$\overline{E}_{1}+\sum_{i=1}^{10}\overline{C}_{i}$ to recover $X$. This yields
an irrational (and hence ugly) singularity whose minimal resolution includes a
genus 2 curve $\overline{C}_{7}$. See Figure 22 for an illustration of this.
Note that $\overline{C}_{6}$ and $\overline{C}_{7}$ intersect twice because
the intersection point $C_{6}\cap C_{7}$ is not on the branch locus. We have
also depicted (dashed) the $-1$-curves $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$
present in $\mathbb{H}\mathbb{P}(29)$ coming from the non-toric blow-ups in
its presentation as an almost toric surface.

$C_{1}$$-5$$C_{2}$$-2$$C_{3}$$-2$$C_{4}$$-2$$C_{5}$$-2$$C_{6}$$-2$$C_{7}$$-10$$C_{8}$$-2$$C_{9}$$-2$$C_{10}$$-2$$\mathcal{E}_{0}\,\,-1$$\mathcal{E}_{1}\,\,-1$$B$$\tilde{Y}$$C_{1}$$-6$$C_{2}$$-2$$C_{3}$$-2$$C_{4}$$-2$$C_{5}$$-2$$C_{6}$$-2$$C_{7}$$-10$$C_{8}$$-2$$C_{9}$$-2$$C_{10}$$-2$$\mathcal{E}_{0}\,\,-1$$\mathcal{E}_{1}\,\,-1$$E_{1}$$-1$$B$$\accentset{\approx}{Y}$$-3$$-4$$-1$$-4$$-1$$-4$$g=2$$-20$$-1$$-4$$-2$$-2$$-1$$-1$$-1$$-1$$-2$$\accentset{\approx}{X}$$\accentset{\approx}{f}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$-2$$-3$$-3$$-2$$-3$$-19$$g=2$$-3$$-2$$-2$
Figure 22: $\mathbb{H}\mathbb{P}(29)$. The minimal resolution $\tilde{Y}$, its
blow-up $\accentset{\approx}{Y}$, and the double cover
$\accentset{\approx}{X}$ of $\accentset{\approx}{Y}$. Thick curves are the
branch locus. Any non-dashed curves in $\accentset{\approx}{X}$ are contracted
to get $X$. The top row shows the dual graph of the minimal resolution of the
singularity of $X$.

### $\mathbb{H}\mathbb{P}(34)$

#### §11.21

The Manetti surface $\mathbb{H}\mathbb{P}(34)$ is obtained from
$\mathbb{P}(1,169,1156)$ by smoothing the $\frac{1}{169}(1,25)$ singularity
and keeping the $\frac{1}{1156}(1,169)$ singularity. The octic tropicalisation
of $\mathbb{H}\mathbb{P}(34)$ has vertices at $\bm{p}_{0}=(0,0)$,
$\bm{p}_{1}=(272/13,0)$ and $\bm{p}_{2}=(0,104/34)$, and a branch cut
emanating from $\bm{p}_{1}$ in the $(-89,13)$-direction with monodromy matrix
$M_{\bm{p}_{1}}=\begin{pmatrix}-1156&-7921\\\ 169&1158\end{pmatrix}$. Below we
show the convex hull $\Pi_{A}^{\mathbb{Z}}$ of the integral points (vertices
at $(0,3)$, $(14,1)$ and $(20,0)$) together with the ambient Markov triangle
in grey. Note that the branch cut lies strictly above the convex hull of the
integer points, and never intersects it.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$(272/13,0)$$(0,104/34)$$e_{10}$$e_{2}$branch
cut $(-89,13)$$\times$$e_{y}$$e_{z}$

The resolution of the $\frac{1}{1156}(1,169)$ singularity introduces edges
$e_{1},\ldots,e_{10}$ with inward normals

$\displaystyle\rho_{1}=(0,-1),\quad\rho_{2}=(-1,-7),\quad\rho_{3}=(-7,-48),\quad\rho_{4}=(-13,-89),\quad$
$\displaystyle\rho_{5}=(-19,-130),\quad\rho_{6}=(-44,-301),\quad\rho_{7}=(-69,-472),\quad$
$\displaystyle\rho_{8}=(-94,-643),\quad\rho_{9}=(-119,-814),\quad\rho_{10}=(-144,-985).$

corresponding to exceptional curves, $C_{i}$, $i=1,\ldots,10$ with

$C_{1}^{2}=-7,\quad C_{2}^{2}=-7,\quad C_{3}^{2}=C_{4}^{2}=-2,\quad
C_{5}^{2}=-3,\quad C_{6}^{2}=\cdots=C_{10}^{2}=-2$

Note that the edge $e_{4}$ is parallel to the branch cut: this means that, as
they move inwards, the edges $e_{5},\ldots,e_{10}$ cross the branch cut, and
appear as $M_{\bm{p}_{1}}^{-1}e_{i}$. Moving the edges normally inwards to
$\Pi^{\mathbb{Z}}_{A}$, edge $e_{1}$ ends up with zero-length concentrated at
$(0,3)$, and $e_{3},\ldots,e_{9}$ end up with zero-length concentrated at
$(20,0)$. The table of affine lengths and displacements becomes:

Edge | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$
---|---|---|---|---|---|---
Affine length | $0$ | $2$ | $0$ | $0$ | $0$ | $0$
Affine displacement | $1/17$ | $7/17$ | $14/17$ | $21/17$ | $11/17$ | $12/17$
Edge | $e_{7}$ | $e_{8}$ | $e_{9}$ | $e_{10}$ |  |
Affine length | $1$ | $0$ | $0$ | $1$ |  |
Affine displacement | $13/17$ | $14/17$ | $15/17$ | $16/17$ |  |

There is an important feature in this example which did not appear in the
other examples. Namely, the curve $\mathcal{E}_{1}$ from the non-toric blow-up
in the construction of $\tilde{Y}$ appears as a part of the branch locus, with
multiplicity $1$. This follows from the discussion in §9.3: the polygon
$\Pi_{A}^{\mathbb{Z}}$ has a zero-length edge concentrated at $(14,1)$ defined
by

$\tilde{b}_{1}\coloneqq\langle u,\rho_{4}\rangle=\left(\begin{pmatrix}8/3\\\
8/3\end{pmatrix}-\begin{pmatrix}14\\\
1\end{pmatrix}\right)\cdot\begin{pmatrix}-13\\\ -89\end{pmatrix}=-1.$

Write our octic as $\Omega=\sum_{p}a_{p}\theta_{p}$ where $\theta_{p}$ is the
GHK theta-function corresponding to the integral point $p$. If we pick
$a_{(0,3)}$, $a_{(7,1)}$, $a_{(14,1)}$ and $a_{(20,0)}$ generically, we find
that the strict transform of $B$ intersects $C_{2}$ transversely at two
points, $C_{10}$ once transversely, and is disjoint from the other curves
$C_{i}$. According to the prescription in §2.4, the branch locus is
$B+C_{1}+C_{2}+C_{4}+C_{5}+C_{7}+C_{9}+\mathcal{E}_{1}$. Let
$\accentset{\approx}{Y}$ be the blow up of $\tilde{Y}$ at $C_{1}\cap C_{2}$,
$C_{2}\cap B$, $C_{4}\cap C_{5}$ and $C_{4}\cap\mathcal{E}_{1}$ with
exceptional divisors $E_{1},E_{2},E_{3},E_{4},E_{5}$ and let
$\accentset{\approx}{X}$ be the double cover of $\accentset{\approx}{Y}$ over
the curve. We obtain the surface $X$ by contracting
$\sum_{i=1}^{10}\overline{C}_{i}+\sum_{i=1}^{5}\overline{E}_{i}$. Thus $X$
contains an ugly singularity (see Figure 23).

$C_{1}$$-7$$C_{2}$$-7$$C_{3}$$-2$$C_{4}$$-2$$C_{5}$$-3$$C_{6}$$-2$$C_{7}$$-2$$C_{8}$$-2$$C_{9}$$-2$$C_{10}$$-2$$\mathcal{E}_{1}\,-1$$B$$\tilde{Y}$$C_{1}$$-8$$E_{1}$$-1$$C_{2}$$-10$$C_{3}$$-2$$C_{4}$$-4$$E_{4}$$-1$$C_{5}$$-4$$C_{6}$$-2$$C_{7}$$-2$$C_{8}$$-2$$C_{9}$$-2$$C_{10}$$-2$$E_{5}\,-1$$\mathcal{E}_{1}$$-2$$B$$E_{2}$$E_{3}$$\accentset{\approx}{Y}$$-4$$-2$$-5$$-4$$-2$$-2$$-2$$-4$$-1$$-4$$-1$$-4$$-2$$-1$$-2$$-2$$\accentset{\approx}{X}$$\accentset{\approx}{f}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$-2$$-2$$-5$$-2$$-4$$-2$$-4$$-2$$-2$$-3$$-2$$-3$$-2$
Figure 23: $\mathbb{H}\mathbb{P}(34)$. The minimal resolution $\tilde{Y}$, its
blow-up $\accentset{\approx}{Y}$, and the double cover
$\accentset{\approx}{X}$ of $\accentset{\approx}{Y}$. Thick curves are the
branch locus. Any non-dashed curves in $\accentset{\approx}{X}$ are contracted
to get $X$. The top row shows the dual graph of the minimal resolution of the
singularity of $X$.

##  12 Elliptic fibrations

#### §12.1

If $X$ is a normal octic double Manetti surface with at worst rational
singularities then its geometric genus is the same as an octic double plane,
namely $p_{g}=3$. The minimal model of $X$ is then an elliptic surface with
$p_{g}=3$. To see where the elliptic fibration comes from, observe that the
minimal resolutions of Manetti surfaces are blow-ups of rational ruled
(Hirzebruch) surfaces: when the branch curve intersects the rulings with total
multiplicity $4$, the ruling lifts to an elliptic fibration on the branched
double cover. Note that the singularities for the octic double
$\mathbb{H}\mathbb{P}(29)$ surfaces are irrational, and indeed the minimal
model in that case is a ruled surface rather than an elliptic surface.

This point of view can be helpful for studying stable surfaces, so in this
section we give a description of the elliptic surfaces which contract down to
give our octic double Manetti surfaces in all the cases where it exists. These
descriptions can be read off easily from the tropical pictures.

#### §12.2

Let $\mathcal{F}$ be a singular fibre of the elliptic fibration on the minimal
resolution $X^{\prime}$ of $X$.

  * •

We call $\mathcal{F}$ an $S$-fibre if it contains an exceptional curve of
$X^{\prime}\to X$.

  * •

Otherwise, we call $\mathcal{F}$ a $B$-fibre. The $B$-fibres arise from taking
double covers of rulings which happen to intersect the branch curve $B$ non-
transversely.

We will first describe the $S$-fibres in our examples, then discuss the
possibilities for $B$-fibres. To relate this with what we did earlier, recall
that we constructed a (not necessarily minimal) resolution
$\accentset{\approx}{X}\to X$ which was a double cover
$\accentset{\approx}{f}\colon\accentset{\approx}{X}\to\accentset{\approx}{Y}$.
Our figures below show $X^{\prime}$, and in each case we will record which
curves (if any) in $\accentset{\approx}{X}$ need to be contracted to get
$X^{\prime}$. The curves $C_{i}$ and $E_{i}$ in $\accentset{\approx}{Y}$ make
reference to the notation and figures from Section 11 and the curves
$\mathcal{E}_{i}\subset\accentset{\approx}{Y}$ are curves coming from the non-
toric blow-ups (defined in §9.2). Recall that $\overline{G}$ denotes
$\accentset{\approx}{f}^{-1}(G)$; we will continue to write $\overline{G}$ for
the projection of this curve to $X^{\prime}$. Whenever we write $F_{i}$ we
will mean the $-1$-curve given by the strict transform of the ruling through
the point whose blow-up created the curve $E_{i}$ (whenever that makes sense).
We use standard notation for the Kodaira types of the fibres, see [32] or [6]
for an explanation of this notation.

