# A Van Kampen type obstruction for string graphs

Moshe J. White
Institute of Mathematics,
Hebrew University, Jerusalem, Israel Research supported in part by the Israel
Science Foundation (Grant No. 2669/21) and in part by the ERC Advanced Grant
(Grant No. 834735)

###### Abstract

In this paper we prove that a graph is a string graph (the intersection graph
of curves in the plane) if and only if it admits a drawing in the plane with
certain properties. This also allows us to define an algebraic obstruction,
similar to the Van Kampen obstruction to embeddability, which must vanish for
every string graph. However, unlike in the case of graph planarity this
obstruction is incomplete.

##  1 Introduction

A graph $\mathsf{G}=(V,E)$ is a _string graph_ , if it is the intersection
graph of a collection of curves in the plane: we may find $\left\\{\gamma_{v}\
:\ v\in V\right\\}$ such that every $\gamma_{v}$ is the image of a continuous
map $[0,1]\to\mathbb{R}^{2}$, and for any two vertices $v\neq w$ we have
$\gamma_{v}\cap\gamma_{w}\neq\emptyset$ if and only if
$\left\\{v,w\right\\}\in E$. The collection of curves
$\left\\{\gamma_{v}\right\\}$ is said to be a representation of $\mathsf{G}$.
For a recent brief exposition of string graphs, see [5, §1].

In some sense, we consider string graphs as a generalization of planar graphs.
In one of the earliest papers dealing with string graphs, Sinden [9] already
determined that every planar graph is a string graph, while some non-planar
graphs are string graphs (such as any complete graph) and some are not (such
as a barycentric subdivision of any non-planar graph). The approach presented
in this paper arose from previous work of the author [11], dealing with
simplicial complexes representable by convex sets in $\mathbb{R}^{d}$, where
representability was described in terms of the existence of certain linear
maps. Furthermore, necessary conditions (but usually not sufficient) for
representability were given. The lowest dimensions in which representability
is not fully classified is for graphs ($1$-dimensional simplicial complexes)
representable in dimension $2$. The topological restrictions on graphs are in
fact satisfied by all string graphs (a class strictly containing graphs
representable in dimension $2$, see [11, Example 4.13]).

In this paper we further explore the necessary topological restrictions on
string graphs. Keeping in mind the analogy between string graphs and planar
graphs, a drawing of $\mathsf{G}$ in the plane where _none_ of the edges
intersect indicates that $\mathsf{G}$ is planar. If we instead allow _some_
intersections among the edges of $\mathsf{G}$, the drawing indicates that our
graph is a string graph. The precise conditions classifying string graphs (in
terms of edge intersections) is given in Theorem 3.6. Continuing our analogy,
we consider the (strong) Hanani-Tutte Theorem (see [7] for more details):

######  Theorem 1.1 (Hanani-Tutte).

A graph $\mathsf{G}$ can be drawn in $\mathbb{R}^{2}$ such that every pair of
disjoint edges cross an even number of times, if and only if $\mathsf{G}$ is
planar.

The “if” part is clear, as an embedding of $\mathsf{G}$ admits no crossings
among edges. The converse may be deduced from Kuratowski’s theorem [10], or
even proved directly [6]. The even crossing condition is equivalent to the
existence of a solution to a system of linear equations over $\mathbb{Z}_{2}$,
described in cohomological terms as the vanishing of the (mod 2) Van Kampen
obstruction. We introduce a modified Van Kampen obstruction that must vanish
for every string graph $\mathsf{G}$ (Theorem 3.2). This theorem essentially
states that if $\mathsf{G}$ is a string graph, then there exists a drawing of
$\mathsf{G}$ in the plane such that certain pairs of edges cross an even
number of times. Unlike with graph planarity, the converse fails, as seen in
Example 3.4.

###  1.1 Notation for graphs

A graph $\mathsf{G}=(V,E)$ is an abstract collection of vertices
$V=V(\mathsf{G})$ and edges $E=E(\mathsf{G})$, where each edge is a set
containing exactly two vertices.

Abusing notation, when we use a graph as the domain or range of a function,
such as $f:\mathsf{G}\to\mathbb{R}^{2}$, this $\mathsf{G}$ is to be understood
as the topological space which is the geometric realization of the abstract
graph $\mathsf{G}$ (a point for each vertex and a line segment for every edge,
attached at the endpoints to the points corresponding to its vertices). We
sometimes identify an edge $\alpha\in E$ with the corresponding line segment,
so $f(\alpha)$ denotes the image of the line segment corresponding to
$\alpha$. We also use the abbreviated notation $vw$ in place of
$\left\\{v,w\right\\}$ (often an edge).

We define _a subdivision_ of $\mathsf{G}$ to be a graph $\mathsf{H}$ together
with a homeomorphism $\mathsf{G}\to\mathsf{H}$ taking vertices of $\mathsf{G}$
to vertices of $\mathsf{H}$. If no edge of $\mathsf{G}$ is mapped to a single
edge of $\mathsf{H}$, we say that $\mathsf{H}$ is a _proper_ subdivision. We
let $\mathsf{G}^{*}$ denote the barycentric subdivision of $\mathsf{G}=(V,E)$:

$V(\mathsf{G}^{*})=V\cup E\quad,\quad E(\mathsf{G}^{*})=\left\\{v\alpha\ :\
v\in\alpha\in E\right\\}.$

We usually require that maps $f:\mathsf{G}\to\mathbb{R}^{2}$ are piecewise
linear and in general position: there exists $\mathsf{H}$ which is a
subdivision of $\mathsf{G}$, such that the composition of $f$ with the
(inverse) homeomorphism $\mathsf{H}\to\mathsf{G}$ is linear (every edge is
mapped to a line segment), and takes no three vertices of $\mathsf{H}$ to the
same line.

##  2 Modified Van Kampen obstructions

We adopt the geometric definition of the (mod $2$) Van Kampen obstruction for
graph planarity from [6], and also borrow notation from [4, Appendix D]. For a
graph $\mathsf{G}=(V,E)$, we let

$P_{\Delta}(\mathsf{G})=\left\\{(\alpha,\beta)\in E^{2}\ :\
\alpha\cap\beta=\emptyset\right\\}$

denote the collection of disjoint pairs of edges in $\mathsf{G}$. The Van
Kampen obstruction can be defined as an equivariant cohomology class (with
coefficients in $\mathbb{Z}/2$) of the deleted product

$\mathsf{G}^{2}_{P}:=\bigcup_{(\alpha,\beta)\in P}F(\alpha)\times F(\beta),$

where $P=P_{\Delta}(\mathsf{G})$ and $F:\mathsf{G}\to\mathbb{R}^{3}$ is some
embedding. The only change we introduce to the ordinary mod 2 Van Kampen
obstruction, is that we allow for $P\subseteq P_{\Delta}$ to be some symmetric
subset. We do not assume familiarity with cohomology or the Van Kampen
obstruction, readers familiar with the argument (using finger moves) that the
obstruction must vanish for planar graphs may skip to Proposition 2.4, which
is proved analogously to the more familiar case of the deleted product.

