j\in\mathcal{J}_{-}}\\}\right]\,.$
(C.22)

We emphasize first that the criteria (C.18) and (C.19) for $n_{0}>0$ are
fulfilled by fine-tuning the Weierstrass coefficients $\mathcal{F}_{i}$ and
$\mathcal{G}_{j}$, respectively, for $i\in\mathcal{I}_{+}$ and
$j\in\mathcal{J}_{+}$. To see this, let us note that $f_{+}$ and $g_{+}$ in
general take the form,

$\displaystyle f_{+}=-3s^{4}F_{+}(s,t)\,,$ (C.23) $\displaystyle
g_{+}=-2s^{6}G_{+}(s,t)\,,$ (C.24)

where $F_{+}$ and $G_{+}$ are homogeneous polynomials of degree $4$ and $6$ in
$[s:t]$. In order for them to factorize as

$\displaystyle
F_{+}\stackrel{{\scriptstyle!}}{{=}}(H_{0}t^{2}+H_{1}s\,t+H_{2}s^{2})^{2}\,,$
(C.25) $\displaystyle
G_{+}\stackrel{{\scriptstyle!}}{{=}}(H_{0}t^{2}+H_{1}s\,t+H_{2}s^{2})^{3}\,,$
(C.26)

the $5$ leading coefficients,

$\mathcal{F}_{i}|_{u=0}\quad\text{for}\quad
i\in\mathcal{I}_{0}\cup\mathcal{I}_{+}\,,$ (C.27)

and the $7$ leading coefficients,

$\mathcal{G}_{j}|_{u=0}\quad\text{for}\quad
j\in\mathcal{J}_{0}\cup\mathcal{J}_{+}\,,$ (C.28)

should then be determined in terms only of a triple $(H_{0},H_{1},H_{2})$ of
complex numbers, which is a fine tuning.

On top of that, in order to also fulfill the criteria (C.20) and (C.21) for
achieving $n_{P}>0$, an additional fine tuning is required. The lower degree
counterparts of $f_{+}$ and $g_{+}$ take the form,

$\displaystyle f_{-}=-3t^{4}F_{-}(s,t;u)\,,$ (C.29) $\displaystyle
g_{-}=-2t^{6}G_{-}(s,t;u)\,,$ (C.30)

where $F_{-}$ and $G_{-}$ are also polynomials of degree $4$ and $6$ in
$[s:t]$. Obviously, they can only factorize as

$\displaystyle
F_{-}\stackrel{{\scriptstyle!}}{{=}}(L_{0}u^{2P}t^{2}+L_{1}u^{P}s\,t+L_{2}s^{2})^{2}\,,$
(C.31) $\displaystyle
G_{-}\stackrel{{\scriptstyle!}}{{=}}(L_{0}u^{2P}t^{2}+L_{1}u^{P}s\,t+L_{2}s^{2})^{3}\,,$
(C.32)

if the $4-a$ leading coefficients,

$\frac{\mathcal{F}_{i}}{u^{(4-a-i)P}}\Big{\rvert}_{u=0}\quad\text{for}\quad
i\in\mathcal{I}_{-}\,,$ (C.33)

and the $6-b$ leading coefficients,

$\frac{\mathcal{G}_{j}}{u^{(6-b-j)P}}\Big{\rvert}_{u=0}\quad\text{for}\quad
j\in\mathcal{J}_{-}\,,$ (C.34)

are also tuned so that they may be fixed by a triple
$(L_{0},L_{1},L_{2})$.323232In fact, of this triple, only $L_{0}$ and $L_{1}$
are the free parameters, provided that the $H_{0}$, $H_{1}$ and $H_{2}$ have
already been fixed after the first fine tuning in achieving $n_{0}>0$. This is
because $L_{2}$ must equal $H_{0}$ so that the middle-degree sectors of $f$
and $g$ may be well-defined.

##  Appendix D Special Fibers at Component Intersections

In the configuration $\mathcal{Y}$ obtained by the chain of $P$ blowups of
Appendix A.1, some of the special fibers may be located at some of the
intersections

$B^{p,p+1}:=B^{p}\cap B^{p+1}\,,\quad p=0,\ldots,P-1$ (D.1)

of adjacent base components. In fact, as will become clearer in the following,
in presence of special fibers at intersections $B^{p,p+1}$, there should exist
some special fiber both at $e_{p+1}=0$ in $B^{p}$ and at $e_{p}=0$ in
$B^{p+1}$. Ambiguities thus arise as to how to interpret the Kodaira type of
associated with this collision of special fibers, or equivalently the gauge
algebra realised on the associated 7-branes in F-theory. However, as it turns
out, this seeming ambiguity is resolved by an appropriate base change in $u$.

To make a precise statement, let us first consider applying the base change

$u\to u^{k}\,,$ (D.2)

of order $k$ to the initial single-component configuration
$\widehat{\mathcal{Y}}$, which leads to an equivalent single-component
configuration, $\widehat{\bar{\mathcal{Y}}}$. In order to remove the non-
minimal fiber in the latter, a chain of $\bar{P}$ blowups should follow as
usual, giving rise to another equivalent configuration, $\bar{\mathcal{Y}}$.
Its central surface decomposes as

$\bar{Y}_{0}=\cup_{\bar{p}=0}^{\bar{P}}\bar{Y}^{\bar{p}}\,,$ (D.3)

where the length of the chain has increased to $\bar{P}=kP$. The four
configurations involved here are schematically described by the following
diagram:

$\begin{array}[]{lllll}&{\mathcal{Y}}&&{\bar{\mathcal{Y}}}&\\\
\pi_{P}\circ\cdots\circ\pi_{1}&\Big{\downarrow}&&\Big{\downarrow}&\bar{\pi}_{\bar{P}}\circ\cdots\circ\bar{\pi}_{1}\\\
&\widehat{\mathcal{Y}}&\xrightarrow{\makebox[71.13188pt]{\text{\tiny base
change of order $k$}}}&\widehat{\bar{\mathcal{Y}}}&\end{array}$ (D.4)

where $\pi_{p}$ denote the blowups at $e_{p}=s=0$ used for constructing
$\mathcal{Y}$ and $\bar{\pi}_{\bar{p}}$ similarly denote the analogous blowups
for $\bar{\mathcal{Y}}$.