$\mathbb{P}(1,1,4)$Case ICase IICase III(a)Case III(b)Case III(c)Case
III(d)$-4$section$-4$section$-4$$-4$$-4$$-4$$-4$ Figure 24: Minimal
resolutions for octic double $\mathbb{P}(1,1,4)$ surfaces as elliptic
fibrations. The exceptional locus is drawn thickly. Any unlabelled curves have
self-intersection $-2$. Any flat horizontal lines are sections.
$\mathbb{P}(1,4,25)$Case ICase IICase
III$-4$$-3$$-3$$-1$$-4$$-1$$-4$$-4$$-4$$-1$$-4$$-1$$-4$$-4$$-3$$-3$$-1$$-4$
Figure 25: Minimal resolutions for octic double $\mathbb{P}(1,4,25)$ surfaces
as elliptic fibrations. The exceptional locus is drawn thickly. Any unlabelled
curves have self-intersection $-2$. Any flat horizontal lines are sections.
$\mathbb{H}\mathbb{P}(5)$Case I$\mathbb{H}\mathbb{P}(5)$Case
II$-4$$-3$$-3$$-1$$-4$$-4$$-1$$-1$$\mathbb{H}\mathbb{P}(13)$$\mathbb{H}\mathbb{P}(34)$$-4$$-4$$-3$$-1$$-1$$-3$$-4$$-5$$-4$$-3$$-3$$-1$
Figure 26: Minimal resolutions for octic double $\mathbb{H}\mathbb{P}(5)$,
$\mathbb{H}\mathbb{P}(13)$, and $\mathbb{H}\mathbb{P}(34)$ surfaces as
elliptic fibrations. The exceptional locus is drawn thickly. Any unlabelled
curves have self-intersection $-2$. Any perfectly horizontal lines are
sections. The $-1$-curves come from non-toric blow-ups.

####  §12.3 $\mathbb{P}(1,1,4)$.

See Figure 24. In these cases, $\accentset{\approx}{X}=X^{\prime}$. The
possible $S$-fibres are as follows.

  * •

In Case I the curve $\overline{C}_{1}$ comprises two disjoint sections of
square $-4$. There are no $S$-fibres.

  * •

In Case II (compare with Figure 13), the curve $\overline{C}_{1}$ is a single
section of square $-4$; the curves $\overline{E}_{i}+\overline{F}_{i}$ form
four $I_{2}$ fibres.

  * •

In Case III we include the subcases (a)–(d) shown in Figure 14: III(a) has two
$I_{2}$ and one $I_{4}$ fibre; III(b) has two $I_{4}$ fibres; III(c) has one
$I_{2}$ and one $I_{6}$ fibre; III(d) has one $I_{8}$ fibre. In all these
cases, $\overline{C}_{1}$ is the section of square $-4$ and the
$\overline{E}_{i}$ and $\overline{F}_{i}$ assemble themselves into $I_{n}$
fibres (the curves $\overline{F}_{i}$ are precisely those which are not drawn
thickly in Figure 24).

####  §12.4 $\mathbb{P}(1,4,25)$.

See Figure 25. The possible $S$-fibres are as follows.

  * •

In Case I (compare with Figure 16), $\overline{C}_{1}$ is a section of square
$-4$, $\overline{E}_{1}+\overline{F}$ forms an $I_{2}$ fibre and
$\overline{D}_{x}+\sum_{i=2}^{5}\overline{C}_{i}$ is a (blown-up) $IV$ fibre.
Note that in $X^{\prime}$ is obtained from $\accentset{\approx}{X}$ by
contracting $\overline{C}_{3}$.

  * •

In Case II (compare with Figure 17), the curve $\overline{C}_{1}$ is a section
of square $-4$, and the curves
$\overline{E}_{1}+\overline{E}_{2}+\sum_{i=2}^{5}\overline{C}_{i}+\overline{D}_{x}$
is a (blown-up) $I^{*}_{1}$ fibre. In this case,
$\accentset{\approx}{X}=X^{\prime}$.

  * •

In Case III (compare with Figure 18), the curve $\overline{C}_{1}$ is a
section of square $-4$, $\overline{E}_{1}+\overline{F}_{1}$ is an $I_{2}$
fibre, and
$\overline{D}_{x}+\sum_{i=2}^{5}\overline{E}_{i}+\sum_{i=2}^{5}\overline{C}_{i}$
is a (blown-up) $I^{*}_{0}$ fibre. In this case, $X^{\prime}$ is obtained from
$\accentset{\approx}{X}$ by contracting $\overline{C}_{3}$.

####  §12.5 $\mathbb{H}\mathbb{P}(5)$.

See Figure 26. The possible $S$-fibres are as follows.

  * •

In Case I (compare with Figure 19), $\overline{C}_{1}$ is a section of square
$-4$, $\overline{E}_{1}+\overline{F}_{1}$ is an $I_{2}$ fibre and
$\overline{\mathcal{E}}_{1}+\sum_{i=2}^{4}\overline{C}_{2}$ is a (blown-up)
$III$ fibre. In this case, $X^{\prime}$ is obtained from
$\accentset{\approx}{X}$ by contracting $\overline{C}_{3}$.

  * •

In Case II (compare with Figure 20), the curve $\overline{C}_{1}$ is a section
of square $-4$ and
$\overline{\mathcal{E}}_{1}+\sum_{i=1}^{2}\overline{E}_{i}+\sum_{i=2}^{4}\overline{C}_{i}$
is a (blown-up) $I^{*}_{1}$ fibre. In this case,
$\accentset{\approx}{X}=X^{\prime}$.

####  §12.6 $\mathbb{H}\mathbb{P}(13)$ (generic situation).

See Figure 26. In this case (compare with Figure 21), the curve
$\overline{C}_{1}$ is a section of square $-4$ and
$\overline{\mathcal{E}}_{1}+\sum_{i=1}^{3}\overline{E}_{i}+\sum_{i=2}^{7}\overline{C}_{i}$
is a (blown-up) $I^{*}_{0}$ fibre. This is the only $S$-fibre. In this case,
$X^{\prime}$ is obtained from $\accentset{\approx}{X}$ by contracting
$\overline{C}_{4}$ and $\overline{C}_{6}$.

####  §12.7 $\mathbb{H}\mathbb{P}(29)$ (generic situation).

The minimal model is ruled rather than elliptic. The genus 2 curve
$\overline{C}_{7}$ is a bi-section and the curves
$\overline{E}_{1}+\overline{\mathcal{E}}_{0}+\overline{C}_{1}+\cdots+\overline{C}_{6}$
and
$\overline{C}_{8}+\overline{C}_{9}+\overline{C}_{10}+\overline{\mathcal{E}}_{1}$
are each fibres.

####  §12.8 $\mathbb{H}\mathbb{P}(34)$ (generic situation).

See Figure 26. For this surface (compare with Figure 23), the curve
$\overline{C}_{1}$ is a section of square $-4$ and
$\overline{\mathcal{E}}_{1}+\sum_{i=1}^{5}\overline{E}_{i}+\sum_{i=2}^{10}\overline{C}_{i}$
is a (blown-up) $I^{*}_{0}$ fibre. This is the only $S$-fibre. In this case,
$X^{\prime}$ is obtained from $\accentset{\approx}{X}$ by contracting
$\overline{C}_{7}$ and $\overline{C}_{9}$.

####  §12.9 $B$-fibres for $\mathbb{P}(1,1,4)$, Case I.

The $B$-fibres appear when the branch locus $B\subset\tilde{Y}$ intersects the
branch locus non-transversely. In $\mathbb{P}(1,1,4)$, Case I, $B$ intersects
each ruling $F$ with total multiplicity four. If we assume that $B$ is smooth
away from the singularities of $Y$, the following possibilities can occur for
the remaining singular elliptic fibres:

  * •

$B$ intersects $F$ at two points transversely and one with multiplicity $2$.
This yields a $B$-fibre of Type $I_{1}$ in the double cover.

  * •

$B$ intersects $F$ at two points with multiplicity $2$. This yields a
$B$-fibre of Type $I_{2}$ in the double cover.

  * •

$B$ intersects $F$ at one point transversely and at one point with
multiplicity $3$. This yields a $B$-fibre of Type $II$ (cuspidal) in the
double cover.

  * •

$B$ intersects $F$ at one point with multiplicity $4$. This yields a $B$-fibre
of Type III in the double cover.

The $B$-fibres of Type $I_{1}$ and $II$ are irreducible, and hence intersect
both components of $\overline{C}_{1}$. The $B$-fibres of Types $I_{2}$ and
$III$ are reducible, but each component necessarily intersects one of the
irreducible components of $\overline{C}_{1}$.

####  §12.10 $B$-fibres in other cases.

For all the other cases except $\mathbb{H}\mathbb{P}(29)$, the branch curve
$B$ intersects each ruling $F$ at three points: the the fourth branch point of
each elliptic fibre over a ruling comes from where the ruling intersects the
section $C_{1}$. If we assume that $B$ is smooth away from the singularities
of $Y$, the following possibilities can occur for the remaining $B$-fibres:

  * •

$B$ intersects $F$ at one point transversely and one point with multiplicity
$2$. All three points are distinct from $F\cap C_{1}$, so we get a $B$-fibre
of Type $I_{1}$ in the double cover.

  * •

$B$ intersects $F$ at one point with multiplicity $3$, distinct from $F\cap
C_{1}$. This yields a $B$-fibre of Type $II$ in the double cover.

This implies that all of the $B$-fibres are irreducible, and hence intersect
$\overline{C}_{1}$. We will use this fact when we study stability of these
surfaces.

##  13 Stability

#### §13.1

In this section, we will discuss the stability of the surfaces we have found.
The surfaces not covered by Theorem §13.2 below are not log canonical, and
hence not stable. We expect that their stable replacements are non-normal; it
would be interesting to know what you get.

####  §13.2 Theorem.

We call an octic curve $B\subset Y$ generic if it is smooth away from the
singularities of $Y$. Assuming genericity of the branch curve, the following
octic double Manetti surfaces are KSBA-stable: $\mathbb{P}(1,1,4)$, Cases I,
II; $\mathbb{P}(1,4,25)$, Cases I, III; $\mathbb{H}\mathbb{P}(5)$, Case I.

####  §13.3 Proof of Theorem §13.2.

The proof of this theorem will occupy the rest of this section. Certainly the
singularities are log canonical, so it remains only to show that $K_{X}$ is
positive on all curves. We will suppose there is an irreducible curve
$D\subset X$ with $K_{X}\cdot D\leq 0$ and derive a contradiction. Let
$\rho\colon X^{\prime}\to X$ be the minimal resolution and write
$A_{1},\ldots,A_{m}$ for the irreducible components of the exceptional locus.
In these cases, the minimal model of $X^{\prime}$ is an elliptic surface, as
discussed in Section 12. We write $S$ for this elliptic surface,
$\varphi\colon X^{\prime}\to S$ for the contraction, and $B_{1},\ldots,B_{n}$
for the exceptional curves of $\varphi$.