The set $P$ should be thought of as a collection of edge pairs whose
intersections are forbidden: our goal will be to find some map
$f:\mathsf{G}\to\mathbb{R}^{2}$ such that $f(\alpha)\cap f(\beta)=\emptyset$
for every $(\alpha,\beta)\in P$ (or prove that no such $f$ exists).

Since any pair $(\alpha,\beta)\in P$ involves disjoint edges, the intersection
$f(\alpha)\cap f(\beta)$ is either empty or consists of finitely many points,
all in the interior of both edges. Assuming $f$ is piecewise linear and in
general position, we define a vector $o_{f}\in\mathbb{Z}_{2}^{P}$ by

$o_{f}(\alpha,\beta)=\left|f(\alpha)\cap f(\beta)\right|\text{ mod }2.$

If $f$ is not piecewise linear and in general position, we say that $o_{f}$ is
undefined. Our next step is to define vectors corresponding to the Van Kampen
“finger moves”. For every vertex $u\in V$ and edge $\omega\in E$ such that
$u\notin\omega$, define $\varphi_{\omega,u}\in\mathbb{Z}_{2}^{P}$ by

$\varphi_{\omega,u}\left(\alpha,\beta\right)=\begin{cases}1&\text{if
$\alpha=\omega$ and $u\in\beta$,}\\\ 1&\text{if $\beta=\omega$ and
$u\in\alpha$,}\\\ 0&\text{otherwise}.\end{cases}$

The collection of all finger move vectors
$\Phi=\left\\{\varphi_{\omega,u}:u\in V,\ \omega\in E,\
u\notin\omega\right\\}$ defines an equivalence relation on
$\mathbb{Z}_{2}^{P}$, where two vectors $\varphi,\psi$ are equivalent if
$\varphi-\psi\in\textnormal{span}_{\mathbb{Z}_{2}}(\Phi)$.

Note that both $o_{f}$ and $\varphi_{\omega,u}$ in fact depend on the set of
pairs $P$, and we will denote them by $o_{f}^{P}$ and $\varphi_{\omega,u}^{P}$
respectively if $P$ isn’t clear from the context. We can now choose some
function $f:\mathsf{G}\to\mathbb{R}^{2}$, and define:

######  Definition 2.1.

The Van Kampen obstruction of a graph $\mathsf{G}$ is defined as the
equivalence class
${\mathfrak{o}_{\mathsf{G}}=\left[o_{f}^{P_{\Delta}(\mathsf{G})}\right]}$.

And more generally:

######  Definition 2.2.

Let $\mathsf{G}=(V,E)$ be a graph, and let $P\subseteq E^{2}$ be a symmetric
collection of disjoint edges: any $(\alpha,\beta)\in P$ satisfies
$(\beta,\alpha)\in P$ and $\alpha\cap\beta=\emptyset$.
The Van Kampen obstruction of $\mathsf{G}$ (modified for $P$) is the
equivalence class

$\mathfrak{o}_{\mathsf{G}}(P)=\left[o_{f}^{P}\right].$

We say that the obstruction _vanishes_ if
$\mathfrak{o}_{\mathsf{G}}(P)=\left[0\right]$.

The definition implies that $\mathfrak{o}_{\mathsf{G}}(P)$ doesn’t depend on
our choice of $f$. Indeed, the vectors equivalent to _any_ $o_{f}$ are
precisely _all_ vectors of the form $o_{g}$. Explicitly:

######  Lemma 2.3.

For $\mathsf{G},P$ as above and $f:\mathsf{G}\to\mathbb{R}^{2}$, we have:

  1. 1.

For every $\varphi_{\omega,u}\in\Phi$, there exists
$g:\mathsf{G}\to\mathbb{R}^{2}$, such that $o_{g}=o_{f}+\varphi_{\omega,u}$,

  2. 2.

For every $g:\mathsf{G}\to\mathbb{R}^{2}$, we have
$o_{f}-o_{g}\in\textnormal{span}_{\mathbb{Z}_{2}}(\Phi)$.

###### Proof.

We achieve (1) by applying a finger move to $f$: take a thin tube around a
path connecting an interior point of $f(\omega)$ to the vertex $u$, such that
$u$ is the only vertex within the tube. Then we only modify the image of
$\omega$ to go around the boundary of the tube. The resulting modified
function $g$ has $o_{g}=o_{f}+\varphi_{\omega,u}$ (see Figure 2).

$1$$2$$3$$4$$5$$6$$1$$2$$3$$4$$5$$6$

Figure 1: a map obtained from a linear $f:\mathsf{G}\to\mathbb{R}^{2}$ (left)
by applying a finger move taking $\omega=12$ around $u=4$ (right). Note that
we ignore intersections of $\omega$ with edges $\beta$ satisfying
$(\omega,\beta)\notin P$ (such as $\beta=24$ in this example, as follows from
$P\subseteq P_{\Delta}$). Self intersections of $\omega$ may also be created
and ignored for the same reason. Intersections of $\omega$ with edges $\beta$
such that $u\notin\beta\neq\omega$ (such as $\beta=36$ above) are created in
pairs and thus have no effect on $o_{f}$.

For (2), we say that a homotopy $h_{t}:\mathsf{G}\to\mathbb{R}^{2}$ is
generic, if $o_{h_{t}}$ is defined for all but finitely many $t\in[0,1]$. In
this case, as $t$ increases from $0$ to $1$, the only occasion where
$o_{h_{t}}$ is modified is when a vertex $u$ passes through an edge $\omega$,
and this modifies $o_{h_{t}}$ by adding $\varphi_{\omega,u}$. For example, if
we take $f=h_{0}$ to be a linear map of the graph with vertices positioned as
in Figure 2 (left), and as $t$ increases modify our function by gradually
pulling vertex $4$ to the left, such that at time $t_{1}>0$ the vertex passes
through the edge $36$, and at time $t_{2}\in(t_{1},1)$ the vertex passes
through the edge $12$. Then we have $o_{h_{t}}=o_{f}+\varphi_{36,4}$ for
$t\in(t_{1},t_{2})$, and $o_{h_{t}}=o_{f}+\varphi_{36,4}+\varphi_{12,4}$ for
$t>t_{2}$. Given any two (general position piecewise linear) functions
$f,g:\mathsf{G}\to\mathbb{R}^{2}$, we can easily find such a generic homotopy
between them, completing our proof. ∎

This clearly implies that $\mathfrak{o}_{\mathsf{G}}$ vanishes if $\mathsf{G}$
is planar. For modified obstructions, we deduce from the lemma:

######  Proposition 2.4.

Let $\mathsf{G}=(V,E)$ be a graph, $P\subseteq E^{2}$ a symmetric collection
of disjoint edges. Then $\mathfrak{o}_{\mathsf{G}}(P)$ vanishes if and only if
there exists $f:\mathsf{G}\to\mathbb{R}^{2}$, such that $\left|f(\alpha)\cap
f(\beta)\right|$ is even for every $(\alpha,\beta)\in P$.