Note that we use the same symbols with a bar above them to denote the objects
in the base-changed configuration which are analogous to the ones in the
original configuration, leaving the information about the order $k$ implicit.
The major goal of this appendix is then to prove that no special fibers are
present at any of the intersections

$\bar{B}^{\bar{p},\bar{p}+1}:=\bar{B}^{\bar{p}}\cap\bar{B}^{\bar{p}+1}\,,\quad\bar{p}=0,\ldots,\bar{P}-1\,,$
(D.5)

of the new base components, if and only if the order $k$ of the base change
(D.2) is divisible by a certain integer $k_{0}$, which we will define later as
(D.60) in terms of the geometry of $\mathcal{Y}$.

To this end, we begin by recalling that the special fibers are read off from
the factored-out discriminant, where the maximal overall powers of the
exceptional coordinates in the discriminant are ignored. For the two
configurations $\mathcal{Y}$ and $\bar{\mathcal{Y}}$, respectively, we thus
have their discriminants factorized as

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\prod_{p=0}^{P}e_{p}^{n_{p}}\Delta^{\prime}\,,$ (D.6)
$\displaystyle\bar{\Delta}$ $\displaystyle=$
$\displaystyle\prod_{\bar{p}=0}^{\bar{P}}\bar{e}_{\bar{p}}^{\bar{n}_{\bar{p}}}\bar{\Delta}^{\prime}\,,$
(D.7)

where $\bar{e}_{\bar{p}}$ denote the $\bar{P}+1$ exceptional coordinates for
$\bar{\mathcal{Y}}$. Our next task is then to prove two lemmas concerning the
structure of these factorizations, in absence and in presence of special
fibers at the intersections $B^{p,p+1}$, respectively, of the configuration
$\mathcal{Y}$.

###### Lemma 1

Suppose that there are no special fibers at $B^{p,p+1}$ for a given
$p\in[0,P-1]$. Then the following holds:

  1. 1.

The exponents $\bar{n}_{\bar{p}}$ in (D.7) for $kp\leq\bar{p}\leq k(p+1)$ are
determined by $n_{p}$ and $n_{p+1}$ as

$\bar{n}_{kp+r}=(k-r)n_{p}+rn_{p+1}\,,\quad\text{for}\quad r=0,\ldots,k\,.$
(D.8)

  2. 2.

There are no special fibers at $\bar{B}^{\bar{p},\bar{p}+1}$ for all
$\bar{p}=kp,\ldots,k(p+1)-1$, regardless of the value of $k$.

Proof.

  1. 1.

Let us first note that the absence of special fibers at $B^{p,p+1}$ is
equivalent to the presence in $\Delta$ of at least one monomial of the form,

$\cdots e_{p}^{n_{p}}e_{p+1}^{n_{p+1}}\cdots\,,$ (D.9)

whose powers in $e_{p}$ and $e_{p+1}$ saturate the overall ones in (D.6). Note
that such a monomial originates either from $f^{3}$ or $g^{2}$. Let us assume
the former (analogous reasoning works also in the latter case); from the
expansions (A.17) and (A.23), we know that the monomial should take the form

$A_{i_{1}}A_{i_{2}}A_{i_{3}}\left(\prod_{q=0}^{P}e_{q}^{\mu_{q,i_{1}}+\mu_{q,i_{2}}+\mu_{q,i_{3}}}+\cdots\right)s^{3a+i_{1}+i_{2}+i_{3}}t^{24-3a-i_{1}-i_{2}-i_{3}}\,,$
(D.10)

indicating that

$\displaystyle\mu_{p,i_{1}}+\mu_{p,i_{2}}+\mu_{p,i_{3}}$ $\displaystyle=$
$\displaystyle n_{p}\,,$ (D.11)
$\displaystyle\mu_{p+1,i_{1}}+\mu_{p+1,i_{2}}+\mu_{p+1,i_{3}}$
$\displaystyle=$ $\displaystyle n_{p+1}\,.$ (D.12)

The leading monomial in (D.10) is the product of the three terms in $f$,

$A_{i_{l}}\prod_{q=0}^{P}e_{q}^{\mu_{q,i_{l}}}s^{a+i_{l}}t^{8-a-i_{l}}\,,\quad\text{for}\quad
l=1,2,3\,,$ (D.13)

of the blown-up configuration $\mathcal{Y}$. They originate from

$A_{i_{l}}u^{\mu_{0,i_{l}}}s^{a+i_{l}}t^{8-a-i_{l}}\,,$ (D.14)

in the initial configuration $\widehat{\mathcal{Y}}$. If we now start off with
the base-changed configuration, they lead in $\bar{\mathcal{Y}}$ to the terms

$A_{i_{l}}\prod_{\bar{q}=0}^{\bar{P}}e_{\bar{q}}^{\bar{\mu}_{\bar{q},i_{l}}}s^{a+i_{l}}t^{8-a-i_{l}}\,,$
(D.15)

where the barred exponents $\bar{\mu}_{\bar{q},i_{l}}$ are related to the
unbarred ones as

$\displaystyle\bar{\mu}_{kp+r,i_{l}}=(k-r)\mu_{p,i_{l}}+r\mu_{p+1,i_{l}}\,,\quad\text{for}\quad
r=0,\ldots,k\,,$ (D.16)

which results simply from the linearity of the recursion relations (A.26) for
the configurations $\mathcal{Y}$ and $\bar{\mathcal{Y}}$. The product of this
new triple gives the monomial in $\bar{\Delta}$ of the form

$A_{i_{1}}A_{i_{2}}A_{i_{3}}\left(\prod_{\bar{q}=0}^{\bar{P}}\bar{e}_{\bar{q}}^{\bar{\mu}_{\bar{q},i_{1}}+\bar{\mu}_{\bar{q},i_{2}}+\bar{\mu}_{\bar{q},i_{3}}}+\cdots\right)s^{3a+i_{1}+i_{2}+i_{3}}t^{24-3a-i_{1}-i_{2}-i_{3}}\,,$
(D.17)

whose exponents in the coordinates,

$\bar{e}_{\bar{p}}\quad\text{for}\quad\bar{p}=kp+r\,,\leavevmode\nobreak\
r=0,\ldots,k\,,$ (D.18)

are given as

$\displaystyle\sum_{l=1}^{3}\bar{\mu}_{kp+r,i_{l}}$ $\displaystyle=$
$\displaystyle(k-r)\sum_{l=1}^{3}\mu_{p,i_{l}}+r\sum_{l=1}^{3}\mu_{p+1,i_{i}}$
(D.19) $\displaystyle=$ $\displaystyle(k-r)n_{p}+rn_{p+1}\,.$ (D.20)

Here the relation (D.16) has been used in the first line and the constraints
(D.11) and (D.22) in the second. This thus sets the upper bounds for
$\bar{n}_{kp+r}$ as (D.20). Furthermore, it is clear that the most general
triple products of the form (D.10) in the configuration $\mathcal{Y}$ are
subject to the inequality version of the constraints (D.11) and (D.22),

$\displaystyle\mu_{p,i_{1}}+\mu_{p,i_{2}}+\mu_{p,i_{3}}$ $\displaystyle\geq$
$\displaystyle n_{p}\,,$ (D.21)
$\displaystyle\mu_{p+1,i_{1}}+\mu_{p+1,i_{2}}+\mu_{p+1,i_{3}}$
$\displaystyle\geq$ $\displaystyle n_{p+1}\,,$ (D.22)

and hence that they correspond in the base-changed configuration
$\bar{\mathcal{Y}}$ to (D.17), for which the exponents in
$\bar{e}_{\bar{p}=kp+r}$ are not less than (D.20). It thus follows that the
upper bounds (D.20) for the exponents are saturated, and hence that the
equation (D.8) holds. $\Box$

  2. 2.