#### §13.4

We have $\rho^{*}K_{X}=K_{X^{\prime}}+\sum\alpha_{i}A_{i}$ (where
$-\alpha_{i}$ are the discrepancies of $\rho$) and
$K_{X^{\prime}}=\varphi^{*}K_{S}+\sum\beta_{i}B_{i}$. Here $\alpha_{i}$ and
$\beta_{i}$ are all positive numbers and $K_{S}$ is homologous to a positive
multiple of an elliptic fibre. Therefore $\rho^{*}K_{X}$ is an effective sum
of curves. Write $\Sigma$ for the support of $\rho^{*}K_{X}$. Let $D$ be a
curve in $X$ and $D^{\prime}=\mathrm{strict}_{\rho}(D)$. We have $K_{X}\cdot
D=\rho^{*}K_{X}\cdot D^{\prime}$. If this quantity is non-positive then either
$D^{\prime}$ is one of the curves in $\Sigma$ or else
$D^{\prime}\cap\Sigma=\emptyset$.

####  §13.5 Lemma.

If $K_{X}\cdot D\leq 0$ then $D^{\prime}$ is an irreducible component of an
elliptic fibre.

###### Proof.

There are two possible cases: either $D^{\prime}$ is disjoint from $\Sigma$ or
it is a component of $\Sigma$. In the first case, since $\Sigma$ contains an
elliptic fibre, $D^{\prime}$ must be contained in a (different) fibre. In the
second case, $D^{\prime}$ is either contained in a fibre or else it is one of
the curves $A_{i}$ or $B_{i}$. It cannot be one of the $A_{i}$, since these
project to points in $X$, and the $B_{i}$ are contained in elliptic fibres.
Therefore $D^{\prime}$ is contained in an elliptic fibre. ∎

#### §13.6

If $D^{\prime}$ is contained in a $B$-fibre then it is a $-2$-curve which
intersects one of the $A_{i}$ (one of the components of $\overline{C}_{1}$).
This implies that $\rho^{*}K_{X}\cdot D^{\prime}>0$. Therefore such a
$D^{\prime}$ cannot satisfy $K_{X}\cdot D\leq 0$.

#### §13.7

If $D^{\prime}$ is contained in an $S$-fibre then it is one of the
uncontracted (thinly-drawn) curves in Figures 24–26. For each of these curves
in the cases of interest, we can check (by calculating discrepancies) that
$\rho^{*}K_{X}\cdot D^{\prime}>0$, so again, it is impossible to find such a
$D$ with $K_{X}\cdot D\leq 0$. This completes the proof of Theorem §13.2.∎

##  14 Smoothability

#### §14.1

We now address the question of smoothability of octic double Manetti surfaces,
and how the various strata of the moduli space we have found fit together. We
have seen that an NPQ limit of octic double planes is an octic double Manetti
surface; it turns out that every octic double Manetti surface is the limit of
a degeneration of octic double planes:

####  §14.2 Theorem.

Any octic double Manetti surface is smoothable.

###### Proof.

Suppose that $Y$ is a Manetti surface and let $n$ be the product of the Markov
numbers associated to the singularities of $Y$, that is:

$n=\begin{cases}abc&\mbox{ if }Y=\mathbb{P}(a^{2},b^{2},c^{2})\\\ ab&\mbox{ if
}Y=\mathbb{H}\mathbb{P}(a,b)\\\ c&\mbox{ if
}Y=\mathbb{H}\mathbb{P}(c).\end{cases}$

Let $B\subset Y$ an octic curve and $f\colon X\to Y$ the double cover of $Y$
branched over $B$. Let $\mathcal{Y}\to\Delta$ be a $\mathbb{Q}$-Gorenstein
smoothing of $Y$ with general fibre $\mathbb{P}^{2}$. By [12, Theorem 1.6(4)],
after possibly making a base change, we can find a divisor
$\mathcal{B}\subset\mathcal{Y}$ extending $B\subset Y$ if and only if the Weil
divisor class $[B]$ is divisible by $n$. Since $B$ is an octic curve,
$[B]=8n$, so $\mathcal{B}$ exists, and we can take the double cover of
$\mathcal{Y}$ branched over $\mathcal{B}$ to get a smoothing of $X$. ∎

#### §14.3

The moduli space of smooth octic double planes is $36$-dimensional: the
projective space of octics is the $44$-dimensional space
$\mathbb{P}\left(\mathrm{Sym}^{8}(V^{*})\right)$ where $V$ is the standard
representation of the $9$-dimensional group $GL(3)$ (whose projectivisation is
the $8$-dimensional $PGL(3)$), so
$\dim\mathbb{P}\left(\mathrm{Sym}^{8}(V^{*})\right)\mathbin{/\mkern-6.0mu/}PGL(3)=36$.
We have the following boundary strata comprising NPQ surfaces:

  * $D_{A}$:

Surfaces with two $\frac{1}{4}(1,1)$ singularities (§1.8(i)).

  * $D_{B}$:

Surfaces with one $\frac{1}{25}(1,4)$ singularity (§1.8(iii)).

  * $D_{AB}$:

Surfaces with two $\frac{1}{4}(1,1)$ singularities and one $\frac{1}{25}(1,4)$
singularity (§1.8(ii)).

This enumeration of NPQ strata is only scratching the surface of the KSBA
boundary: there are also strata corresponding to the surfaces mentioned in
§1.14, which can also be understood similarly, and many deeper strata which we
left untouched. There are also degenerations corresponding to double covers of
singular limits of $\mathbb{P}^{2}$ with irrational singularities, which are
not amenable to the (almost) toric methods from this paper.

####  §14.4 Proposition.

These strata have dimensions:

$\dim(D_{A})=\dim(D_{B})=35,\quad\dim(D_{AB})=34$

and $D_{AB}$ is in the intersection of the closures of $D_{A}$ and $D_{B}$.

###### Proof.

For any Manetti surface, the space of generic “octics” is $45$-dimensional,
which becomes $44$ after projectivising. This is because a (GHK)-basis is in
bijection with the integral points of the corresponding Markov triangle; this
is easily seen to be $45$ for $\Pi(1,1,1)$, and the number of integral points
is invariant under mutation.

The cases $A$, $B$ and $AB$ respectively correspond to general octic double
covers of $\mathbb{P}(1,1,4)$, $\mathbb{H}\mathbb{P}(5)$ and
$\mathbb{P}(1,4,25)$. Since $\mathbb{P}(1,4,25)$ is a common degeneration of
both $\mathbb{H}\mathbb{P}(5)$ and $\mathbb{P}(1,1,4)$, we find that $D_{AB}$
is in the closure of both $D_{A}$ and $D_{B}$. It remains to compute the
dimensions, and for that it is sufficient to show:

$\dim(\mathrm{Aut}(\mathbb{P}(1,1,4))=\dim(\mathrm{Aut}(\mathbb{H}\mathbb{P}(5)))=9,\qquad\dim(\mathrm{Aut}(\mathbb{P}(1,4,25)))=10.$

The general automorphism of $\mathbb{P}(1,1,4)$ is:

$[x:y:z]\mapsto[a_{1}x+a_{2}y:a_{3}x+a_{4}y:a_{5}z+a_{6}x^{4}+a_{7}x^{3}y+\cdots+a_{10}y^{5}]$

(we thank Sönke Rollenske for pointing the five terms we originally missed)
which has $9$ free parameters up to an overall scale factor.

The general automorphism of $\mathbb{P}(1,4,25)$ is:

$[x:y:z]\mapsto[a_{1}x:a_{2}y+a_{3}x^{4}:a_{4}z+a_{5}x^{25}+a_{6}x^{21}y+\cdots+a_{11}xy^{6}]$

which has $10$ free parameters up to scale.

To understand the automorphism group of $\mathbb{H}\mathbb{P}(5)$, recall that
its minimal resolution is obtained from $\mathbb{F}_{7}$ by a sequence of two
toric and one non-toric blow-ups (see §7.1–we use the notation from that
paragraph). Let $Y$ denote the result of performing the two toric blow-ups to
$\mathbb{F}_{7}$. An automorphism of $Y$ which fixes the $-1$-curve denoted
$F$ in Figure 6 lifts to a unique automorphism of $\mathbb{H}\mathbb{P}(5)$
and every automorphism of $\mathbb{H}\mathbb{P}(5)$ arises this way, because
the three points $F\cap B$, $F\cap E$ and $F\cap G$ must be fixed, so any
automorphism of $\mathbb{H}\mathbb{P}(5)$ fixes $F$ pointwise and can be
blown-down.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$z$$w$$v$$y$$u$$x$
Figure 27: The moment polygon of $Y$.

It therefore suffices to find the group of automorphisms of $Y$ fixing $F$
pointwise. Since $Y$ is toric, with hexagonal moment polygon, there is a GIT
model for $Y$ as $\mathbb{C}^{6}\mathbin{/\mkern-6.0mu/}\mathbb{G}_{m}^{4}$.
Let $x,y,z,u,v,w$ be the Cox variables associated to the edges of the moment
hexagon as shown in Figure 27. The inward normals associated to these edges
are

$\rho_{x}=(0,1),\,\rho_{y}=(-1,-7),\,\rho_{z}=(0,1),\,\rho_{u}=(0,-1),\,\rho_{v}=(-2,-13),\,\rho_{w}=(-1,-6),$

from which we can read off the weights of the Cox variables under the
$\mathbb{G}_{m}^{4}$-action:

$x$ | $y$ | $z$ | $u$ | $v$ | $w$
---|---|---|---|---|---
$1$ | $1$ | $7$ | $0$ | $0$ | $0$
$0$ | $0$ | $1$ | $1$ | $0$ | $0$
$2$ | $0$ | $13$ | $0$ | $1$ | $0$
$1$ | $0$ | $6$ | $0$ | $0$ | $1$

The unstable locus is

$vwyz=vwxz=uwxz=uxyz=uvxy=uvwy=0.$

An automorphism has the form

$[x:y:z:u:v:w]\mapsto\left[ax+byv^{2}w:cy:dz+\sum_{i=0}^{6}e_{i}uv^{13-2i}w^{6-i}x^{i}y^{7-i}:fu:gv:hw\right]$

for some $a,b,c,d,e_{0},\ldots,e_{6},f,g,h$. Invertibility implies that
$c,f,g,h$ are all nonzero, and using the $\mathbb{G}_{m}^{4}$-action we can
assume they are all equal to $1$. This leaves $10$ free parameters
$a,b,d,e_{0},\ldots,e_{6}$. The points $F\cap B=[1:0:1:1:0:1]$ and $F\cap
E=[1:1:1:1:0:0]$ are automatically fixed, so it suffices to fix another point
on $F$, say $[1:1:1:1:0:1]$. This reduces to the condition that

$[a:1:d:1:0:1]=[1:1:1:1:0:1].$

If we pick a square root of $a$ then act using
$(1,1,1/\sqrt{a},1)\in\mathbb{G}_{m}^{4}$ we get

$[1:1:d/\sqrt{a}:1:0:1]=[1:1:1:1:0:1]$

and have used up all of the freedom in the group action, so the relevant
subgroup is defined by the condition that $d^{2}=a$. This leaves $9$ free
parameters, as required. ∎

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# A higher dimensional Hilbert irreducibility theorem

Giulio Bresciani  Freie Universität Berlin, Arnimallee 3, 14195, Berlin,
Germany<EMAIL_ADDRESS>
###### Abstract.

Assuming the weak Bombieri-Lang conjecture, we prove that a generalization of
Hilbert’s irreducibility theorem holds for families of geometrically mordellic
varieties (for instance, families of hyperbolic curves). As an application we
prove that, assuming Bombieri-Lang, there are no polynomial bijections
$\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$.

The author is supported by the DFG Priority Program "Homotopy Theory and
Algebraic Geometry" SPP 1786

###### Contents

  1. 1 Introduction
  2. 2 Pulling families to maximal Kodaira dimension
  3. 3 Higher dimensional HIT
  4. 4 Polynomial bijections $\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$

##  1\. Introduction

Serre reformulated Hilbert’s irreducibility theorem as follows [Ser97, Chapter
9].