##  3 The string graph obstruction

We are now ready to define the string obstruction of a graph:

######  Definition 3.1.

For a graph $\mathsf{G}=(V,E)$, denote

$P=P_{s}(\mathsf{G})=\left\\{(\alpha,\beta)\in E^{2}\ :\ vw\notin E\text{ for
every }v\in\alpha,w\in\beta\right\\}$

and define the _string obstruction_ of $\mathsf{G}$ as

$\mathfrak{o}^{s}_{\mathsf{G}}=\mathfrak{o}_{\mathsf{G}}(P).$

We claim:

######  Theorem 3.2.

Let $\mathsf{G}$ be a string graph. Then $\mathfrak{o}^{s}_{\mathsf{G}}$
vanishes.

The proof will be postponed until after establishing Lemma 3.7, and first we
consider some examples:

######  Corollary 3.3.

The Heawood graph (see Figure 3 below) is not a string graph.

123456789abcde

Figure 2: the Heawood graph.

###### Proof.

Let $\mathsf{G}$ denote the Heawood graph and $P=P_{s}(\mathsf{G})$. Denote

$P_{\text{set}}=\left\\{\left\\{\alpha,\beta\right\\}\ :\ (\alpha,\beta)\in
P\right\\},$

which is well defined because $P$ is symmetric. Also note that if
$\psi\in\mathbb{Z}_{2}^{P}$ is either a finger move vector or some $o_{f}$,
then $\psi(\alpha,\beta)=\psi(\beta,\alpha)$ for every $(\alpha,\beta)\in P$.
We may therefore say that such $\psi$ is _even_ , if

$\sum_{(\alpha,\beta)\in
P_{\text{set}}}\psi(\alpha,\beta)=0\in\mathbb{Z}_{2},$

and otherwise say that $\psi$ is _odd_. Furthermore, for any
$f:\mathsf{G}\to\mathbb{R}^{2}$ we have

$\sum_{(\alpha,\beta)\in
P_{\text{set}}}o_{f}(\alpha,\beta)=\sum_{(\alpha,\beta)\in
P_{\text{set}}}\left|f(\alpha)\cap f(\beta)\right|\text{ mod }2.$

For the function $f$ displayed in Figure 3, we obtain that $o_{f}$ is odd:
there are $7$ crossings in total among edge pairs belonging to
$P_{\text{set}}$ \- those which are closest to the center of the figure, such
as $f(1a)\cap f(3c)$. The other $7$ crossings such as $f(1a)\cap f(5e)$
involve pairs of edges not belonging to $P_{\text{set}}$.

Our claim will now follow by proving that every finger move for $\mathsf{G}$
is even: this implies that $o_{g}$ is odd for every
$g:\mathsf{G}\to\mathbb{R}^{2}$, therefore $\mathfrak{o}_{\mathsf{G}}^{s}$
doesn’t vanish and Theorem 3.2 implies that $\mathsf{G}$ is not a string
graph.

Consider a finger move vector $\varphi_{\omega,u}$, where $\omega=12$ and
$u\notin\omega$. If $\varphi_{\omega,u}(\alpha,\beta)=1$ then one of the edges
$\alpha,\beta$ equals $\omega$ and the other edge contains $u$ (w.l.o.g.
$\alpha=\omega$ and $u\in\beta$). Furthermore, $(\omega,\beta)\in P$ only for
edges $\beta$ whose vertices are not adjacent to either of the vertices of
$\omega$, in this case the vertices of $\beta$ must be two of
$\left\\{4,5,6,8,9,b,c,d\right\\}$. These vertices form a cycle of length $8$
in $\mathsf{G}$, in the cyclic order $456bcd89$. We may now deduce that
$\varphi_{\omega,u}$ must be even: if $u\in\left\\{3,7,a,e\right\\}$ then
$\varphi_{\omega,u}$ is the zero vector. Otherwise
$u\in\left\\{4,5,6,8,9,b,c,d\right\\}$ and there are exactly two vertices $v$
for which $\beta=uv$ has $\varphi_{\omega,u}(\omega,\beta)=1$ (those $v$ which
are adjacent to $u$ in the cycle), and again $\varphi_{\omega,u}$ is even.

For other edges $\omega$ the proof is analogous: if $\omega=1a$ then non-zero
values are obtained when $u$ and $\beta$ are part of the cycle $345678dc$, and
if $\omega=1e$ then non-zero values are obtained when $u$ and $\beta$ are part
of the cycle $349876bc$. The proof for any other $\omega$ easily follows from
the previous cases by rotations and reflections fixing Figure 3. ∎

There are many more cases of graphs for which the string obstruction doesn’t
vanish, and Theorem 3.2 gives us a unified approach in proving that they
aren’t string graphs. For example, if $\mathsf{H}$ is a proper barycentric
subdivision of a non-planar graph $\mathsf{G}$, the non-vanishing of
$\mathfrak{o}_{\mathsf{H}}^{s}$ follows from Hananni-Tutte (Theorem 1.1):
every map $f:\mathsf{H}\to\mathbb{R}^{2}$ determines a map
$\mathsf{G}\to\mathbb{R}^{2}$, so there are two disjoint $\mathsf{G}$-edges
which intersect an odd number of times. In $\mathsf{H}$ these are divided into
pairs of edges in $P_{s}(\mathsf{G})$, so at least one such pair must
intersect an odd number of times, thus $o_{f}$ is non-zero.

In [3], Goljan, Kratochvíl, and Kučera explored string graphs and gave a
number of examples of small non-string graphs, including two with $12$
vertices [3, Example 3.9], also proving that this is the minimal number of
vertices in any string graphs. We won’t include all such examples and proofs -
for any graph $\mathsf{G}$, determining if the string graph obstruction
vanishes can be done by solving a linear system of equations over
$\mathbb{Z}_{2}$: $\mathfrak{o}_{\mathsf{G}}^{s}$ vanishes if and only if
$o_{f}\in\text{span}_{\mathbb{Z}_{2}}(\Phi)$ for some
$f:\mathsf{G}\to\mathbb{R}^{2}$. This paper also provides an example in which
the converse of Theorem 3.2 doesn’t hold:

######  Example 3.4.

Consider the graph GP defined [3], and an important infinite class of non-
string graphs denoted $C*\overline{C}_{n}$ for every $n\geq 5$ [1]. The two
functions $f:\mathsf{G}\to\mathbb{R}^{2}$ displayed in Figure 3.4 for
$\mathsf{G}=GP$ and $\mathsf{G}=C*\overline{C}_{6}$ have $o_{f}=0$, as all
crossings involve pairs of edges not in $P_{s}(\mathsf{G})$.