We have seen that $\bar{\Delta}$ contains a monomial of the form (D.17) with
the exponents in $\bar{e}_{\bar{p}}$ given as (D.20). Furthermore, this
monomial saturates the overall factors $\bar{e}_{\bar{p}}^{\bar{n}_{\bar{p}}}$
of the factorization (D.7) for all $\bar{p}=kp,\ldots,k(p+1)$, so that
$\Delta^{\prime}$ contains a non-trivial monomial of the form,

$\cdots\prod_{\bar{p}=kp}^{k(p+1)}\bar{e}_{\bar{p}}^{0}\cdots\,,$ (D.23)

independent of all $\bar{p}=kp,\ldots,k(p+1)$. We thus learn that there arise
no special fibers at $\bar{B}^{\bar{p},\bar{p}+1}$ for all
$\bar{p}=kp,\ldots,k(p+1)-1$, as claimed. $\Box$

###### Lemma 2

Suppose that there are no special fibers at $B^{p,p+1}$ for a given
$p\in[0,P-1]$. Let the factored-out discriminant
$\Delta^{\prime}(s,t;e_{0},\ldots,e_{P})$ be given by a
$\mathbb{C}^{*}$-linear combination of the monomials
$M_{m}(s,t;e_{0},\ldots,e_{P})$ as

$\Delta^{\prime}=\sum_{m=1}^{N}c_{m}M_{m}\,,\quad
c_{m}\in\mathbb{\mathbb{C}}^{*}\,,$ (D.24)

where $M_{m}$ are written as

$M_{m}=t^{\tau_{m}}\left(\prod_{q=0}^{P}e_{q}^{\epsilon_{q,m}}\right)s^{\sigma_{m}}\,,$
(D.25)

and $N$ denotes the number of non-trivial terms in $\Delta^{\prime}$. Then,
the following holds:

  1. 1.

The quantities $\epsilon_{p}^{(0)}$ and $\epsilon_{p}^{(1)}$ defined as

$\displaystyle\epsilon_{p}^{(0)}$ $\displaystyle:=$ $\displaystyle{\rm
min}\\{\,\epsilon_{p,m}\;|\;\epsilon_{p+1,m}=0\,,\;m=1,\ldots,N\,\\}\,,$
(D.26) $\displaystyle\epsilon_{p}^{(1)}$ $\displaystyle:=$ $\displaystyle{\rm
min}\\{\,\epsilon_{p+1,m}\;|\;\epsilon_{p,m}=0\,,\;m=1,\ldots,N\,\\}\,.$
(D.27)

are both strictly positive, i.e., there exists a non-trivial brane stack both
at $e_{p+1}=0$ in $B^{p}$ and at $e_{p}=0$ in $B^{p+1}$.

  2. 2.

The exponents $\bar{n}_{\bar{p}}$ in (D.7) for $kp\leq\bar{p}\leq k(p+1)$ are
determined by

$\bar{n}_{kp+r}=(k-r)n_{p}+rn_{p+1}+\bar{\mathfrak{M}}(r)\,,\quad
r=0,\ldots,k\,,$ (D.28)

where $\bar{\mathfrak{M}}:[0,k]\to{\mathbb{R}}_{\geq 0}$ is a function defined
as the minimum

$\bar{\mathfrak{M}}(r):={\rm
min}\\{\,\bar{\mathfrak{L}}_{m}(r)\;|\;m=1,\ldots,N\,\\}\,,$ (D.29)

of the $N$ linear functions $\bar{\mathfrak{L}}_{m}:[0,k]\to{\mathbb{R}}_{\geq
0}$,

$\bar{\mathfrak{L}}_{m}(r):=(k-r)\epsilon_{p,m}+r\epsilon_{p+1,m}\,,\quad
m=1,\ldots,N\,.$ (D.30)

  3. 3.

The function $\bar{\mathfrak{M}}(r)$ defined as (D.29) is piece-wise-linear
and concave. If we denote by $\mathfrak{M}$ the $\bar{\mathfrak{M}}$ with
$k=1$, i.e., for the original configuration, the two functions are related as

$\bar{\mathfrak{M}}(kr)=k\,\mathfrak{M}(r)\,,$ (D.31)

for $r$ in the unit interval $I:=[0,1]$. The piece-wise linearity of
$\mathfrak{M}$ prompts us to naturally split its domain $I$ into a finite
number of sub-intervals,

$I_{\iota}=[\mathcal{R}_{\iota},\mathcal{R}_{\iota+1}]\,,\quad\iota=0,\ldots,\mathcal{N}-1\,,$
(D.32)

in each of which $\mathfrak{M}$ is linear, with rational boundary values
$\mathcal{R}_{\iota}$. Furthermore, $\mathfrak{M}$ vanishes at the boundary of
the domain, i.e.,

$\mathfrak{M}(0)=0=\mathfrak{M}(1)\,.$ (D.33)

  4. 4.

There appear no special fibers at $\bar{B}^{\bar{p},\bar{p}+1}$ for all
$\bar{p}=kp,\ldots,k(p+1)-1$ if and only if the base-change order $k$ is a
multiple of $k_{0}^{(p)}$, where the latter is defined as the smallest
positive integer for which

$k\mathcal{R}_{\iota}\in\mathbb{Z}\,,\quad\forall\iota=1,\ldots,\mathcal{N}-1\,.$
(D.34)

Proof. Before we start giving our proof, let us point out that Fig. -1014
exemplifies the variety of objects defined in stating the Lemma for a concrete
geometry, to which the readers are referred along the way for illustrative
purposes.