######  Theorem (Hilbert’s irreducibility, Serre’s form).

Let $k$ be finitely generated over $\mathbb{Q}$, and let
$f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ be a morphism with $X$
a scheme of finite type over $k$. Suppose that the generic fiber is finite,
and that there are no generic sections
$\operatorname{Spec}k(\mathbb{P}^{1})\to X$. Then $X(k)\to\mathbb{P}^{1}(k)$
is not surjective.

Recall that the weak Bombieri-Lang conjecture states that, if $X$ is a
positive dimensional variety of general type over a field $k$ finitely
generated over $\mathbb{Q}$, then $X(k)$ is not dense in $X$.

A variety $X$ over a field $k$ is _geometrically mordellic_ , or _GeM_ , if
every subvariety of $X_{\bar{k}}$ is of general type. This generalizes to
defining a scheme $X$ as geometrically mordellic, or GeM, if it is of finite
type over $k$ and every subvariety of $X_{\bar{k}}$ is of general type. If the
weak Bombieri-Lang conjecture holds and $k$ is a field finitely generated over
$\mathbb{Q}$, then the set of rational points of a GeM scheme over $k$ is
finite, since its Zariski closure cannot have positive dimension.

Assuming Bombieri-Lang, we prove that Hilbert’s irreducibility theorem
generalizes to morphisms whose generic fiber is GeM.

######  Theorem A.

Let $k$ be finitely generated over $\mathbb{Q}$, and let
$f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ be a morphism with $X$
a scheme of finite type over $k$. Suppose that the generic fiber is GeM, and
that there are no generic sections $\operatorname{Spec}k(\mathbb{P}^{1})\to
X$.

Assume either that the weak Bombieri-Lang conjecture holds in every dimension,
or that it holds up to dimension equal to $\dim X$ and that there exists an
$N$ such that $|X_{v}(k)|\leq N$ for every rational point
$v\in\mathbb{P}^{1}(k)$. Then $X(k)\to\mathbb{P}^{1}(k)$ is not surjective.

There is a version of Hilbert’s irreducibility theorem over non-rational
curves, and the same is true for the higher dimensional generalization.

######  Theorem B.

Assume that the weak Bombieri-Lang conjecture holds in every dimension. Let
$k$ be finitely generated over $\mathbb{Q}$, and let
$f\mathrel{\mathop{\ordinarycolon}}X\to C$ be a morphism with $X$ any scheme
of finite type over $k$ and $C$ a geometrically connected curve. Assume that
the generic fiber is GeM, and that there are no generic sections
$\operatorname{Spec}k(C)\to X$. Then $X(h)\to C(h)$ is not surjective for some
finite extension $h/k$.

As an application of Theorem A we give an answer to a long-standing
Mathoverflow question [Mat19] which asks whether there exists a polynomial
bijection $\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$, conditional on the weak
Bombieri-Lang conjecture.

######  Theorem C.

Assume that the weak Bombieri-Lang conjecture for surfaces holds, and let $k$
be a field finitely generated over $\mathbb{Q}$. There are no polynomial
bijections $k\times k\to k$.

We remark that B. Poonen has proved that, assuming the weak Bombieri-Lang
conjecture for surfaces, there are polynomials giving _injective_ maps
$\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$, see [Poo10].

In 2019, T. Tao suggested on his blog [Tao19] a strategy to try to solve the
problem of polynomial bijections $\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$
conditional on Bombieri-Lang, let us summarize it. Given a morphism
$\mathbb{A}^{2}\to\mathbb{A}^{1}$ and a cover
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{A}^{1}\dashrightarrow\mathbb{A}^{1}$,
denote by $P_{c}$ the pullback of $\mathbb{A}^{2}$. If $P_{c}$ is of general
type, by Bombieri-Lang $P_{c}(\mathbb{Q})$ is not dense in $P_{c}$ and hence
by Hilbert irreducibility a generic section $\mathbb{A}^{1}\dashrightarrow
P_{c}$ exists. If $P_{c}$ is of general type for "many" covers $c$, one might
expect this to force the existence a generic section
$\mathbb{A}^{1}\dashrightarrow\mathbb{A}^{2}$, which would be in contradiction
with the bijectivity of
$\mathbb{A}^{2}(\mathbb{Q})\to\mathbb{A}^{1}(\mathbb{Q})$.

The strategy had some gaps, though. There were no results showing that the
pullback $P_{c}$ is of general type for "many" covers $c$, and it was not
clear how this would force a generic section of
$\mathbb{A}^{2}\to\mathbb{A}^{1}$. Tao started a so-called "polymath project"
in order to crowdsource a formalization. The project was active for roughly
one week in the comments section of the blog but didn’t reach a conclusion.
Partial progress was made, we cite the two most important contributions. W.
Sawin showed that $\mathbb{A}^{2}(\mathbb{Q})\to\mathbb{A}^{1}(\mathbb{Q})$
can’t be bijective if the generic fiber has genus $0$ or $1$. H. Pasten showed
that, for some morphisms $\mathbb{A}^{2}\to\mathbb{A}^{1}$ with generic fiber
of genus at least $2$, the base change of $\mathbb{A}^{2}$ along the cover
$z^{2}-b\mathrel{\mathop{\ordinarycolon}}\mathbb{A}^{1}\to\mathbb{A}^{1}$ is
of general type for a generic $b$.

Theorem A is far more general than Theorem C, but it is possible to extract
from the proof of the former the minimal arguments needed in order to prove
the latter. These minimal arguments are a formalization of the ideas described
above, hence as far as Theorem C is concerned we have essentially filled in
the gaps in Tao’s strategy.

### Acknowledgements

I would like to thank Hélène Esnault for reading an earlier draft of the paper
and giving me a lot of valuable feedback, and Daniel Loughran for bringing to
my attention the problem of polynomial bijections
$\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$.

### Conventions

A variety over $k$ is a geometrically integral scheme of finite type over $k$.
A smooth, projective variety is of general type if its Kodaira dimension is
equal to its dimension: in particular, a point is a variety of general type.

We say that a variety is of general type if it is birational to a smooth,
projective variety of general type. More generally, we define the Kodaira
dimension of any variety $X$ as the Kodaira dimension of any smooth projective
variety birational to $X$.

Curves are assumed to be smooth, projective and geometrically connected. Given
a variety $X$ (resp. a scheme of finite type $X$) and $C$ a curve, a morphism
$X\to C$ is a family of varieties of general type (resp. of GeM schemes) if
the generic fiber is a variety of general type (resp. a GeM scheme). Given a
morphism $f\mathrel{\mathop{\ordinarycolon}}X\to C$, a generic section of $f$
is a morphism $s\mathrel{\mathop{\ordinarycolon}}\operatorname{Spec}k(C)\to X$
(equivalently, a rational map
$s\mathrel{\mathop{\ordinarycolon}}C\dashrightarrow X$) such that $f\circ s$
is the natural morphism $\operatorname{Spec}k(C)\to C$ (equivalently, the
identity $C\dashrightarrow C$).

##  2\. Pulling families to maximal Kodaira dimension

This section is of purely geometric nature, thus we may assume that $k$ is
algebraically closed of characteristic $0$ for simplicity. The results then
descend to non-algebraically closed fields with standard arguments.

Given a family $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ of
varieties of general type and
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$ a finite
covering, let $f_{c}\mathrel{\mathop{\ordinarycolon}}X_{c}\to\mathbb{P}^{1}$
be the fiber product and, by abuse of notation,
$c\mathrel{\mathop{\ordinarycolon}}X_{c}\to X$ the base change of $c$. The
goal of this section is to obtain sufficient conditions on $c$ such that
$X_{c}$ is of general type. This goal will be reached in 2.13, which contains
all the geometry we’ll need for arithmetic applications.

Let us say that $X\to\mathbb{P}^{1}$ is birationally trivial if there exists a
birational morphism $X\dashrightarrow F\times\mathbb{P}^{1}$ which commutes
with the projection to $\mathbb{P}^{1}$. If $f$ is birationally trivial, then
clearly our goal is unreachable, since $X_{c}$ will have Kodaira dimension
$-\infty$ no matter which cover
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$ we choose.
We will show that this is in fact the only exception.

Assume that $X$ is smooth and projective (we can always reduce to this case),
then the relative dualizing sheaf $\omega_{f}$ exists [Kle80, Corollary 24].
First, we show that for _every_ non-birationally trivial family there exists
an integer $m$ such that $f_{*}\omega_{f}^{m}$ has _some_ positivity 2.10.
Second, we show that if $f_{*}\omega_{f}^{m}$ has _enough_ positivity, then
$X$ is of general type 2.11. We then pass from "some" to "enough" positivity
by base changing along a cover
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$.

###  2.1. Positivity of $f_{*}\omega_{f}^{m}$

There are two cases: either there exists some finite cover
$c\mathrel{\mathop{\ordinarycolon}}C\to\mathbb{P}^{1}$ such that $X_{d}\to C$
is birationally trivial, or not. Let us say that
$f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ is _birationally
isotrivial_ in the first case, and non-birationally isotrivial in the second
case.

The non-birationally isotrivial case has been extensively studied by Viehweg
and Kollár, we don’t need to do any additional work.

######  Proposition 2.1 (Kollár, Viehweg [Kol87, Theorem p.363]).

Let $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ be a non-
birationally isotrivial family of varieties of general type, with $X$ smooth
and projective. There exists an $m>0$ such that, in the decomposition of
$f_{*}\omega_{f}^{m}$ in a direct sum of line bundles, each factor has
positive degree.∎

We are thus left with studying the positivity of $f_{*}\omega_{f}^{m}$ in the
birationally isotrivial, non-birationally trivial case. We’ll have to deal
with various equivalent birational models of families, not always smooth, so
let us first compare their relative pluricanonical sheaves.

####  2.1.1. Morphisms of pluricanonical sheaves

In this subsection, fix a base scheme $S$. If a morphism to $S$ is given, it
is tacitly assumed to be flat, locally projective, finitely presentable, with
Cohen-Macauley equidimensional fibers of dimension $n$. For such a morphism
$f\mathrel{\mathop{\ordinarycolon}}X\to S$, the relative dualizing sheaf
$\omega_{f}$ exists and is coherent, see [Kle80, Theorem 21]. Recall that
$\omega_{f}$ satisfies the functorial isomorphism

$f_{*}\underline{\operatorname{Hom}}_{X}(F,\omega_{f}\otimes_{X}f^{*}N)\simeq\underline{\operatorname{Hom}}_{S}(R^{n}f_{*}F,N)$

for every quasi-coherent sheaf $F$ on $X$ and every quasi-coherent sheaf $N$
on $S$. Write $\omega_{f}^{\otimes m}$ for the $m$-th tensor power, we may
drop the superscript $\\_^{\otimes}$ and just write $\omega_{f}^{m}$ if
$\omega_{f}$ is a line bundle.

Every flat, projective map $f\mathrel{\mathop{\ordinarycolon}}X\to S$ of
smooth varieties over $k$ satisfies the above, see [Kle80, Corollary 24], and
in this case we can compute $\omega_{f}$ as $\omega_{X}\otimes
f^{*}\omega_{S}^{-1}$, where $\omega_{X}$ and $\omega_{S}$ are the usual
canonical bundles. Moreover, the relative dualizing sheaf behaves well under
base change along morphisms $S^{\prime}\to S$, see [Kle80, Proposition 9.iii].