Figure 3: the graphs $GP$ (left) and $C*\overline{C}_{6}$ (right).

A similar $C*\overline{C}_{n}\to\mathbb{R}^{2}$ exists for every $n\geq 5$,
proving that the string obstruction fails to classify infinitely many non-
string graphs.

###  3.1 The obstruction via barycentric subdivisions

######  Definition 3.5.

Let $\mathsf{G}=(V,E)$ be a graph, and recall that $\mathsf{G}^{*}$ denotes
the barycentric subdivision of $\mathsf{G}$: the edges of $\mathsf{G}^{*}$ are
$v\alpha$ where $v\in\alpha\in E$. Denote

$P=P_{\textnormal{sd}}(\mathsf{G})=\left\\{(v\alpha,w\beta)\in
E\left(\mathsf{G}^{*}\right)^{2}\ :\ v\neq w\text{ and }vw\notin E\right\\},$

and define the _subdivided string obstruction_ of $\mathsf{G}$ as

$\mathfrak{o}_{\mathsf{G}}^{\textnormal{sd}}=\mathfrak{o}_{\mathsf{G}^{*}}(P).$

Note that $\mathfrak{o}_{\mathsf{G}}^{\textnormal{sd}}$ is in fact a Van
Kampen obstruction of $\mathsf{G}^{*}$ modified for $P$ (in the sense of
Definition 2.2). We associate it with $\mathsf{G}$, however, because it
vanishes when $\mathsf{G}$ is a string graph. In fact,
$P_{\textnormal{sd}}(\mathsf{G})$ contains precisely the pairs of edges whose
intersections are forbidden if $\mathsf{G}$ is a string graph, in the
following sense:

######  Theorem 3.6.

Let $\mathsf{G}=(V,E)$ be a graph. Then $\mathsf{G}$ is a string graph if and
only if there exists a drawing of the barycentric subdivision
$g:\mathsf{G}^{*}\to\mathbb{R}^{2}$, such that

$g(v\alpha)\cap g(w\beta)=\emptyset$

for every $(v\alpha,w\beta)\in P_{\textnormal{sd}}(\mathsf{G})$.

###### Proof.

Suppose $g:\mathsf{G}^{*}\to\mathbb{R}^{2}$ is as above. For every $v\in E$
let $\mathsf{T}_{v}$ denote the subgraph of $\mathsf{G}^{*}$ induced on the
collection (of $\mathsf{G}^{*}$-vertices)
$\left\\{v\right\\}\cup\left\\{\alpha\in E\ :\ v\in\alpha\right\\}$. From our
assumption on the properties of $g$, it is clear that for every $v,w\in V$
with $v\neq w$ we have $g(\mathsf{T}_{v})\cap g(\mathsf{T}_{w})\neq\emptyset$
if and only if $vw\in E$. For every $v\in V$ we know that $g(\mathsf{T}_{v})$
is connected (in fact $\mathsf{T}_{v}$ is a tree), so let
$\gamma_{v}=g(\mathsf{T}_{v})$ be a path tracing along the line segments
$\bigcup_{\alpha\ni v}g(v\alpha)$. This gives us a representation of
$\mathsf{G}$ as a string graph.

For the converse, suppose that $\mathsf{G}$ is a string graph represented by a
collection of paths $\gamma_{v}\subseteq\mathbb{R}^{2}$ for every $v\in V$. So
for every $v\neq w$ in $V$, the intersection $\gamma_{v}\cap\gamma_{w}$ is
non-empty if and only if $vw\in E$. For every $v\in V$ define $g(v)$ as an
arbitrary point in $\gamma_{v}$, and for every $\alpha=vw\in E$ define
$g(\alpha)$ as an arbitrary point in $\gamma_{v}\cap\gamma_{w}$. For every
$v\in\alpha\in E$, note that both $g(v)$ and $g(\alpha)$ are in $\gamma_{v}$.
We may therefore connect them within this image, and obtain
$g:\mathsf{G}^{*}\to\mathbb{R}^{2}$ such that $g(v\alpha)\subseteq\gamma_{v}$
for every $v\in\alpha\in E$. For every $(v\alpha,w\beta)\in
P_{\textnormal{sd}}(\mathsf{G})$, we clearly have $vw\notin E$, therefore
$g(v\alpha)\cap g(w\beta)\subseteq\gamma_{v}\cap\gamma_{w}=\emptyset$
completing our proof. ∎

It might seem that we have described a second obstruction for string graphs.
However, we will now prove in the following lemma that they vanish together,
therefore once we know $\mathfrak{o}_{\mathsf{G}}^{s}$, computing
$\mathfrak{o}_{\mathsf{G}}^{\textnormal{sd}}$ tells us nothing new about
whether $\mathsf{G}$ is a string graph or not. Half of the statement is also
our final piece for the proof of Theorem 3.2.

######  Lemma 3.7.

Let $\mathsf{G}=(V,E)$ be a graph. Then $\mathfrak{o}_{\mathsf{G}}^{s}$
vanishes if and only if $\mathfrak{o}_{\mathsf{G}}^{\textnormal{sd}}$
vanishes.

###### Proof.

We will prove that there exists a function $f:\mathsf{G}\to\mathbb{R}^{2}$
such that $\left|f(\alpha)\cap f(\beta)\right|$ is even for every
$(\alpha,\beta)\in P_{s}(\mathsf{G})$, if and only if there exists a function
$g:\mathsf{G}^{*}\to\mathbb{R}^{2}$ such that $\left|g(v\alpha)\cap
g(w\beta)\right|$ is even for every $(v\alpha,w\beta)\in
P_{\textnormal{sd}}(\mathsf{G})$. This statement is equivalent to our lemma,
due to Proposition 2.4 (applied for both obstructions).