Figure -1014: The minimum functions $\mathfrak{M}$ and $\bar{\mathfrak{M}}$
defined by (D.29) for $k=1$ and $6$, respectively, in their domains $I:[0,1]$
and $\bar{I}:[0,k]$. Sketched in the left figure is for an example
configuration where a specific triple of linear functions $\mathfrak{L}_{m}$,
$m=m_{1,2,3}$, determine the minimum function $\mathfrak{M}$ globally, and
hence, $\mathcal{N}=3$, with the corresponding two-vectors
$(\epsilon_{p,m},\epsilon_{p+1,m})=(0,9),(1,4)$ and $(5,0)$, which indicates
that $\epsilon_{p}^{(0)}=5$ and $\epsilon_{p}^{(1)}=9$; the dashed lines in
red, green and blue are for the triple $\mathfrak{L}_{m_{1,2,3}}$ and the
solid black line is for their minimum $\mathfrak{M}$. Note that
$\mathcal{R}_{1}=\frac{1}{6}$ and $\mathcal{R}_{2}=\frac{1}{2}$ so that their
denominators $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ determine $k_{0}^{(p)}$
as ${\rm lcm}(\mathcal{Q}_{1},\mathcal{Q}_{2})={\rm lcm}(6,2)=6$, which is
taken as the value of $k$ for the right figure; the analogous objects defined
for the base-changed configuration with $k=6$ are depicted on the right.

  1. 1.

Note first that the vanishing of $\epsilon_{p}^{(0)}$ implies that of
$\epsilon_{p}^{(1)}$ and vice versa. Let us thus suppose that they both
vanish,

$\epsilon_{p}^{(0)}=0=\epsilon_{p}^{(1)}\,.$ (D.35)

This means that $\Delta^{\prime}$ contains a monomial of the form

$\cdots e_{p}^{0}e_{p+1}^{0}\cdots$ (D.36)

independent of $e_{p}$ and $e_{p+1}$, which contradicts the starting
assumption that branes are present at $B^{p,p+1}$. $\Box$

  2. 2.

We can use essentially the same reasoning as in the proof of Lemma 1-1 to show
the monomials

$\prod_{p=0}^{P}e_{p}^{n_{p}}M_{m}$ (D.37)

appear in $\Delta$ if and only if the associated monomials

$\prod_{\bar{p}=0}^{\bar{P}}\bar{e}_{\bar{p}}^{\bar{n}_{\bar{p}}}\bar{M}_{m}$
(D.38)

appear in $\bar{\Delta}$. Here

$\bar{M}_{m}=t^{\tau_{m}}\left(\prod_{\bar{q}=0}^{\bar{P}}\bar{e}_{\bar{q}}^{\bar{\epsilon}_{\bar{q},m}}\right)s^{\sigma_{m}}$
(D.39)

has the same single-component-configuration origin as $M_{m}$ in that

$\Big{[}\prod_{\bar{p}=0}^{\bar{P}}\bar{e}_{\bar{p}}^{\bar{n}_{\bar{p}}}\bar{M}_{m}\Big{]}_{\bar{e}_{1}=\ldots=\bar{e}_{\bar{P}}=1}=\Big{[}\prod_{p=0}^{P}e_{p}^{n_{p}}M_{m}\Big{]}_{e_{0}=\bar{e}_{0}^{k};\;e_{1}=\ldots=e_{P}=1}\,.$
(D.40)

Here, once again, essentially due to the linearity of the recursion relations
(A.26) and (A.27) for $\mathcal{Y}$ and $\bar{\mathcal{Y}}$, the exponents in
the coordinates,

$\bar{e}_{\bar{p}}\quad\text{for}\quad\bar{p}=kp+r\,,\leavevmode\nobreak\
r=0,\ldots,k\,,$ (D.41)

for the monomials (D.38) are given by

$(k-r)(n_{p}+\epsilon_{p,m})+r(n_{p+1}+\epsilon_{p+1,m})\,.$ (D.42)

This in turn indicates that $\bar{n}_{kp+r}$ are determined by

$\bar{n}_{kp+r}=(k-r)n_{p}+rn_{p+1}+\bar{\mathfrak{M}}(r)\quad\text{for}\quad
r=0,\ldots,k\,,$ (D.43)

where $\bar{\mathfrak{M}}:[0,k]\to{\mathbb{R}}_{\geq 0}$ is a function defined
as

$\displaystyle\bar{\mathfrak{M}}(r)$ $\displaystyle:=$ $\displaystyle{\rm
min}\\{\,(k-r)\epsilon_{p,m}+r\epsilon_{p+1,m}\;|\;m=1,\ldots,N\,\\}$ (D.44)
$\displaystyle=$ $\displaystyle{\rm
min}\\{\,\bar{\mathfrak{L}}_{m}(r)\;|\;m=1,\ldots,N\,\\}\,,$ (D.45)

to extract the minimum of the $m$-dependent contributions to the exponents
(D.42). $\Box$

  3. 3.

While the symbol $\bar{\mathfrak{M}}$ for the function does not make it
explicit, we emphasize that the following set of quantities are assumed to
have been fixed in order for $\bar{\mathfrak{M}}$ to be well-defined via
(D.29):

     * •

$k$ for the base-change order;

     * •

$p$ for the specific intersection $B^{p,p+1}$, the location of the special
fiber in scrutiny;

     * •

$N$ for the number of monomials in $\Delta$;

     * •

${\vec{\epsilon}_{m}}:=(\epsilon_{p,m},\epsilon_{p+1,m})\in\mathbb{Z}_{\geq
0}^{2}$ where $m=1,\ldots,N$.

Since $\bar{\mathfrak{M}}(r)$ is the minimum of the $N$ linear functions
$\bar{\mathfrak{L}}_{m}:[0,k]\to{\mathbb{R}}_{\geq 0}$,

$\bar{\mathfrak{L}}_{m}(r)=(k-r)\epsilon_{p,m}+r\epsilon_{p+1,m}\,,\quad
m=1,\ldots,N\,,$ (D.46)

it is clearly a piece-wise linear function. Let us now consider the $k=1$
version, $\mathfrak{M}(r)$, which is the minimum of
${\mathfrak{L}}_{m}:[0,1]\to{\mathbb{R}}_{\geq 0}$ defined analogously as

${\mathfrak{L}}_{m}(r)=(1-r)\epsilon_{p,m}+r\epsilon_{p+1,m}\,,\quad\quad
m=1,\ldots,N\,.$ (D.47)

Given that the $N$ pairs of the individual linear functions
$(\bar{\mathfrak{L}}_{m},\mathfrak{L}_{m})$ are identified through the scaling
$r\to kr$ as