Given a morphism $g\mathrel{\mathop{\ordinarycolon}}Y\to X$ over $S$ and a
quasi-coherent sheaf $F$ over $Y$, then $R^{n}f_{*}(g_{*}F)$ is the
$E^{n,0}_{2}$ term of the Grothendieck spectral sequence $(R^{p}f\circ
R^{q}g)(F)\Rightarrow R^{p+q}(f\circ g)(F)$, thus there is a natural morphism
$R^{n}f_{*}(g_{*}F)\to R^{n}(fg)_{*}F$. This induces a natural map

$\operatorname{Hom}_{Y}(F,\omega_{fg})=\operatorname{Hom}_{S}(R^{n}(fg)_{*}F,\mathcal{O}_{S})\to\operatorname{Hom}_{S}(R^{n}f_{*}(g_{*}F),\mathcal{O}_{S})=\operatorname{Hom}_{X}(g_{*}F,\omega_{f}).$

######  Definition 2.2.

If $g\mathrel{\mathop{\ordinarycolon}}Y\to X$ is a morphism over $S$, define
$g_{\scaleto{\triangle}{0.5em},f}\mathrel{\mathop{\ordinarycolon}}g_{*}(\omega_{fg})\to\omega_{f}$
as the sheaf homomorphism induced by the identity of $\omega_{fg}$ via the
homomorphism

$\operatorname{Hom}_{Y}(\omega_{fg},\omega_{fg})\to\operatorname{Hom}_{X}(g_{*}\omega_{fg},\omega_{f})$

given above for $F=\omega_{fg}$. With an abuse of notation, call
$g_{\scaleto{\triangle}{0.5em},f}$ the induced sheaf homomorphism
$g_{*}(\omega_{fg}^{\otimes m})\to\omega_{f}^{\otimes m}$ for every $m\geq 0$.
If there is no risk of confusion, we may drop the subscript $\\__{f}$ and just
write $g_{\scaleto{\triangle}{0.5em}}$.

The following facts are straightforward, formal consequences of the definition
of $g_{\scaleto{\triangle}{0.5em}}$, we omit proofs.

######  Lemma 2.3.

Let $g\mathrel{\mathop{\ordinarycolon}}Y\to X$ be a morphism over $S$ and
$s\mathrel{\mathop{\ordinarycolon}}S^{\prime}\to S$ any morphism,
$f^{\prime}\mathrel{\mathop{\ordinarycolon}}X^{\prime}\to S^{\prime}$,
$g^{\prime}\mathrel{\mathop{\ordinarycolon}}Y^{\prime}\to X^{\prime}$ the
pullbacks to $S^{\prime}$. By abuse of notation, call $s$ the morphisms
$Y^{\prime}\to Y$, $X^{\prime}\to X$, too. Then

$g^{\prime}_{\scaleto{\triangle}{0.5em}}=g_{\scaleto{\triangle}{0.5em}}|_{X^{\prime}}\in\operatorname{Hom}_{X^{\prime}}(g^{\prime}_{*}\omega_{f^{\prime}g^{\prime}},\omega_{f^{\prime}})=\operatorname{Hom}_{X^{\prime}}(s^{*}g_{*}\omega_{fg},s^{*}\omega_{f}).$

∎

######  Lemma 2.4.

For every quasi-coherent sheaf $F$ on $Y$, the natural map

$\operatorname{Hom}_{Y}(F,\omega_{fg})\to\operatorname{Hom}_{X}(g_{*}F,\omega_{f})$

constructed above is given by

$\varphi\mapsto g_{\scaleto{\triangle}{0.5em}}\circ
g_{*}\varphi\mathrel{\mathop{\ordinarycolon}}g_{*}F\to
g_{*}\omega_{fg}\to\omega_{f}.$

∎

######  Corollary 2.5.

Let $h\mathrel{\mathop{\ordinarycolon}}Z\to Y$,
$g\mathrel{\mathop{\ordinarycolon}}Y\to X$ be morphisms over $S$. Then, for
every $m\geq 0$,

$g_{\scaleto{\triangle}{0.5em}}\circ
g_{*}h_{\scaleto{\triangle}{0.5em}}=(gh)_{\scaleto{\triangle}{0.5em}}\mathrel{\mathop{\ordinarycolon}}gh_{*}\omega_{fgh}^{\otimes
m}\to g_{*}\omega_{fg}^{\otimes m}\to\omega_{f}^{\otimes m}.$

∎

######  Corollary 2.6.

Let $g\mathrel{\mathop{\ordinarycolon}}Y\to X$ be a morphism over $S$. Suppose
that a group $H$ acts on $Y,X,S$ and $g,f$ are $H$-equivariant. Then
$g_{*}\omega_{fg}^{\otimes m},\omega_{f}^{\otimes m}$ are $H$-equivariant
sheaves and
$g_{\scaleto{\triangle}{0.5em}}\mathrel{\mathop{\ordinarycolon}}g_{*}\omega_{fg}^{\otimes
m}\to\omega_{f}^{\otimes m}$ is $H$-equivariant.∎

######  Lemma 2.7.

Let $g\mathrel{\mathop{\ordinarycolon}}Y\to X$ be a morphism over $S$. Assume
that $Y,X$ are smooth varieties over a field $k$, and that $g$ is birational.
Then $g_{\scaleto{\triangle}{0.5em}}$ is an isomorphism.

###### Proof.

We have $\omega_{f}=\omega_{X}\otimes f^{*}\omega_{S}^{-1}$ and
$\omega_{fg}=\omega_{Y}\otimes(fg)^{*}\omega_{S}^{-1}$. Moreover,
$\omega_{Y}=g^{*}\omega_{X}\otimes\mathcal{O}_{Y}(R)$ where $R$ is some
effective divisor whose irreducible components are contracted by $g$, hence
$\omega_{fg}=g^{*}\omega_{f}\otimes\mathcal{O}_{Y}(R)$. Since
$g_{*}\mathcal{O}_{Y}(mR)\simeq\mathcal{O}_{X}$, we have a natural isomorphism
$g_{*}(\omega_{fg}^{m})\simeq\omega_{f}^{m}$ by projection formula. This is
easily checked to correspond to $g_{\scaleto{\triangle}{0.5em}}$, which is
then an isomorphism as desired. ∎

####  2.1.2. Birationally isotrivial families

Let $C$ be a smooth projective curve and
$f\mathrel{\mathop{\ordinarycolon}}X\to C$ a birationally isotrivial family of
varieties of general type, and let $F/k$ be a smooth projective variety such
that the generic fiber of $f$ is birational to $F$. Let $H$ be the finite
group of birational automorphisms of $F$. The scheme of fiberwise birational
isomorphisms $\operatorname{Bir}(X/C,F)\to C$ restricts to an $H$-torsor on
some non-empty open subset $V$ of $C$. The action of $H$ on
$\operatorname{Bir}(X/C,F)|_{V}$ is transitive on the connected components,
thus they are all birational.

######  Definition 2.8.

In the situation above, define $b\mathrel{\mathop{\ordinarycolon}}B_{f}\to C$
as the smooth completion of any connected component of
$\operatorname{Bir}(X/C,F)|_{V}$, and $G_{f}\subseteq H$ as the subgroup of
elements mapping $B_{f}$ to itself. Let us call $B_{f}\to C$ and $G_{f}$ the
_monodromy cover_ and the _monodromy group_ of $f$ respectively.

We have that $B_{f}\to C$ is a $G_{f}$-Galois covering characterized by the
following universal property: if $C^{\prime}$ is a smooth projective curve
with a finite morphism $c\mathrel{\mathop{\ordinarycolon}}C^{\prime}\to C$,
then $X_{c}\to C^{\prime}$ is birationally trivial if and only if there exists
a factorization $C^{\prime}\to B_{f}\to C$.

######  Proposition 2.9.

Let $f\mathrel{\mathop{\ordinarycolon}}X\to C$ be a birationally isotrivial
family of varieties of general type, with $X$ smooth and projective. If $p\in
B_{f}$ is a ramification point of the monodromy cover
$b\mathrel{\mathop{\ordinarycolon}}B_{f}\to C$, then for some $m$ there exists
an injective sheaf homomorphism $\mathcal{O}_{B_{f}}(p)\to
f_{b*}\omega_{f_{b}}^{m}$.

###### Proof.

The statement is equivalent to the existence of a non-trivial section of
$\omega_{f_{b}}^{m}$ which vanishes on the fiber $X_{b,p}$. Let $F$ be as
above, $G_{f}$ acts faithfully with birational maps on $F$. By equivariant
resolution of singularities, we may assume that $G_{f}$ acts faithfully by
isomorphisms on $F$. We have that $X$ is birational to $(F\times B_{f})/G_{f}$
where $G_{f}$ acts diagonally.

By resolution of singularities, let $X^{\prime}$ be a smooth projective
variety with birational morphisms $X^{\prime}\to X$, $X^{\prime}\to(F\times
B_{f})/G_{f}$: thanks to 2.7 we may replace $X$ with $X^{\prime}$ and assume
we have a birational morphism $X\to(F\times B_{f})/G_{f}$. By equivariant
resolution of singularities again, we may find a smooth projective variety $Y$
with an action of $G_{f}$, a birational morphism
$g\mathrel{\mathop{\ordinarycolon}}Y\to X_{b}$ and a birational,
$G_{f}$-equivariant morphism $y\mathrel{\mathop{\ordinarycolon}}Y\to F\times
B_{f}$. Call $\pi\mathrel{\mathop{\ordinarycolon}}F\times B_{f}\to B_{f}$ the
projection.

${Y}$${X_{b}}$${X}$${F\times B_{f}}$${(F\times
B_{f})/G_{f}}$${B_{f}}$${C}$$\scriptstyle{y}$$\scriptstyle{g}$$\scriptstyle{b}$$\scriptstyle{\pi}$$\scriptstyle{b}$$\scriptstyle{\pi
y}$$\scriptstyle{f_{b}}$

Recall that we are trying to find a global section of $\omega_{f_{b}}^{m}$
that vanishes on $X_{b,p}$, where $p$ is a ramification point of $b$. Thanks
to 2.7, we have that $\pi y_{*}\omega_{\pi
y}^{m}\simeq\pi_{*}\omega_{\pi}^{m}\simeq\mathcal{O}_{B_{f}}\otimes\operatorname{H}^{0}(F,\omega_{F}^{m})$,
thus $\operatorname{H}^{0}(Y,\omega_{\pi
y}^{m})=\operatorname{H}^{0}(F,\omega_{F}^{m})=\operatorname{H}^{0}(Y_{p},\omega_{Y_{p}}^{m})$.

The sheaf homomorphism
$g_{\scaleto{\triangle}{0.5em}}=g_{\scaleto{\triangle}{0.5em},f_{b}}\mathrel{\mathop{\ordinarycolon}}g_{*}\omega_{\pi
y}^{m}\to\omega_{f_{b}}^{m}$ induces a linear map

$g_{\scaleto{\triangle}{0.5em}}(p)\mathrel{\mathop{\ordinarycolon}}\operatorname{H}^{0}(Y_{p},\omega_{Y_{p}}^{m})=\operatorname{H}^{0}(Y,\omega_{\pi
y}^{m})\xrightarrow{g_{\scaleto{\triangle}{0.5em}}}\operatorname{H}^{0}(X_{b},\omega_{f_{b}}^{m})\xrightarrow{\bullet|_{p}}\operatorname{H}^{0}(X_{b,p},\omega_{X_{b,p}}^{m})$

where the last map is the restriction to the fiber. Let $V\subseteq B_{f}$ be
the étale locus of $b\mathrel{\mathop{\ordinarycolon}}B_{f}\to C$. Since
$X_{b}|_{V}$ is smooth, then $g_{\scaleto{\triangle}{0.5em}}$ restricts to an
isomorphism on $X_{b}|_{V}$ thanks to 2.7 and thus the map
$\operatorname{H}^{0}(Y,\omega_{\pi
y}^{m})\to\operatorname{H}^{0}(X_{b},\omega_{f_{b}}^{m})$ is injective.