Suppose such $g:\mathsf{G}^{*}\to\mathbb{R}^{2}$ exists. Let
$f:\mathsf{G}\to\mathbb{R}^{2}$ denote the composition of the homeomorphism
$\mathsf{G}\to\mathsf{G}^{*}$ with $g$. Let $(\alpha,\beta)\in
P_{s}(\mathsf{G})$, where $\alpha=v_{1}v_{2}$ and $\beta=w_{1}w_{2}$. As line
segments, we have $\alpha=v_{1}\alpha\cup v_{2}\alpha$ and
$\beta=w_{1}\beta\cup w_{2}\beta$. Clearly we have
$(v_{i}\alpha,w_{j}\beta)\in P_{\textnormal{sd}}(\mathsf{G})$ for every
$i,j=1,2$. We deduce that

$\left|f(\alpha)\cap f(\beta)\right|=\sum_{i,j=1}^{2}\left|g(v_{i}\alpha)\cap
g(w_{j}\beta)\right|$ (3.1)

is even as a sum of even terms.
Conversely, if $f:\mathsf{G}\to\mathbb{R}^{2}$ exists, let
$g:\mathsf{G}^{*}\to\mathbb{R}^{2}$ be obtained from $f$ by identifying
$\mathsf{G}$ and $\mathsf{G}^{*}$ as homeomorphic, taking care that for every
$\alpha\in E$, the point $g(\alpha)$ is chosen as a point of $f(\alpha)$ where
$f(\alpha)$ doesn’t cross itself or any other edges. Suppose that
$\left|g(v\alpha)\cap g(w\beta)\right|$ is odd, for some $v\in\alpha\in E$ and
$w\in\beta\in E$ where $(v\alpha,w\beta)\in P_{\textnormal{sd}}(\mathsf{G})$.
We say that the pair $(\alpha,\beta)$ is _problematic_ , and denote
$\alpha=v_{1}v_{2}$ and $\beta=w_{1}w_{2}$. For every such problematic pair,
we will modify $g$ using the finger moves (for $\mathsf{G}^{*}$):

$\varphi_{v_{1}\alpha,\beta}\ ,\ \varphi_{v_{2}\alpha,\beta}\ ,\
\varphi_{w_{1}\beta,\alpha}\ ,\ \varphi_{w_{2}\beta,\alpha}$

to obtain some modified $g^{\prime}:\mathsf{G}^{*}\to\mathbb{R}^{2}$. Note
that these finger moves only effect the parity of $\left|g(v_{i}\alpha)\cap
g(w_{j}\beta)\right|$ for $i,j=1,2$, and applying some or all of these finger
moves allows us to freely move any number of intersection points between the
four sets $g(v_{i}\alpha)\cap g(w_{j}\beta)$ ($i,j=1,2$). We proceed as
follows: if $v_{k}w_{l}\in E$ for some $k,l\in\left\\{1,2\right\\}$, then
$(v_{k}\alpha,w_{l}\beta)\notin P_{\textnormal{sd}}(\mathsf{G})$, and we may
apply the finger moves to guarantee that $\left|g(v_{i}\alpha)\cap
g(w_{j}\beta)\right|$ is even for every $(i,j)\neq(k,l)$. If $v_{i}w_{j}\notin
E$ for every $i,j=1,2$ then $(\alpha,\beta)\in P_{s}(\mathsf{G})$. Clearly
(3.1) still holds, and the sum is even by our assumption on $f$. We may
therefore apply the finger moves to guarantee that $\left|g(v_{i}\alpha)\cap
g(w_{j}\beta)\right|$ is even for every $i,j=1,2$. Proceeding similarly for
every problematic pair $(\alpha,\beta)$, we obtain our modified $g^{\prime}$
as required. ∎

###### Proof of Theorem 3.2.

Let $\mathsf{G}$ be a string graph. By Theorem 3.6, and Proposition 2.4,
$\mathfrak{o}_{\mathsf{G}}^{\textnormal{sd}}$ vanishes. Our proof now follows
from Lemma 3.7. ∎

##  4 Final remarks

In an attempt to strengthen Theorem 3.2, we may consider an _integer valued_
modified Van Kampen obstruction for string graphs. This can be achieved using
cohomology with integer coefficients, or as a variation of the geometric
definitions we used. Given $f:\mathsf{G}\to\mathbb{R}^{2}$, every intersection
point in $f(\alpha)\cap f(\beta)$ (where $\alpha,\beta\in E(\mathsf{G})$) has
either a positive or negative sign. Defining $\widetilde{o}_{f}(\alpha,\beta)$
as the sum of these contributions gives us an element
$\widetilde{o}_{f}\in\mathbb{Z}^{P}$ such that $o_{f}$ is the modulo $2$
reduction of $\widetilde{o}_{f}$. This was already considered (in a different
form) by Tutte [10], see [4, Appendix D] for the integer Van Kampen
obstruction. Clearly there are functions $f$ such that $\widetilde{o}_{f}$ is
non-zero while $o_{f}=0\in\mathbb{Z}_{2}^{P}$. However, the author is
currently unaware of any example $\mathsf{G}$ for which
$\mathfrak{o}_{\mathsf{G}}^{s}$ vanishes but the integer valued obstruction
does not ($\widetilde{o}_{f}\neq 0\in\mathbb{Z}^{P_{s}(\mathsf{G})}$ for every
$f$).

Our final remark relates to the computational complexity of recognizing string
graphs. Determining if $\mathfrak{o}_{\mathsf{G}}^{s}$ vanishes can be done in
polynomial time (even if we consider the integer valued obstruction), however
Kratochvíl [2] proved that recognizing string graphs is NP-hard (later
determined to be NP-complete [8]). Proof of NP-hardness is done by reducing
the planar 3-satisfiability problem to determining whether certain graphs are
string graphs. Avoiding almost all details, a formula $\varphi$ (in
conjunctive normal form) with certain properties is satisfiable if and only if
some graph $\mathsf{G}$ derived from $\varphi$ is a string graph. String
representations of $\mathsf{G}$ are in one-to-one correspondence with variable
assignments that satisfy $\varphi$.
Obtaining a string graph representation for $\mathsf{G}$ from an assignment
which doesn’t satisfy $\varphi$ would be possible if $C*\overline{C}_{6}$ (see
Figure 3.4) were a string graph: representing each subgraph of $\mathsf{G}$
corresponding to a false clause in $\varphi$ in a manner consistent with the
assignment, is equivalent to obtaining a string representation of
$C*\overline{C}_{6}$. Since we have $f:C*\overline{C}_{6}\to\mathbb{R}^{2}$
for which $o_{f}=0$, any arbitrary assignment to the variables of $\varphi$
indicates that $\mathfrak{o}_{\mathsf{G}}^{s}$ vanishes, whether $\varphi$ is
satisfiable or not. We thus deduce, perhaps unsurprisingly, that the
computationally easy task of determining whether the string obstruction
vanishes is of no use to the NP-hard problem described by Kratochvíl.

## Acknowledgments

This work was completed as part of the PhD thesis of the author. I would like
to thank my advisor Gil Kalai, as well as Jan Kratochvíl and Geva Yashfe.