$\bar{\mathfrak{L}}_{m}(kr)=k\,\mathfrak{L}_{m}(r)\,,$ (D.48)

their minimums should also be identified in the same manner,

$\bar{\mathfrak{M}}(kr)=k\,\mathfrak{M}(r)\,.$ (D.49)

Therefore, it suffices to analyze the structure of $\mathfrak{M}$ as a
reference function, on which we will focus now. Since $\mathfrak{M}$ is the
minimum of $N$ linear functions $\mathfrak{L}_{m}$, it is natural to decompose
the unit interval $I$ into a finite number of sub-intervals

$I_{\iota}:=[\mathcal{R}_{\iota},\mathcal{R}_{\iota+1}]\,,\quad\iota=1,\ldots,\mathcal{N}-1\,,$
(D.50)

in each of which $\mathfrak{M}$ is given by a fixed linear function
$\mathfrak{L}_{m_{\iota}}$ and hence linear. Obviously, the number of such
sub-intervals is not bigger than that of the individual linear functions,
i.e.,

$\mathcal{N}\leq N\,,$ (D.51)

since a fixed $\mathfrak{L}_{m}$ can only provide the minimal value of the
total of $N$ functions in at most one $I_{\iota}$. Furthermore, each boundary
value $r=\mathcal{R}_{\iota}$ at the intersection $I_{\iota-1}\cap I_{\iota}$
for $1\leq\iota\leq\mathcal{N}-1$ arises from the collision of the pair of
linear functions $\mathfrak{L}_{m_{\iota-1}}$ and $\mathfrak{L}_{m_{\iota}}$,
serving as the minimum of the $N$ functions in the respective sub-intervals.
Since the associated two-vectors
$\vec{\epsilon}_{m}=(\epsilon_{p,m},\epsilon_{p+1,m})$ for
$m=m_{\iota-1},m_{\iota}$ are both integral, it follows that the linear graphs
of $\mathfrak{L}_{m_{\iota-1}}$ and $\mathfrak{L}_{m_{\iota}}$ meet at a
rational value of $r$, i.e., that $\mathcal{R}_{\iota}$ is rational.
Furthermore, since there exists at least one two-vector $\vec{\epsilon}_{m}$
for which the first component vanishes, it follows that

$\epsilon_{p,m}=0\,,$ (D.52)

since, otherwise, the overall power of $e_{p}$ in $\Delta$ would be strictly
bigger than $n_{p}$. Hence, $\mathfrak{L}_{m}(0)=0$ for such $m$, which
results in the vanishing of $\mathfrak{M}$ at $r=0$,

$\mathfrak{M}(0)=0\,.$ (D.53)

Similarly, the presence of a two-vector $\vec{\epsilon}_{m}$ for which
$\epsilon_{p+1,m}=0$ results in

$\mathfrak{M}(1)=0\,.$ (D.54)

$\Box$

  4. 4.

First of all, since the end points $\mathcal{R}_{\iota}$ are rational, there
exist integral orders $k$ with the property (D.34). Specifically, if we write

$\mathcal{R}_{\iota}=\frac{\mathcal{P}_{\iota}}{\mathcal{Q}_{\iota}}\,,$
(D.55)

where $\mathcal{P}_{\iota}$ and $\mathcal{Q}_{\iota}$ are relatively prime
integers, the minimal such order, $k_{0}^{(p)}$, is obtained as

$k_{0}^{(p)}={\rm lcm}\\{\mathcal{Q}_{\iota}\\}_{\iota=1}^{\mathcal{N}}\,,$
(D.56)

and the condition (D.34) is satisfied if and only if the base-change order $k$
is divisible by $k_{0}^{(p)}$. Let us now consider the fibers for a fixed $k$
and test if some special fibers sit at some of the $k$ intersections,

$\bar{B}^{\bar{p},\bar{p}+1}\,,\quad\text{for}\quad\bar{p}=kp+r\,,\leavevmode\nobreak\
r=0,\ldots,k-1\,.$ (D.57)

Obviously, no special fibers are found at $\bar{B}^{kp+r,kp+r+1}$ for a fixed
$r$ if and only if there exists a monomial $\bar{M}_{m_{r}}$ in
$\Delta^{\prime}$ with the property

$\bar{\epsilon}_{kp+r,m_{r}}=0=\bar{\epsilon}_{kp+r+1,m_{r}}\,.$ (D.58)

Such a monomial $\bar{M}_{m_{r}}$ can in turn be seen to exist if and only if
the associated linear function $\mathfrak{L}_{m_{r}}$, defined for the
configuration with $k=1$ as (D.47), serves as the minimum in the sub-interval
$I_{\iota_{r}}$ that contains both $\frac{r}{k}$ and $\frac{r+1}{k}$, i.e.,
iff

$\exists\iota_{r}\in[0,\mathcal{N}-1]\quad\text{such that}\quad
k\mathcal{R}_{\iota_{r}}\leq r<r+1\leq k\mathcal{R}_{\iota_{r}+1}\,.$ (D.59)

Clearly, demanding (D.59) for all values of $r=0,\ldots,k-1$ is equivalent to
requiring that $k\mathcal{R}_{\iota}$ be integral for all
$\iota=1,\ldots,\mathcal{N}-1$. $\Box$

Note that Lemma 1 is in fact follow as a corollary of Lemma 2. In any case,
upon combining the explicit statements of Lemma 1-2 and Lemma 2-4, we
immediately learn that

###### Proposition 7

There are no special fibers at any $\bar{B}^{\bar{p},\bar{p}+1}$ for
$\bar{p}=0,\ldots,\bar{P}-1$ if and only if the base-change order $k$ is a
multiple of $k_{0}$ defined as

$k_{0}:=\prod_{p=0}^{P-1}k_{0}^{(p)}\,,$ (D.60)

where $k_{0}^{(p)}$ are the smallest positive integers with the defining
property (D.34), given explicitly as (D.56).

While Proposition 7 teaches us that the ambiguities in interpreting the
Kodaira type of the special fibers at intersections can be resolved by an
appropriate base change, one may still question if base changes do not change
the Kodaira type of the special fibers away from the intersections, which we
will now call “interior fibers” for brevity. We observe that essentially they
do not. To be more precise, we claim the following

###### Proposition 8

Suppose there are no special fibers at any of the intersections $B^{p,p+1}$ in
the configuration $\mathcal{Y}$. Then, the elliptic fiber of the central
surface $Y_{0}$ at a point $[e_{p-1}:e_{p+1}]$ in $B^{p}$ has the same Kodaira
type as that of $\bar{Y}_{0}$ at the point
$[\bar{e}_{kp-1}:\bar{e}_{kp+1}]=[e_{p-1}:e_{p+1}]$ in $\bar{B}^{kp}$. Here,
$e_{p}$ for $p=-1$ and $p=P+1$ respectively denote $t$ and $s$.