We want to show that the restriction map
$\operatorname{H}^{0}(X_{b},\omega_{f_{b}}^{m})\to\operatorname{H}^{0}(X_{b,p},\omega_{X_{b,p}}^{m})$
is not injective for some $m$, it is enough to show that
$g_{\scaleto{\triangle}{0.5em}}(p)$ is not injective. Thanks to 2.3, we have
that
$g_{\scaleto{\triangle}{0.5em}}(p)=g_{p,\scaleto{\triangle}{0.5em}}\mathrel{\mathop{\ordinarycolon}}\operatorname{H}^{0}(Y_{p},\omega_{Y_{p}}^{m})\to\operatorname{H}^{0}(X_{b,p},\omega_{X_{b,p}}^{m})$.

Recall now that $G_{f}$ acts on $Y$. Let $G_{f,p}$ be the stabilizer of $p\in
B_{f}$, it is a non-trivial group since $p$ is a ramification point. Thanks to
2.6, the stabilizer $G_{f,p}$ acts naturally on
$\operatorname{H}^{0}(Y_{p},\omega_{Y_{p}}^{m})$,
$\operatorname{H}^{0}(F,\omega_{F}^{m})$,
$\operatorname{H}^{0}(X_{b,p},\omega_{X_{b,p}}^{m})$, and the maps
$y_{p,\scaleto{\triangle}{0.5em}}\mathrel{\mathop{\ordinarycolon}}\operatorname{H}^{0}(Y_{p},\omega_{Y_{p}}^{m})\simeq\operatorname{H}^{0}(F,\omega_{F}^{m})$,
$g_{p,\scaleto{\triangle}{0.5em}}\mathrel{\mathop{\ordinarycolon}}\operatorname{H}^{0}(Y_{p},\omega_{Y_{p}}^{m})\to\operatorname{H}^{0}(X_{b,p},\omega_{X_{b,p}}^{m})$
are $G_{f,p}$-equivariant. Moreover, the action on
$\operatorname{H}^{0}(X_{b,p},\omega_{X_{b,p}}^{m})$ is trivial since the
action on $X_{b,p}$ is trivial. It follows that
$g_{\scaleto{\triangle}{0.5em}}(p)$ is $G_{f,p}$-invariant, and hence to show
that it is not injective for some $m$ it is enough to show that the action of
$G_{f,p}$ on
$\operatorname{H}^{0}(F,\omega_{F}^{m}))=\operatorname{H}^{0}(Y_{p},\omega_{Y_{p}}^{m})$
is not trivial for some $m$.

Since $F$ is of general type,
$F\dashrightarrow\mathbb{P}(\operatorname{H}^{0}(F,\omega_{F}^{m}))$ is
generically injective for some $m$, fix it. Since the action of $G_{f,p}$ on
$F$ is faithful, for every non-trivial $g\in G_{f,p}$ there exists a section
$s\in\operatorname{H}^{0}(F,\omega_{F}^{m})$ and a point $v\in F$ such that
$s(v)=0$ and $s(g(v))\neq 0$, in particular the action of $G_{f,p}$ on
$\operatorname{H}^{0}(F,\omega_{F}^{m})$ is not trivial and we conclude. ∎

######  Corollary 2.10.

Let $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ be a non-
birationally trivial family of varieties of general type, with $X$ smooth and
projective. Then there exists an $m$ with an injective homomorphism
$\mathcal{O}(1)\to f_{*}\omega_{f}^{m}$.

###### Proof.

If $f$ is not birationally isotrivial, apply 2.1. Otherwise, $f$ is
birationally isotrivial and not birationally trivial, thus the monodromy cover
$b\mathrel{\mathop{\ordinarycolon}}B_{f}\to\mathbb{P}^{1}$ is not trivial.
Since $\mathbb{P}^{1}$ has no non-trivial étale covers, we have that
$B_{f}\to\mathbb{P}^{1}$ has at least one ramification point $p$. Let $m$ be
the integer given by 2.9, and write
$f_{*}\omega_{f}^{m}=\bigoplus_{i}\mathcal{O}_{\mathbb{P}^{1}}(d_{i})$. Since
$\mathcal{O}_{B_{f}}(p)\subseteq f_{b*}\omega_{f_{b}}^{m}$ and
$\omega_{f_{b}}=b^{*}\omega_{f}$, see [Kle80, Proposition 9.iii], there exists
an $i$ with $d_{i}>0$. ∎

###  2.2. Pulling families to maximal Kodaira dimension

Now that we have established a positivity result for $f_{*}\omega_{f}^{m}$ of
any non-birationally trivial family
$f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$, let us use this to
pull families to maximal Kodaira dimension.

######  Proposition 2.11.

Let $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ be a family of
varieties of general type, with $X$ smooth and projective. Then $X$ is of
general type if and only if there exists an injective homomorphism
$\mathcal{O}_{\mathbb{P}^{1}}(1)\to f_{*}\omega_{X}^{m_{0}}$, or equivalently
$\mathcal{O}_{\mathbb{P}^{1}}(2m_{0}+1)\to f_{*}\omega_{f}^{m_{0}}$, for some
$m_{0}>0$.

###### Proof.

By resolution of singularities, there exists a birational morphism
$g\mathrel{\mathop{\ordinarycolon}}X^{\prime}\to X$ with $X^{\prime}$ smooth
and projective such that the generic fiber of $X^{\prime}\to\mathbb{P}^{1}$ is
smooth and projective. We have
$\omega_{X^{\prime}}=g^{*}\omega_{X}\otimes\mathcal{O}_{X^{\prime}}(R)$ where
$R$ is some effective divisor whose irreducible components are contracted by
$g$, hence $g_{*}\omega_{X^{\prime}}^{m}=\omega_{X}^{m}\otimes
g_{*}O(mR)=\omega_{X}^{m}$ for every $m\geq 0$. We may thus replace $X$ with
$X^{\prime}$ and assume that the generic fiber is smooth. This guarantees that
$\operatorname{rank}f_{*}\omega_{X}^{m}=\operatorname{rank}f_{*}\omega_{f}^{m}$
has growth $O(m^{\dim X-1})$.

If there are no injective homomorphisms $\mathcal{O}_{\mathbb{P}^{1}}(1)\to
f_{*}\omega_{X}^{m}$ for every $m>0$, then
$\operatorname{h}^{0}(\omega_{X}^{m})\leq\operatorname{rank}f_{*}\omega_{X}^{m}=\operatorname{rank}f_{*}\omega_{f}^{m}$,
and this has growth $O(m^{\dim X-1})$.

On the other hand, let $\mathcal{O}_{\mathbb{P}^{1}}(1)\to
f_{*}\omega_{X}^{m_{0}}$ be an injective homomorphism for some $m_{0}>0$. In
particular, $X$ has Kodaira dimension $\geq 0$.

For some $m$, the closure $Y$ of the image of
$X\dashrightarrow\mathbb{P}(\operatorname{H}^{0}(X,\omega_{X}^{mm_{0}}))$ has
dimension equal to the Kodaira dimension of $X$ and $k(Y)$ is algebraically
closed in $k(X)$, see [Iit71, §3]. If $X^{\prime}$ is a smooth projective
variety birational to $X$, then there is a natural isomorphism
$\operatorname{H}^{0}(X,\omega_{X}^{mm_{0}})=\operatorname{H}^{0}(X^{\prime},\omega_{X^{\prime}}^{mm_{0}})$,
see [Har77, Ch. 2, Theorem 8.19]. Thus, up to replacing $X$ with some other
smooth, projective variety birational to $X$, we may assume that
$X\dashrightarrow
Y\subseteq\mathbb{P}(\operatorname{H}^{0}(X,\omega_{X}^{mm_{0}}))$ is defined
everywhere and has smooth, projective generic fiber $Z$ by resolution of
singularities. Iitaka has then shown that $Z$ has Kodaira dimension $0$, see
[Iit71, Theorem 5]. This is easy to see in the case in which
$\omega_{X}^{mm_{0}}$ is base point free, since then $\omega_{X}^{mm_{0}}$ is
the pullback of $\mathcal{O}(1)$ and thus
$\omega_{Z}^{mm_{0}}=\omega_{X}^{mm_{0}}|_{Z}$ is trivial.

Let us recall briefly Grothendieck’s convention that, if $V$ is a vector
bundle, then $\mathbb{P}(V)$ is the set (or scheme) of linear quotients $V\to
k$ up to a scalar. A non-trivial linear map $W\to V$ thus induces a rational
map $\mathbb{P}(V)\dashrightarrow\mathbb{P}(W)$ by restriction. If $L$ is a
line bundle with non-trivial global sections, the rational map
$X\dashrightarrow\mathbb{P}(\operatorname{H}^{0}(X,L))$ is defined by sending
a point $x\in X$ outside the base locus to the quotient
$\operatorname{H}^{0}(X,L)\to L_{x}\simeq k$. If $L$ embeds in another line
bundle $M$, then there is a natural factorization
$X\dashrightarrow\mathbb{P}(\operatorname{H}^{0}(X,M))\dashrightarrow\mathbb{P}(\operatorname{H}^{0}(X,L))$,
and any point of $X$ outside the support of $M/L$ and outside the base locus
of $L$ maps to the locus of definition of
$\mathbb{P}(\operatorname{H}^{0}(X,M))\dashrightarrow\mathbb{P}(\operatorname{H}^{0}(X,L))$.

Let $F\subseteq X$ be the fiber over any rational point of $\mathbb{P}^{1}$.
The injective homomorphism $\mathcal{O}_{\mathbb{P}^{1}}(1)\to
f_{*}\omega_{X}^{m_{0}}$ induces an injective homomorphism
$\mathcal{O}_{\mathbb{P}^{1}}(m)\to f_{*}\omega_{X}^{mm_{0}}$, choose any
embedding $\mathcal{O}_{\mathbb{P}^{1}}(1)\to\mathcal{O}_{\mathbb{P}^{1}}(m)$,
these induce an injective homomorphism
$\mathcal{O}_{X}(F)\to\omega_{X}^{mm_{0}}$. Since $\mathcal{O}_{X}(F)$ induces
the morphism $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$, the
composition

$X\to
Y\subseteq\mathbb{P}(\operatorname{H}^{0}(X,\omega_{X}^{mm_{0}}))\dashrightarrow\mathbb{P}^{1}$

coincides with $f$. Observe that the right arrow depends on the choice of the
embedding $\mathcal{O}_{X}(F)\to\omega_{X}^{mm_{0}}$, but the composition
doesn’t.

Let $\xi$ be the generic point of $\mathbb{P}^{1}$, $U\subseteq Y$ an open
subset such that $U\to\mathbb{P}^{1}$ is defined, $Y_{\xi}$ the closure of
$U_{\xi}$ in $Y$. Then the generic fiber $Z$ of $X\to Y$ is the generic fiber
of $X_{\xi}\to Y_{\xi}$, too. By hypothesis, $X_{\xi}$ is of general type,
thus by adjunction $\omega_{X_{\xi}}|_{Z}=\omega_{Z}$ is big and hence $Z$ is
of general type.

Since $Z$ is a variety of general type of Kodaira dimension $0$ over
$\operatorname{Spec}k(Y)$, then $Z=\operatorname{Spec}k(Y)$, the morphism
$X\to Y$ is generically injective and thus $X$ is of general type. ∎

######  Remark 2.12.

We don’t actually need the precision of 2.11: for our purposes it is enough to
show that, if $f_{*}\omega_{X}^{m_{0}}$ has a positive _enough_ sub-line
bundle for some $m_{0}$, then $X$ is of general type. This weaker fact has a
more direct proof, let us sketch it.