## References

  * [1] J. Kratochvíl, String graphs. I. The number of critical nonstring graphs is infinite, Journal of Combinatorial Theory, Series B, 52 (1991), 53–66.
  * [2] J. Kratochvíl, String graphs. II. Recognizing string graphs is NP-hard, Journal of Combinatorial Theory, Series B, 52 (1991), 67–78.
  * [3] J. Kratochvíl, M. Goljan, P. Kučera, String graphs, Academia, Prague, 1986.
  * [4] J. Matoušek, M. Tancer, U. Wagner, Hardness of embedding simplicial complexes in $\mathbb{R}^{d}$, arxiv:0807.0336 (2008).
  * [5] J. Pach, B. Reed, Y. Yuditsky, Almost all string graphs are intersection graphs of plane convex sets, Discrete & Computational Geometry, 63 (2020), 888–917.
  * [6] K. S. Sarkaria, A one-dimensional Whitney trick and Kuratowski’s graph planarity criterion, Israel Journal of Mathematics, 73 (1991), 79–89.
  * [7] M. Schaefer, Hanani-Tutte and related results, In: Geometry - intuitive, discrete, and convex, Springer, Berlin, Heidelberg, 2013, pp. 259–299.
  * [8] M. Schaefer, E. Sedgwick, D. Štefankovič, Recognizing string graphs in NP, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, (2002), 1–6.
  * [9] F. W. Sinden, Topology of thin film RC circuits, Bell System Technical Journal, 45 (1966), 1639–1662.
  * [10] W. T. Tutte, Toward a theory of crossing numbers, Journal of Combinatorial Theory, 8 (1970), 45–53.
  * [11] M. J. White, $d$-representability as an embedding problem, (preprint), arxiv:2205.10099.


# Algebraic hyperbolicity of very general hypersurfaces in products of
projective spaces

Wern Yeong<EMAIL_ADDRESS>Department of Mathematics
University of Notre Dame, Notre Dame, IN 46556

###### Abstract.

We study the algebraic hyperbolicity of very general hypersurfaces in
$\mathbb{P}^{m}\times\mathbb{P}^{n}$ by using three techniques that build on
past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As a result,
we completely answer the question of whether or not a very general
hypersurface of bidegree $(a,b)$ in $\mathbb{P}^{m}\times\mathbb{P}^{n}$ is
algebraically hyperbolic, except in $\mathbb{P}^{3}\times\mathbb{P}^{1}$ for
the bidegrees $(a,b)=(7,3),(6,3)$ and $(5,b)$ with $b\geq 3.$ As another
application of these techniques, we improve the known result that very general
hypersurfaces in $\mathbb{P}^{n}$ of degree at least $2n-2$ are algebraically
hyperbolic when $n\geq 6$ to $n\geq 5$, leaving $n=4$ as the only open case.

##  1\. Introduction

A complex projective variety is _algebraically hyperbolic_ if for some
$\varepsilon>0$ and some ample divisor $H$, every integral curve $C$ that lies
in the variety satisfies

(1) $2g(C)-2\geq\varepsilon\cdot\deg_{H}C,$

where $g(C)$ is the geometric genus of the curve. It follows that varieties
that contain any rational or elliptic curves are not algebraically hyperbolic.

Algebraic hyperbolicity was introduced by Demailly in [De95] as an algebraic
analogue to Kobayashi hyperbolicity for complex manifolds, which was shown to
be equivalent to Brody hyperbolicity when the manifold is compact [Br78]. A
compact complex manifold $X$ is _Brody hyperbolic_ if all holomorphic maps
$\mathbb{C}\rightarrow X$ are constant, i.e. $X$ contains no entire curves.
Demailly [De95] proved that for smooth projective varieties Kobayashi
hyperbolicity implies algebraic hyperbolicity, and conjectured that the
converse holds.

The algebraic and Brody hyperbolicity of very general degree $d$ hypersurfaces
$X_{d}\subseteq\mathbb{P}^{n}$ is well-studied. (See for instance the surveys
[Co05, De18, Vo03].) For $n=3$, Xu [Xu94] proved that very general surfaces
$X_{d}\subseteq\mathbb{P}^{3}$ of degree $d\geq 6$ are algebraically
hyperbolic by using a degeneration argument. Then Coskun and Riedl [CR19a]
improved Xu’s bound to $d\geq 5$ by considering the degrees of surface scrolls
that contain any particular curve in $X_{d}$. Their result completes the
classification in $\mathbb{P}^{3}$ since surfaces of degree at most $4$
contain rational curves. For larger $n$, Clemens [Cl86] and Ein [Ei88] proved
that $X_{d}$ is algebraically hyperbolic when $d\geq 2n$ for $n\geq 4$. The
bound was subsequently improved by Voisin [Vo96, Vo98] to $d\geq 2n-1$ for
$n\geq 4$, and then by Pacienza [Pa04] and Clemens and Ran [ClR04] to $d\geq
2n-2$ for $n\geq 6$. We confirm in this article that very general degree 8
hypersurfaces in $\mathbb{P}^{5}$ are algebraically hyperbolic, hence
improving the previously-known result to $d\geq 2n-2$ for $n\geq 5$ (see
Theorem 3.21). When $d\leq 2n-3$, $X_{d}$ contains lines and hence, is not
algebraically hyperbolic. Thus, the only remaining open question in
$\mathbb{P}^{n}$ is the case of sextic threefolds.

The hyperbolicity of very general hypersurfaces has also been studied in other
ambient varieties besides $\mathbb{P}^{n}$. Haase and Ilten [HI19] studied
very general surfaces in Gorenstein toric threefolds by building on the focal
loci techniques of Chiantini and Lopez [CL99]. As an application, they almost
completely classified very general surfaces in the threefolds
$\mathbb{P}^{2}\times\mathbb{P}^{1}$,
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$,
$\mathbb{F}_{e}\times\mathbb{P}^{1}$ and the blowup of $\mathbb{P}^{3}$ at a
point. Then, Coskun and Riedl completed this classification in [CR19b] as a
result of their techniques for three-dimensional complex projective varieties
that admit a group action with dense orbit. More recently, Robins [Ro21]
almost completely classified very general surfaces in smooth projective toric
threefolds with Picard rank 2 or 3 by combining Haase and Ilten’s focal loci
techniques with a study of the combinatorics of such threefolds.

In this paper, we study very general hypersurfaces in
$\mathbb{P}^{m}\times\mathbb{P}^{n}$ and prove the following main theorem. We
assume throughout that $m\geq n.$

######  Theorem 1.1.

A very general hypersurface in $\mathbb{P}^{m}\times\mathbb{P}^{n}$ of
bidegree $(a,b)$ is

  1. (1)

algebraically hyperbolic if

    1. (a)

$m+n\geq 5$: $a\geq 2m+n-2$ and $b\geq m+2n-2$; or

    2. (b)

$(m,n)=(2,2)$: $a\geq 5$ and $b\geq 5$; or

    3. (c)

$(m,n)=(3,1)$: $a\geq 6$ and $b\geq 4$, or $a\geq 8$ and $b=3$,

  2. (2)

not algebraically hyperbolic if

    1. (a)

$m+n\geq 5$: either $a<2m+n-2$ or $b<m+2n-2$; or

    2. (b)

$(m,n)=(2,2)$: either $a<5$ or $b<5$; or

    3. (c)

$(m,n)=(3,1)$: either $a<5$ or $b<3$.