Proof. Recall that, from the perspective of the central surface $Y_{0}$, the
fiber over a point in $B^{p}$ has its Kodaira type determined by the
K3-vanishing orders, i.e., the vanishing orders of the restrictions $f_{p}$,
$g_{p}$ and $\Delta^{\prime}_{p}$ of the sections $f$, $g$ and
$\Delta^{\prime}$. We will thus consider those sections for the base
components $B^{p}$ in the left end ($p=0$), in the middle ($0<p<P$) and in the
right end ($p=P$) of the chain in turn.

  * •

For the left-end component, let us analyze how the restricted sections behave
around the point with homogeneous coordinates $[t:e_{1}]$ in $B^{0}$. Since we
are interested in relating the behavior of those sections to that of their
analogues in $\bar{B}^{0}$, we start by noting that terms in $f$ and $g$
divisible (resp., not divisible) by $e_{0}$ turn into those in $\bar{f}$ and
$\bar{g}$ divisible (resp., not divisible) by $\bar{e}_{0}$. Therefore, once
we set $e_{0}=0$, $e_{p}=1$ for $p>1$ and $\bar{e}_{0}=0$,
$\bar{e}_{\bar{p}}=1$ for $\bar{p}>1$, it immediately follows that the
functional behavior of $(f_{0},g_{0})$ around $[t:e_{1}]$ is precisely the
same as that of $(\bar{f}_{0},\bar{g}_{0})$ around $[t:\bar{e}_{1}]$.

While a similar statement can be made about all terms in $\Delta$, there may
arise a slight complication due to the common factors $e_{0}^{n_{0}}$ and
$e_{1}^{n_{1}}$ that are to be factored out in defining $\Delta^{\prime}$. In
order to address this issue, let us first write, as in (D.25) and (D.39), the
monomials appearing in $\Delta^{\prime}$ and $\bar{\Delta}^{\prime}$,
respectively, as

$\displaystyle M_{m}$ $\displaystyle=$ $\displaystyle
t^{\tau_{m}}\left(\prod_{q=0}^{P}e_{q}^{\epsilon_{q,m}}\right)s^{\sigma_{m}}\,,$
(D.61) $\displaystyle\bar{M}_{m}$ $\displaystyle=$ $\displaystyle
t^{\tau_{m}}\left(\prod_{\bar{q}=0}^{\bar{P}}\bar{e}_{\bar{q}}^{\bar{\epsilon}_{\bar{q},m}}\right)s^{\sigma_{m}}\,,$
(D.62)

upon factoring out the overall factors $\prod_{p=0}^{P}e_{p}^{n_{p}}$ and
$\prod_{\bar{p}=0}^{\bar{P}}\bar{e}_{\bar{p}}^{\bar{n}_{\bar{p}}}$ from
$\Delta$ and $\Delta^{\prime}$. As addressed by (D.8), the overall exponents
transform linearly as

$\bar{n}_{0}=kn_{0}\,,\qquad\bar{n}_{1}=(k-1)n_{0}+n_{1}\,.$ (D.63)

Furthermore, the way the corresponding overall factors are taken out from the
discriminant sections is uniform in that, for any monomial $M_{m}$ with
$\epsilon_{0,m}=0$, the corresponding monomial $\bar{M}_{m}$ has the
properties

$\displaystyle\bar{\epsilon}_{0,m}$ $\displaystyle=$ $\displaystyle 0\,,$
(D.64) $\displaystyle\bar{\epsilon}_{1,m}$ $\displaystyle=$
$\displaystyle\epsilon_{1,m}\,.$ (D.65)

This guarantees that the functional behavior of $\Delta^{\prime}_{0}$ around
$[s:e_{1}]$ is identical to that of $\bar{\Delta}^{\prime}_{0}$ around
$[s:\bar{e}_{1}]=[s:e_{1}]$.

All in all, the relevant sections, $f$, $g$ and $\Delta^{\prime}$, restricted
to $B^{0}$ in the configuration $\mathcal{Y}$ and their counterparts in
$\bar{\mathcal{Y}}$ may be viewed as being completely identical, and hence,
the same applies to their vanishing orders, as well as the associated fiber
types.

  * •

Next, for the middle components $B^{p}$ with $0<p<P$, it suffices to analyze
the behavior of the discriminants, since we have already established in
Proposition 3 that only special fibers of Type In may arise there. Then, as
for the discriminant behaviors, the rationale used for the left-end component
persists; the overall exponents transform linearly as

$\bar{n}_{kp}=kn_{p}\,,\qquad\bar{n}_{kp\pm 1}=(k-1)n_{0}+n_{p\pm 1}\,,$
(D.66)

and the way the overall factors are taken out is uniform so that, for $M_{m}$
with $\epsilon_{p,m}=0$, the corresponding $\bar{M}_{m}$ has the properties

$\displaystyle\bar{\epsilon}_{kp,m}$ $\displaystyle=$ $\displaystyle 0\,,$
(D.67) $\displaystyle\bar{\epsilon}_{kp\pm 1,m}$ $\displaystyle=$
$\displaystyle\epsilon_{p\pm 1,m}\,.$ (D.68)

This guarantees once again that the functional behavior of
$\Delta^{\prime}_{p}$ around $[e_{p-1}:e_{p+1}]$ is identical to that of
$\bar{\Delta}^{\prime}_{kp}$ around
$[\bar{e}_{kp-1}:\bar{e}_{kp+1}]=[e_{p-1}:e_{p+1}]$. Therefore, the
corresponding vanishing orders of $\Delta^{\prime}_{p}$ and
$\bar{\Delta}^{\prime}_{kp}$ are identical there, and hence, the fibers
thereon have the same Kodaira type.