First, let us mention an elementary fact about injective sheaf homomorphisms.
Let $P,Q$ be vector bundles on $\mathbb{P}^{1}$ and $M,N$ vector bundles on
$X$, with $P$ of rank $1$. Suppose we are given injective homomorphisms
$m\in\operatorname{Hom}(P,f_{*}M)$, $n\in\operatorname{Hom}(Q,f_{*}N)$. Then
$m^{a}\otimes n\in\operatorname{Hom}(P^{\otimes a}\otimes Q,f_{*}(M^{\otimes
a}\otimes N))$ is injective for every $a>0$: this can be checked on the
generic point of $\mathbb{P}^{1}$ and thus on the generic fiber
$X_{k(\mathbb{P}^{1})}$, where the fact that $P$ has rank $1$ allows us to
reduce to the fact that the tensor product of non-zero sections of vector
bundles is non-zero on an integral scheme.

Assume we have an injective homomorphism
$\mathcal{O}_{\mathbb{P}^{1}}(3m_{0})\to f_{*}\omega_{X}^{m_{0}}$, or
equivalently $\mathcal{O}_{\mathbb{P}^{1}}(5m_{0})\to
f_{*}\omega_{f}^{m_{0}}$, we want to prove that $X$ is of general type. Let
$r(m)$ be the rank $f_{*}\omega_{f}^{mm_{0}}$ for every $m$. Since the generic
fiber is of general type, up to replacing $m_{0}$ by a multiple
$m_{0}^{\prime}$ we may assume that the growth of $r(m)$ is $O(m^{\dim X-1})$.
The induced morphism $\mathcal{O}_{\mathbb{P}^{1}}(5m_{0}^{\prime})\to
f_{*}\omega_{f}^{m_{0}^{\prime}}$ is injective thanks to the above.

Thanks to [Vie83, Theorem III], every line bundle in the factorization of
$f_{*}\omega_{f}^{mm_{0}}$ has non-negative degree, we may thus choose an
injective homomorphism $\mathcal{O}_{\mathbb{P}^{1}}^{r(m)}\to
f_{*}\omega_{f}^{mm_{0}}$. Taking the tensor product with the $m$-th power of
the homomorphism given by hypothesis, we get an homomorphism
$\mathcal{O}_{\mathbb{P}^{1}}(5mm_{0})^{r(m)}\to f_{*}\omega_{f}^{2mm_{0}}$
which is injective thanks to the above.

Since
$f_{*}\omega_{X}^{2mm_{0}}=f_{*}\omega_{f}^{2mm_{0}}\otimes\mathcal{O}_{\mathbb{P}^{1}}(-4mm_{0})$,
we thus have an injective homomorphism

$\mathcal{O}_{\mathbb{P}^{1}}(mm_{0})^{r(m)}\to f_{*}\omega_{X}^{2mm_{0}}.$

In particular, we have
$\operatorname{h}^{0}(\omega_{X}^{2mm_{0}})\geq(mm_{0}+1)r(m)$ which has
growth $O(m^{n})$, hence $X$ is of general type.

######  Corollary 2.13.

Let $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ be a non-
birationally trivial family of varieties of general type. Then there exists an
integer $d_{0}$ and a non-empty open subset $U\subseteq\mathbb{P}^{1}$ such
that, for every finite cover
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$ with $\deg
c\geq d_{0}$ and such that the branch points of $c$ are contained in $U$, we
have that $X_{c}$ is of general type. If $X$ is smooth and projective, $U$ can
be chosen as the largest open subset such that $f|_{f^{-1}(U)}$ is smooth.

###### Proof.

By resolution of singularities, we may assume that $X$ is smooth and
projective. By generic smoothness, there exists an open subset
$U\subseteq\mathbb{P}^{1}$ be such that $f|_{X_{U}}$ is smooth. We have that
$X_{c}$ is smooth for every
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$ whose
branch points are contained in $U$ since each point of $X_{c}$ is smooth
either over $X$ or over $\mathbb{P}^{1}$.

Let $m_{0}$ be the integer given by 2.10, we have an injective homomorphism
$\mathcal{O}(1)\to f_{*}\omega_{f}^{m_{0}}$. Set $d_{0}=2m_{0}+1$, for every
finite cover $c$ of degree $\deg c\geq d_{0}=2m_{0}+1$ we have an induced
homomorphism $\mathcal{O}(2m_{0}+1)\to f_{c*}\omega_{f_{c}}^{m_{0}}$ and thus
$\mathcal{O}(1)\to f_{c*}\omega_{X_{c}}^{m_{0}}$. It follows that $X_{c}$ is
of general type thanks to 2.11. ∎

##  3\. Higher dimensional HIT

###  3.1. Pulling fat sets

Recall that Serre [Ser97, Chapter 9] defined a subset $S$ of
$\mathbb{P}^{1}(k)$ as _thin_ if there exists a morphism
$f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$ with $X$ of finite type
over $k$, finite generic fiber and no generic sections
$\operatorname{Spec}k(\mathbb{P}^{1})\to X$ such that $S\subseteq f(X(k))$.
It’s immediate to check that a subset of a thin set is thin, and a finite
union of thin sets is thin. Serre’s form of Hilbert’s irreducibility theorem
says that, if $k$ is finitely generated over $\mathbb{Q}$, then
$\mathbb{P}^{1}(k)$ is not thin.

######  Definition 3.1.

A subset $S\subseteq\mathbb{P}^{1}(k)$ is _fat_ if the complement
$\mathbb{P}^{1}(k)\setminus S$ is thin.

Given a subset $S\subseteq\mathbb{P}^{1}(k)$, a finite set of finite morphisms
$D=\\{d_{i}\mathrel{\mathop{\ordinarycolon}}D_{i}\to\mathbb{P}^{1}\\}_{i}$
each of degree $>1$ with $D_{i}$ smooth, projective and geometrically
connected is a _scale_ for $S$ if
$S\cup\bigcup_{i}d_{i}(D_{i}(k))=\mathbb{P}^{1}(k)$. The set of branch points
of the scale $D$ is the union of the sets of branch points of $d_{i}$.

Using the fact that a connected scheme with a rational point is geometrically
connected [Sta20, Lemma 04KV], it’s immediate to check that a subset of
$\mathbb{P}^{1}$ is fat if and only if it has a scale. The set of branch
points of a scale gives valuable information about a fat set.

######  Lemma 3.2.

Let $S\subseteq\mathbb{P}^{1}$ be a fat set, and let
$D=\\{d_{i}\mathrel{\mathop{\ordinarycolon}}D_{i}\to\mathbb{P}^{1}\\}_{i}$ be
a scale for $S$. Let
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$ be a
morphism such that the sets of branch points of $c$ and $D$ are disjoint. Then
$c^{-1}(S)$ is fat.

###### Proof.

Let
$d_{i}^{\prime}\mathrel{\mathop{\ordinarycolon}}D_{i}^{\prime}\to\mathbb{P}^{1}$
be the base change of $d_{i}$ along
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$. By
construction,
$c^{-1}(S)\cup\bigcup_{i}d_{i}^{\prime}(D_{i}^{\prime}(k))=\mathbb{P}^{1}(k)$.
Since the sets of branch points of $c$ and $d_{i}$ are disjoint, we have that
$D_{i}^{\prime}$ is geometrically connected, see for instance [Str20, Lemma
2.8]. Moreover, $D_{i}^{\prime}$ is smooth since each point of
$D_{i}^{\prime}$ is étale either over $\mathbb{P}^{1}$ or $D_{i}$. It follows
that $d_{i}^{\prime}$ has degree $>1$ and
$\\{d_{i}^{\prime}\mathrel{\mathop{\ordinarycolon}}D_{i}^{\prime}\to\mathbb{P}^{1}\\}_{i}$
is a scale for $c^{-1}(S)$, which is thus fat. ∎

###  3.2. Decreasing the fiber dimension

Let us now prove Theorem A. Using Hilbert’s irreducibility, it’s easy to check
that Theorem A is equivalent to the following statement.

If the generic fiber of $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{P}^{1}$
is GeM and $f(X(k))$ is fat, there exists a section
$\operatorname{Spec}k(\mathbb{P}^{1})\to X$.

We prove this statement by induction on the dimension of the generic fiber. If
the generic fiber has dimension $0$, this follows from the definition of fat
set. Let us prove the inductive step.

We define recursively a sequence of closed subschemes $X_{i+1}\subseteq X_{i}$
with $X_{0}=X$ and such that $f(X_{i}(k))\subseteq\mathbb{P}^{1}_{k}$ is fat.

  * •

Define $X^{\prime}_{i}$ as the closure of $X_{i}(k)$ with the reduced scheme
structure, $f(X^{\prime}_{i}(k))=f(X_{i}(k))\subseteq\mathbb{P}^{1}_{k}$ is
fat.

  * •

Define $X^{\prime\prime}_{i}$ as the union of the irreducible components of
$X^{\prime}_{i}$ which dominate $\mathbb{P}^{1}$,
$f(X^{\prime\prime}_{i}(k))\subseteq\mathbb{P}^{1}_{k}$ is fat since
$f(X_{i}^{\prime}(k))\setminus f(X_{i}^{\prime\prime}(k))$ is finite.

  * •

Write $X^{\prime\prime}_{i}=\bigcup_{j}Y_{i,j}$ as union of irreducible
components, $Y_{i,j}\to\mathbb{P}^{1}$ is dominant for every $j$. For every
$j$, there exists a finite cover $C_{i,j}\to\mathbb{P}^{1}$ with $C_{i,j}$
smooth projective and a rational map $Y_{i,j}\dashrightarrow C_{i,j}$ with
geometrically irreducible generic fiber. If $C_{i,j}\to\mathbb{P}^{1}$ is an
isomorphism, define $Z_{i,j}=Y_{i,j}$. Otherwise, there exists a non-empty
open subset $V_{i,j}\subseteq Y_{i,j}$ such that $Y_{i,j}\dashrightarrow
C_{i,j}$ is defined on $V_{i,j}$. In particular,
$f(V_{i,j}(k))\subseteq\mathbb{P}^{1}(k)$ is thin. Define
$Z_{i,j}=Y_{i,j}\setminus V_{i,j}$ and $X_{i+1}=\bigcup_{j}Z_{i,j}\subseteq
X_{i}$. By construction, $f(X_{i+1}(k))\subseteq\mathbb{P}^{1}(k)$ is fat
since $f(X_{i}^{\prime\prime}(k))\setminus f(X_{i+1}(k))$ is thin.

By noetherianity, the sequence is eventually stable, let $r$ be such that
$X_{r+1}=X_{r}$. Since $X_{r+1}=X_{r}$, then $X_{r}(k)$ is dense in $X_{r}$,
thus every irreducible component is geometrically irreducible, see [Sta20,
Lemma 0G69]. Moreover, every irreducible component of $X_{r}$ dominates
$\mathbb{P}^{1}$ with geometrically irreducible generic fiber. Replace $X$
with $X_{r}$ and write $X=\bigcup_{j}Y_{j}$ as union of irreducible
components, we may assume that $Y_{j}\to\mathbb{P}^{1}$ is a family of GeM
varieties for every $j$ and $Y_{j}(k)$ is dense in $Y_{j}$.

If $Y_{j}\to\mathbb{P}^{1}$ is birationally trivial for some $j$, since
$Y_{j}(k)$ is dense in $Y_{j}$ and a generic fiber of $Y_{j}\to\mathbb{P}^{1}$
has a finite number of rational points, then $\dim Y_{j}=0$,
$Y_{j}\to\mathbb{P}^{1}$ is birational and we conclude. Otherwise, thanks to
2.13, there exists an integer $d_{0}$ and a non-empty open subset
$U\subseteq\mathbb{P}^{1}$ such that, for every finite cover
$c\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$ with $\deg
c\geq d_{0}$ such that the branch points of $c$ are contained in $U$, we have
that $Y_{j,c}$ is of general type for every $j$.