This completely resolves algebraic hyperbolicity of very general hypersurfaces
in $\mathbb{P}^{m}\times\mathbb{P}^{n}$, except in
$\mathbb{P}^{3}\times\mathbb{P}^{1}$ for bidegrees $(a,b)=(7,3),(6,3)$ and
$(5,b)$ with $b\geq 3.$

Note that the classification of very general surfaces in
$\mathbb{P}^{2}\times\mathbb{P}^{1}$ was already done by Haase and Ilten
[HI19] and Coskun and Riedl [CR19b], while hypersurfaces in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ are curves, which are easy to classify.
We give a description of the remaining open cases in
$\mathbb{P}^{3}\times\mathbb{P}^{1}$ in Remark 4.3.

###  1.1. Organization

In §2, we introduce the setup developed by Ein, Voisin, Pacienza, and others.
In §3, we present the three techniques that are used in §4 to prove the main
theorem. As another application of these techniques, we give a streamlined
proof of the algebraic hyperbolicity of very general hypersurfaces in
$\mathbb{P}^{n}$ of degree at least $2n-1$ when $n\geq 4$ and degree at least
$2n-2$ when $n\geq 5$.

###  1.2. Acknowledgements

I would like to thank my adviser Eric Riedl for suggesting this problem to me,
and for many valuable discussions and revisions of my drafts.

##  2\. Setup

We use a similar setup as [ClR04, CR19a, CR19b, Pa03, Pa04, Vo96]. Denote by
$\pi_{1}$ and $\pi_{2}$ the projection maps from
$\mathbb{P}^{m}\times\mathbb{P}^{n}$. Let
$H_{1}=\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1)$ and
$H_{2}=\pi_{2}^{*}\mathcal{O}_{\mathbb{P}^{n}}(1)$. Then
$\operatorname{Pic}(\mathbb{P}^{m}\times\mathbb{P}^{n})=\mathbb{Z}\cdot
H_{1}\oplus\mathbb{Z}\cdot H_{2}$. For $a,b>0$, denote the line bundle
$aH_{1}+bH_{2}$ by $\mathcal{E}$. Suppose that a very general section $X$ of
$\mathcal{E}$ contains a curve $Y$ of geometric genus $g$ and degree $e$ with
respect to the ample class $H_{1}+H_{2}$. Denote
$V=H^{0}(\mathbb{P}^{m}\times\mathbb{P}^{n},\mathcal{E})$, and let
$\mathcal{X}_{V}\subseteq(\mathbb{P}^{m}\times\mathbb{P}^{n})\times V$ be the
universal hypersurface of bidegree $(a,b)$. Let
$T\subseteq\text{Hilb}(\mathbb{P}^{m}\times\mathbb{P}^{n})\times V$ be an
irreducible variety such that

  1. (1)

a general element $(Y,F)$ of $T$ projects to a point in
$\text{Hilb}(\mathbb{P}^{m}\times\mathbb{P}^{n})$ representing a curve $Y$ of
geometric genus $g$ and $(H_{1}+H_{2})$-degree $e$ that is contained in the
hypersurface defined by $F$, and

  2. (2)

the projection map $T\rightarrow V$ is dominant, and

  3. (3)

$T$ is stable under the $G:=GL_{m+1}\times GL_{n+1}$-action on
$\text{Hilb}(\mathbb{P}^{m}\times\mathbb{P}^{n})\times V.$

Let $\mathcal{Y}_{T}\subseteq\mathcal{X}_{T}:=\mathcal{X}_{V}\times_{V}T$
denote the universal curve over $T$. By (3), there is a $G$-invariant
subvariety $U\subseteq T$ such that $U\rightarrow V$ is étale, and we restrict
the universal curve to $\mathcal{Y}_{U}$. Finally after taking a
$G$-equivariant resolution of $\mathcal{Y}_{U}$ and restricting $U$ to a
$G$-invariant open set, we obtain a smooth family $\mathcal{Y}\rightarrow U$
whose fibers are smooth curves mapping to curves of geometric genus $g$ and
$(H_{1}+H_{2})$-degree $e$. Similarly, pull back $\mathcal{X}_{T}$ to $U$ to
get a smooth family $\mathcal{X}\rightarrow U$ of hypersurfaces. Denote the
projection maps by $p_{1}:\mathcal{X}\rightarrow U$ and
$p_{2}:\mathcal{X}\rightarrow\mathbb{P}^{m}\times\mathbb{P}^{n}$. Denote by
$h:\mathcal{Y}\rightarrow\mathcal{X}$ the natural map, which is generically
injective. Note that

(2) $\operatorname{codim}(\mathcal{Y}\subseteq\mathcal{X})=m+n-2.$

By $G$-invariance, we can define the _vertical tangent sheaves_
$T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}$ and
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}$ by the following short
exact sequences:

$0\rightarrow T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}\rightarrow
T_{\mathcal{X}}\rightarrow
p_{2}^{*}T_{\mathbb{P}^{m}\times\mathbb{P}^{n}}\rightarrow 0$ $0\rightarrow
T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}\rightarrow
T_{\mathcal{Y}}\rightarrow
h^{*}p_{2}^{*}T_{\mathbb{P}^{m}\times\mathbb{P}^{n}}\rightarrow 0$

Let $t\in U$ be a general element. Denote the fibers of $h^{*}\mathcal{Y}$ and
$\mathcal{X}$ over $t$ by $Y_{t}$ and $X_{t}$ respectively, and the
restriction of $h$ by $h_{t}:Y_{t}\rightarrow X_{t}.$ Let us also define the
normal bundles $N_{h/\mathcal{X}}$ and $N_{h_{t}/X_{t}}$ by the following
short exact sequences:

$0\rightarrow T_{\mathcal{Y}}\rightarrow h^{*}T_{\mathcal{X}}\rightarrow
N_{h/\mathcal{X}}\rightarrow 0$ $0\rightarrow T_{Y_{t}}\rightarrow
h_{t}^{*}T_{X_{t}}\rightarrow N_{h_{t}/X_{t}}\rightarrow 0$

Taking degrees in the second short exact sequence above, we have

(3) $2g(Y_{t})-2-K_{X_{t}}\cdot h_{t}(Y_{t})=\deg N_{h_{t}/X_{t}}.$

Hence, one way to confirm algebraic hyperbolicity is to find a suitable lower
bound for $\deg N_{h_{t}/X_{t}}$.