  * •

Finally, for the right-end component $B^{P}$, it is by now obvious that the
same rationale is valid once again for the discriminants, so that the
functional behavior of $\Delta^{\prime}_{P}$ around $[e_{P-1}:s]$ is identical
to that of $\bar{\Delta}^{\prime}_{\bar{P}}$ around
$[\bar{e}_{\bar{P}-1}:s]=[e_{P-1}:s]$. However, we need to carefully compare
the behavior of $(f_{P},g_{P})$ with that of
$(\bar{f}_{\bar{P}},\bar{g}_{\bar{P}})$. In order to also identify them, it
suffices to make sure that terms in $f$ and $g$ are divisible by $e_{P}$ if
and only if the corresponding terms in $\bar{f}$ and $\bar{g}$ are by
$\bar{e}_{\bar{P}}$. While this is not true in general, recall that we are
already presuming the base change (A.128) in order to avoid physical non-
minimality at $s=0$ in $B^{P}$, as discussed in Proposition 5-1. As already
argued in the proof of this proposition, either of the two index subsets,

$\displaystyle\mathcal{I}_{-}^{(0)}=\\{\,i\in\mathcal{I}_{-}\,|\,\mu_{P,i}=0\,\\}\,,$
(D.69)
$\displaystyle\mathcal{J}_{-}^{(0)}=\\{\,j\in\mathcal{J}_{-}\,|\,\nu_{P,j}=0\,\\}\,,$
(D.70)

should thus be non-trivial (see Proof 5). Then, the invariance of the
definitions (D.69) and (D.70) is guaranteed for any further base changes, so
that

$\displaystyle\mu_{P,i}=0\quad$ $\displaystyle\Longleftrightarrow$
$\displaystyle\quad\bar{\mu}_{\bar{P},i}=0\,,$ (D.71)
$\displaystyle\nu_{P,j}=0\quad$ $\displaystyle\Longleftrightarrow$
$\displaystyle\quad\bar{\nu}_{\bar{P},j}=0\,.$ (D.72)

Therefore, once we set $e_{P}=0$, $e_{p}=1$ for $p<P$ and
$\bar{e}_{\bar{P}}=0$, $\bar{e}_{\bar{p}}=1$ for $\bar{p}<\bar{P}$, it
immediately follows that the functional behavior of $(f_{P},g_{P})$ around
$[e_{P-1}:s]$ coincides with that of $(\bar{f}_{\bar{P}},\bar{g}_{\bar{P}})$
around $[\bar{e}_{\bar{P}-1}:s]$.

With all of this combined, we thus conclude that the types of the fibers of
$Y_{0}$ are invariant under the base change in the sense of Proposition 8.
Hence, the same is true of the gauge algebras carried by the corresponding
branes, as desired. $\Box$

To summarize, according to Propositions 7 and 8, the base change of an
appropriately chosen order $k$ (so that it may be divisible by $k_{0}$ defined
in (D.60)) reconfigures the brane arrangement in $\mathcal{Y}$ into another in
$\bar{\mathcal{Y}}$ in such a way that

  * 1.

branes at intersections, if present in $\mathcal{Y}$, are “scattered” not to
sit at any base-component intersections in $\bar{\mathcal{Y}}$;

  * 2.

interior brane stacks in $\bar{B}^{\bar{p}}$, for the configuration
$\bar{\mathcal{Y}}$ free of branes at intersections, are only “rescaled” to
interior stacks carrying the same gauge algebra, if a further base change is
applied.333333In fact, we do not need to assume, in Proposition 8, that the
configuration $\mathcal{Y}$ does not have special fibers at component
intersections. The content of Proposition 8 generalizes in a straightforward
manner even without this assumption. In case there were branes at $B^{p-1,p}$
and/or $B^{p,p+1}$, the restriction $\Delta_{p}^{\prime}$ may admit non-
trivial overall factors of $e_{p\pm 1}$. Similarly, the restriction
$\bar{\Delta}_{kp}^{\prime}$ may also admit factors of $\bar{e}_{kp\pm 1}$
unless $k$ is divisible by $k_{0}^{(p)}$. However, we may safely ignore such
overall factors, even if present, away from the component intersections. Then,
modulo this, the two restrictions $\Delta_{p}^{\prime}$ and
$\bar{\Delta}_{kp}^{\prime}$ turn out once again to have exactly the same
functional behavior.

To complete our understanding of the brane arrangement, we will end this
appendix with a precise account of how exactly the branes at intersections
$B^{p,p+1}$ transform into interior branes in the base-changed configuration
$\bar{\mathcal{Y}}$. As is intuitively clear, it turns out that the
configuration $\bar{\mathcal{Y}}$ has the following property on top of the
above two.

  * 3.

the branes at $B^{p,p+1}$ turn into interior branes in $\bar{B}^{kp+r}$ with
$r=1,\ldots,k-1$.

To be precise, we have the following refinement of Lemma 2-4:

###### Proposition 9

Let us consider the configuration $\mathcal{Y}$ containing
$\epsilon_{p}^{(0)}+\epsilon_{p}^{(1)}$ branes at the intersection $B^{p,p+1}$
for a fixed $p\in[0,P-1]$, where $\epsilon_{p}^{(0)}$ and $\epsilon_{p}^{(1)}$
are defined by (D.26) and (D.27) as the number of branes at $e_{p+1}=0$ in
$B^{p}$ and at $e_{p}=0$ in $B^{p+1}$, respectively. Let $\bar{\mathcal{Y}}$
be the configuration obtained for $k$ divisible by $k_{0}^{(p)}$ defined by
(D.34), or practically by (D.56). Then, the
$\epsilon_{p}^{(0)}+\epsilon_{p}^{(1)}$ branes at $B^{p,p+1}$ split into
$\bar{\epsilon}^{(r)}$ interior branes in $\bar{B}_{kp+r}$ for
$r=1,\ldots,k-1$, so that

$\sum_{r=1}^{k-1}\bar{\epsilon}^{(r)}=\epsilon_{p}^{(0)}+\epsilon_{p}^{(1)}\,.$
(D.73)

In fact, the individual $\bar{\epsilon}^{(r)}$are given as

$\bar{\epsilon}^{(r)}=2\,\bar{\mathfrak{M}}(r)-\bar{\mathfrak{M}}(r-1)-\bar{\mathfrak{M}}(r+1)\,,\quad
r=1,\ldots,k-1\,,$ (D.74)

from which the relation (D.73) follows. Here, $\bar{\mathfrak{M}}$ is the
minimum function defined by (D.29).