Let $D=\\{d_{l}\mathrel{\mathop{\ordinarycolon}}D_{l}\to\mathbb{P}^{1}\\}$ be
a scale for $f(X(k))$. Up to shrinking $U$ furthermore, we may assume that the
set of branch points of $D$ is disjoint from $U$. Since we are assuming that
the weak Bombieri-Lang conjecture holds up to dimension $\dim X$, the
dimension of $\overline{Y_{j,c}(k)}\subseteq Y_{j,c}$ is strictly smaller than
$\dim Y_{j}$ for every $j$. Moreover, we have that
$f_{c}(X_{c}(k))=m_{c}^{-1}(f(X(k)))$ is fat thanks to 3.2. It follows that,
by induction hypothesis, there exists a generic section
$\operatorname{Spec}k(\mathbb{P}^{1})\to X_{c}$ for _every_ finite cover $c$
as above. There are a lot of such covers: let us show that we can choose them
so that the resulting sections "glue" to a generic section
$\operatorname{Spec}k(\mathbb{P}^{1})\to X$.

###  3.3. Gluing sections

Choose coordinates on $\mathbb{P}^{1}$ so that $0,\infty\in U$, let $p$ be any
prime number greater than $d_{0}$. For any positive integer $n$, let
$m_{n}\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\to\mathbb{P}^{1}$ be the
$n$-th power map. We have shown above that there exists a rational section
$\mathbb{P}^{1}\dashrightarrow X_{m_{p}}$ for every prime $p\geq d_{0}$, call
$s_{p}\mathrel{\mathop{\ordinarycolon}}\mathbb{P}^{1}\dashrightarrow
X_{m_{p}}\to X$ the composition.

We either assume that there exists an integer $N$ such that, for every
rational point $v\in\mathbb{P}^{1}(k)$, we have $|X_{v}(k)|\leq N$ or that the
Bombieri-Lang conjecture holds in every dimension. In the second case, the
uniform bound $N$ exists thanks to a theorem of Caporaso-Harris-Mazur and
Abramovich-Voloch [CHM97, Theorem 1.1] [AV96, Theorem 1.5] [Abr97]. Choose
$N+1$ prime numbers $p_{0},\dots,p_{N}$ greater than $d_{0}$, for each one we
have a rational section

${X}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$$\scriptstyle{f}$$\scriptstyle{m_{p}}$$\scriptstyle{s_{p}}$

Let $Q=\prod_{i=0}^{N}p_{i}$, for every $i=0,\dots,N$, we get a rational
section $S_{p_{i}}$ by composition with $s_{p_{i}}$:

${X}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$$\scriptstyle{f}$$\scriptstyle{m_{Q/p_{i}}}$$\scriptstyle{S_{p_{i}}}$$\scriptstyle{m_{Q}}$$\scriptstyle{m_{p_{i}}}$$\scriptstyle{s_{p_{i}}}$

Let $V\subseteq\mathbb{P}^{1}$ be an open subset such that $S_{p_{i}}$ is
defined on $V$ for every $i$. For every rational point $v\in V(k)$, we have
$|X_{v}(k)|\leq N$ and thus there exists a couple of different indexes $i\neq
j$ such that $S_{p_{i}}(v)=S_{p_{j}}(v)$ for infinitely many $v\in V(k)$,
hence $S_{p_{i}}=S_{p_{j}}$. Let $Z\subseteq X$ be the image
$S_{p_{i}}=S_{p_{j}}$, by construction we have

$k(\mathbb{P}^{1})=k(t)\subseteq k(Z)\subseteq k(t^{-p_{i}})\cap
k(t^{-p_{j}})\subseteq k(t^{-Q}).$

Using Galois theory on the cyclic extension $k(t^{-Q})/k(t)$, it is immediate
to check that $k(t^{-p_{i}})\cap k(t^{-p_{j}})=k(t)\subseteq k(t^{-Q})$ since
$p_{i},p_{j}$ are coprime, thus $k(Z)=k(t)$ and $Z\to\mathbb{P}^{1}$ is
birational. This concludes the proof of Theorem A.

###  3.4. Non-rational base

Let us show how Theorem A implies Theorem B. Let $C$ be a geometrically
connected curve over a field $k$ finitely generated over $\mathbb{Q}$, and let
$f\mathrel{\mathop{\ordinarycolon}}X\to C$ be a morphism of finite type whose
generic fiber is a GeM scheme. Assume that there exists a non-empty open
subset $V\subseteq C$ such that $X|_{V}(h)\to V(h)$ is surjective for every
finite extension $h/k$. We want to prove that there exists a generic section
$C\dashrightarrow X$. It’s easy to reduce to the case in which $C$ is smooth
and projective, so let us make this assumption.

Observe that, up to replacing $X$ with an affine covering, we may assume that
$X$ is affine. Choose $C\to\mathbb{P}^{1}$ any finite map: since $X$ is
affine, the Weil restriction $R_{C/\mathbb{P}^{1}}(X)\to\mathbb{P}^{1}$ exists
[BLR90, §7.6, Theorem 4]. Recall that
$R_{C/\mathbb{P}^{1}}(X)\to\mathbb{P}^{1}$ represents the functor on
$\mathbb{P}^{1}$-schemes
$S\mapsto\operatorname{Hom}_{C}(S\times_{\mathbb{P}^{1}}C,X)$.

If $L/k(C)/k(\mathbb{P}^{1})$ is a Galois closure and $\Sigma$ is the set of
embeddings $\sigma\mathrel{\mathop{\ordinarycolon}}k(C)\to L$ as
$k(\mathbb{P}^{1})$ extensions, the scheme $R_{C/\mathbb{P}^{1}}(X)_{L}$ is
isomorphic to the product
$\prod_{\Sigma}X\times_{\operatorname{Spec}k(C),\sigma}\operatorname{Spec}L$
and hence is a GeM scheme, see [Bre20, Lemma 3.3]. It follows that the generic
fiber $R_{C/\mathbb{P}^{1}}(X)_{k(\mathbb{P}^{1})}$ is a GeM scheme, too.

Let $U\subseteq\mathbb{P}^{1}$ be the image of $V\subseteq C$. The fact that
$X|_{V}(h)\to V(h)$ is surjective for every finite extension $h/k$ implies
that $R_{C/\mathbb{P}^{1}}(X)|_{U}(k)\to U(k)$ is surjective. By Theorem A, we
get a generic section $\mathbb{P}^{1}\dashrightarrow R_{C/\mathbb{P}^{1}}(X)$,
which in turn induces generic section $C\dashrightarrow X$ by the universal
property of $R_{C/\mathbb{P}^{1}}(X)$. This concludes the proof of Theorem B.

##  4\. Polynomial bijections $\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$

Let us prove Theorem C. Let $k$ be finitely generated over $\mathbb{Q}$, and
let $f\mathrel{\mathop{\ordinarycolon}}\mathbb{A}^{2}\to\mathbb{A}^{1}$ be any
morphism. Assume by contradiction that $f$ is bijective on rational points.

First, let us show that the generic fiber of $f$ is geometrically irreducible.
This is equivalent to saying that $\operatorname{Spec}k(\mathbb{A}^{2})$ is
geometrically connected over $\operatorname{Spec}k(\mathbb{A}^{1})$, or that
$k(\mathbb{A}^{1})$ is algebraically closed in $k(\mathbb{A}^{2})$. Let
$k(\mathbb{A}^{1})\subseteq L\subseteq k(\mathbb{A}^{2})$ a subextension
algebraic over $k(\mathbb{A}^{1})$. Let $C\to\mathbb{A}^{1}$ be a finite cover
with $C$ regular and $k(C)=L$. The rational map $\mathbb{A}^{2}\dashrightarrow
C$ is defined in codimension $1$, thus there exists a finite subset
$S\subseteq\mathbb{A}^{2}$ and an extension $\mathbb{A}^{2}\setminus S\to C$.
Since the composition $\mathbb{A}^{2}\setminus S(k)\to
C(k)\to\mathbb{A}^{1}(k)$ is surjective up to a finite number of points, by
Hilbert’s irreducibility theorem we have that $C=\mathbb{A}^{1}$, i.e.
$L=k(\mathbb{A}^{1})$.

This leaves us with three cases: the generic fiber is a geometrically
irreducible curve of geometric genus $0$, $1$, or $\geq 2$. The first two have
been settled by W. Sawin in the polymath project [Tao19], while the third
follows from Theorem A. Let us give details for all of them.

###  4.1. Genus 0

Assume that the generic fiber of $f$ has genus $0$. By generic smoothness,
there exists an open subset $U\subseteq\mathbb{A}^{2}$ such that $f|_{U}$ is
smooth. For a generic rational point $u\in U(k)$, the fiber $f^{-1}(f(u))$ is
birational to a Brauer-Severi variety of dimension $1$ and has a smooth
rational point, thus it is birational to $\mathbb{P}^{1}$ and
$f^{-1}(f(u))(k)$ is infinite. This is absurd.

###  4.2. Genus 1

Assume now that the generic fiber has genus $1$. By resolution of
singularities, there exists an open subset $V\subseteq\mathbb{A}^{1}$, a
variety $X$ with a smooth projective morphism
$g\mathrel{\mathop{\ordinarycolon}}X\to V$ whose fibers are smooth genus $1$
curves and a compatible birational map $X\dashrightarrow\mathbb{A}^{2}$. Up to
shrinking $V$, we may suppose that the fibers of $f|_{V}$ are geometrically
irreducible. Let $U$ be a variety with open embeddings $U\subseteq X$,
$U\subseteq\mathbb{A}^{2}$, replace $V$ with $g(U)\subseteq V$ so that
$g|_{U}$ is surjective.

The morphism $X\setminus U\to V$ is finite, let $N$ be its degree. Since the
fibers of $U\to V$ have at most one rational point, it follows that
$|X_{v}(k)|\leq N+1$ for every $v\in V(k)$.

Every smooth genus $1$ fibration is a torsor for a relative elliptic curve
(namely, its relative $\underline{\operatorname{Pic}}^{0}$), thus there exists
an elliptic curve $E\to V$ such that $X$ is an $E$-torsor. Moreover, every
torsor for an abelian variety is torsion, thus there exists a finite morphism
$\pi\mathrel{\mathop{\ordinarycolon}}X\to E$ over $V$ induced by the
$n$-multiplication map $E\to E$ for some $n$.

If $v\in V(k)$ is such that $X_{v}(k)$ is non-empty, then
$|X_{v}(k)|=|E_{v}(k)|\leq N+1$. This means that, up to composing $\pi$ with
the $(N+1)!$ multiplication $E\to E$, we may assume that $\pi(X(k))\subseteq
V(k)\subseteq E(k)$, where $V\to E$ is the identity section. In particular,
$X(k)\subseteq\pi^{-1}(V(k))$ is not dense. This is absurd, since $X$ is
birational to $\mathbb{A}^{2}$.

###  4.3. Genus $\geq 2$

Thanks to Theorem A, there exists an open subset $V\subseteq\mathbb{A}^{1}$
and a section $s\mathrel{\mathop{\ordinarycolon}}V\to\mathbb{A}^{2}$. It
follows that $\mathbb{A}^{2}|_{V}(k)=s(V(k))$, which is absurd since $s(V)$ is
a proper closed subset and $\mathbb{A}^{2}|_{V}(k)$ is dense.

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