Proof. We will first confirm that the formulas (D.74) do lead to the sum rule
(D.73). According to (D.74), the sum of the individual countings
$\bar{\epsilon}^{(r)}$ is computed as

$\displaystyle\sum_{r=1}^{k-1}\bar{\epsilon}^{(r)}$ $\displaystyle=$
$\displaystyle\bar{\mathfrak{M}}(0)+\bar{\mathfrak{M}}(1)+\bar{\mathfrak{M}}(k-1)+\bar{\mathfrak{M}}(k)$
(D.75) $\displaystyle=$
$\displaystyle\bar{\mathfrak{M}}(1)+\bar{\mathfrak{M}}(k-1)\,,$ (D.76)

where in the second step the vanishing (D.33) has been used (together with the
scaling relation (D.31)). Let us now note that the linear functions
$\mathfrak{L}_{m_{1}}$ and $\mathfrak{L}_{m_{\mathcal{N}}}$ that govern the
minimum function $\mathfrak{M}$ in the first and the last sub-intervals
$I_{0}$ and $I_{\mathcal{N}-1}$, respectively, are related to their
counterparts $\bar{\mathfrak{L}}_{m_{1}}$ and
$\bar{\mathfrak{L}}_{m_{\mathcal{N}}}$ for $\bar{\mathfrak{M}}$ in the base-
changed configuration by the rescaling (D.48) as

$\bar{\mathfrak{L}}_{m_{1}}(kr)=k\,{\mathfrak{L}}_{m_{1}}(r)\,,\qquad\bar{\mathfrak{L}}_{m_{\mathcal{N}}}(kr)=k\,{\mathfrak{L}}_{m_{\mathcal{N}}}(r)\,.$
(D.77)

Furthermore, according to the definitions (D.26) and (D.27), we have

$\mathfrak{L}_{m_{1}}(1)=\epsilon_{p}^{(1)}\,,\qquad\mathfrak{L}_{m_{\mathcal{N}}}(0)=\epsilon_{p}^{(0)}\,.$
(D.78)

It thus follows that the RHS of the sum (D.75) further simplifies as

$\displaystyle\bar{\mathfrak{L}}_{m_{1}}(1)+\bar{\mathfrak{L}}_{m_{\mathcal{N}}}(k-1)$
$\displaystyle=$ $\displaystyle
k(\mathfrak{L}_{m_{1}}(1/k)+\mathfrak{L}_{m_{\mathcal{N}}}((k-1)/k))$ (D.79)
$\displaystyle=$
$\displaystyle\mathfrak{L}_{m_{1}}(1)+\mathfrak{L}_{m_{\mathcal{N}}}(0)$
(D.80) $\displaystyle=$
$\displaystyle\epsilon_{p}^{(1)}+\epsilon_{p}^{(0)}\,,$ (D.81)

where we have used the rescaling relation (D.77), the linearity of
$\mathfrak{L}_{m}$’s, and the defining property (D.78), respectively, in the
three steps. Therefore, the sum rule (D.73) does follow from the formulas
(D.74), which we will prove in the following.

For a given $r\in[1,k-1]$, the branes in $\bar{B}^{kp+r}$ are located at the
zeroes of $\bar{\Delta}^{\prime}_{kp+r}$. Therefore, they are counted by the
degree of the section $\bar{\Delta}^{\prime}_{kp+r}$ in the homogeneous
coordinates $\bar{e}_{kp+r\pm 1}$ of $\bar{B}^{kp+r}$, and hence,
$\bar{\epsilon}^{(r)}$ can be determined, e.g., from the two monomials
$\bar{M}_{m_{r}}$ and $\bar{M}_{m_{r+1}}$ present in
$\bar{\Delta}^{\prime}_{kp+r}$. The former, for instance, has exponents $0$
and
$\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r+1)-\bar{\mathfrak{L}}_{m_{\iota_{+}}}(r+1)$
in $\bar{e}_{kp-1}$ and in $\bar{e}_{kp+1}$, respectively, where $\iota_{\pm}$
are the sub-interval labels such that

$r\pm 1\in I_{\iota_{\pm}}\,.$ (D.82)

We therefore have

$\displaystyle\bar{\epsilon}^{(r)}$ $\displaystyle=$
$\displaystyle\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r+1)-\bar{\mathfrak{L}}_{m_{\iota_{+}}}(r+1)$
(D.83) $\displaystyle=$
$\displaystyle(\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r+1)-\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r))+(\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r)-\bar{\mathfrak{L}}_{m_{\iota_{+}}}(r+1))$
(D.84) $\displaystyle=$
$\displaystyle(\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r)-\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r-1))+(\bar{\mathfrak{L}}_{m_{\iota_{+}}}(r)-\bar{\mathfrak{L}}_{m_{\iota_{+}}}(r+1))$
(D.85) $\displaystyle=$
$\displaystyle(\bar{\mathfrak{M}}(r)-\bar{\mathfrak{M}}(r-1))+(\bar{\mathfrak{M}}(r)-\bar{\mathfrak{M}}(r+1))$
(D.86) $\displaystyle=$ $\displaystyle
2\,\bar{\mathfrak{M}}(r)-\bar{\mathfrak{M}}(r-1)-\bar{\mathfrak{M}}(r+1)\,.$
(D.87)

Here, in the second step $\bar{\mathfrak{L}}_{m_{r}}(r)$ has been subtracted
and then added back in and in the third step
$\bar{\mathfrak{L}}_{m_{\iota_{-}}}(r)=\bar{\mathfrak{L}}_{m_{\iota_{+}}}(r)$
has been used (as the two linear functions, if $m_{\iota_{-}}$ and
$m_{\iota_{+}}$ are distinct, should meet at $r$). Furthermore, we have used
in the fourth step that $\bar{\mathfrak{L}}_{m_{\iota_{\pm}}}$ govern the
minimum function $\bar{\mathfrak{M}}$ at $r\pm 1$ while both do at $r$. $\Box$

Let us end this appendix by a final remark on the validity of our results. For
simplicity of the presentation, we have so far been working under the
assumption that the configuration $\mathcal{Y}$ is obtained by blowing up
$\widehat{\mathcal{Y}}$ repeatedly at the “same” point, i.e., at $e_{p}=s=0$
for $p=0,\ldots,P-1$. As already addressed, however, in certain non-generic
situations, several subchains of blowups, each at a different point, are to be
performed in order to arrive at a configuration free of physical non-
minimality; recall that Proposition 5 describes how one can deal with this
potential complication. However, we can easily see how our results are also
applicable to those non-generic situations, in which case $P$ blowups at a
point should be followed by additional $Q$ blowups at another point, as
suggested by Proposition 5 and its proof. All we need to do then is to blow
down towards $B^{P}$ the $P$ base components $B^{p}$ for $0\leq p<P$, as well
as the $Q$ components $B^{P+q}$ with $0<q\leq Q$. Viewing the geometry
obtained by such blown-downs as the starting single-component configuration,
we can for instance scatter all the special fibers at intersections
$B^{p,p+1}$ for $p=0,\ldots,P-1$ and $B^{P+q,P+q+1}$ for $q=0,\ldots,Q-1$ by
an appropriate base change; the criterion for this to work is, once again,
that the base-change order $k$ is divisible by all of the minimal orders with
the property (D.34), each defined for a given base-point intersection,
regardless of the fact that two subchains are emanating from the two distinct
points in $B^{P}$.

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