We denote by $M_{\mathcal{E}}$ the _Lazarsfeld-Mukai bundle associated to
$\mathcal{E}$_, which is defined by the short exact sequence

$0\rightarrow M_{\mathcal{E}}\rightarrow
H^{0}(\mathbb{P}^{m}\times\mathbb{P}^{n},\mathcal{E})\otimes\mathcal{O}_{\mathbb{P}^{m}\times\mathbb{P}^{n}}\xrightarrow{\operatorname{ev}}\mathcal{E}\rightarrow
0.$

We can define similarly the Lazarsfeld-Mukai bundles $M_{H_{1}}$ and
$M_{H_{2}}$ associated to $H_{1}$ and $H_{2}$ respectively. These bundles play
a central role in the proof techniques in §3.

The following result follows immediately from Proposition 2.1 in [CR19b] and
summarizes the relationship between the vertical tangent sheaves, the normal
bundles, and the Lazarsfeld-Mukai bundle $M_{\mathcal{E}}$.

######  Proposition 2.1 (cf Proposition 2.1 in [CR19b]).

  1. (1)

$N_{h/\mathcal{X}}|_{Y_{t}}\simeq N_{h_{t}/X_{t}}$.

  2. (2)

The quotient
$\left(h^{*}T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}\right)\big{/}\left(T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}\right)$
of vertical tangent sheaves is isomorphic to $N_{h/\mathcal{X}}$.

  3. (3)

The vertical tangent sheaf
$T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}$ on $\mathcal{X}$ is
isomorphic to the pullback $p_{2}^{*}M_{\mathcal{E}}$ of the Lazarsfeld-Mukai
bundle.

##  3\. Proof techniques

In order to estimate the positivity of $N_{h_{t}/X_{t}},$ we start with the
notion of a section-dominating collection of line bundles introduced in
[CR19b].

######  Definition 3.1 (cf Definition 2.3 in [CR19b]).

Let $\mathcal{E}$ be a vector bundle on a smooth, complex projective variety
$A$. A collection of non-trivial, globally generated line bundles
$L_{1},\ldots,L_{u}$ is called a _section-dominating collection_ of line
bundles for $\mathcal{E}$ if

  1. (1)

$\mathcal{E}\otimes L_{i}^{\vee}$ is globally generated for every $1\leq i\leq
u$, and

  2. (2)

the map

$\bigoplus^{u}_{i=1}\left(H^{0}(L_{i}\otimes\mathcal{I}_{p})\otimes
H^{0}(\mathcal{E}\otimes L_{i}^{\vee})\right)\rightarrow
H^{0}(\mathcal{E}\otimes\mathcal{I}_{p})$

is surjective at every point $p\in A$.

We can view a section-dominating collection of line bundles for $\mathcal{E}$
as a collection of simpler building blocks for $\mathcal{E}$, in the sense of
the following proposition.

######  Proposition 3.2 (cf Proposition 2.6 in [CR19b]).

Let $\mathcal{E}$ be a globally generated vector bundle and $M_{\mathcal{E}}$
the Lazarsfeld-Mukai bundle associated to $\mathcal{E}.$ Let
$L_{1},\ldots,L_{u}$ be a section-dominating collection of line bundles for
$\mathcal{E}.$ Then for some integers $s_{i}$, there is a surjection

(4) $\oplus_{i=1}^{u}M_{L_{i}}^{\oplus s_{i}}\longrightarrow M_{\mathcal{E}}$

induced by multiplication by some choice of basis elements of
$H^{0}(\mathcal{E}\otimes L_{i}^{\vee})$ for $0\leq i\leq u.$

Example 2.4 in [CR19b] shows that on $A=\mathbb{P}^{m}\times\mathbb{P}^{n}$,
the line bundles $H_{1},H_{2}$ form a section-dominating collection for
$\mathcal{E}=aH_{1}+bH_{2}.$ By Proposition 3.2, this gives a surjection (4)
induced by multiplication by generic polynomials in $H^{0}(\mathcal{E}\otimes
H_{i}^{\vee})$ for $i=1,2$. We can use this surjection to obtain a first pair
of lower bounds $a\geq a_{0}$ and $b\geq b_{0}$ such that a very general
hypersurface of bidegree $(a,b)$ is algebraically hyperbolic.

######  Lemma 3.3.

Let $h_{t}:Y_{t}\rightarrow X_{t}$ be a curve in a very general hypersurface
$X_{t}\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ of bidegree $(a,b)$. Let
$N$ be a vector bundle on the curve. Suppose that there is a surjective map

$\beta:M_{H_{1}}\;^{\oplus t_{1}}\oplus M_{H_{2}}\;^{\oplus
t_{2}}\longrightarrow N$

for some $t_{1},t_{2}$. Then,

(5) $\deg N\geq-t_{1}\cdot({Y_{t}}\cdot H_{1})-t_{2}\cdot({Y_{t}}\cdot
H_{2}).$

In particular, setting $N=N_{h_{t}/X_{t}},$ we obtain

(6) $2g(Y_{t})-2\geq(a-(m+1+t_{1}))\cdot({Y_{t}}\cdot
H_{1})+(b-(n+1+t_{2}))\cdot({Y_{t}}\cdot H_{2}).$

###### Proof.

Since $\beta$ is surjective, we have

$\begin{split}\deg N&\geq\deg\beta\left(M_{H_{1}}\;^{\oplus
t_{1}})+\deg\beta(M_{H_{2}}\;^{\oplus t_{2}}\right)\\\ &\geq t_{1}\cdot(-\deg
H_{1}|_{Y_{t}})+t_{2}\cdot(-\deg H_{2}|_{Y_{t}})\end{split}$

where the second inequality follows from Proposition 2.7 in [CR19b]. If
$N=N_{h_{t}/X_{t}}$, then by (3), we obtain

$\begin{split}&2g(Y_{t})-2\\\ &=(K_{X_{t}}\cdot Y_{t})+\deg N_{h_{t}/X_{t}}\\\
&\geq(a-(m+1+t_{1}))\cdot(Y_{t}\cdot H_{1})+(b-(n+1+t_{2}))\cdot(Y_{t}\cdot
H_{2})\end{split}$

∎

######  Proposition 3.4.

Suppose $a\geq 2m+n$ and $b\geq m+2n$, then a very general hypersurface
$X\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ of bidegree $(a,b)$ is
algebraically hyperbolic.

###### Proof.

Suppose $X$ is parametrized by $t\in U$ and let $h_{t}:Y_{t}\rightarrow X_{t}$
be a curve in $X$. By Propositions 2.1 and 3.2, we have a surjection

(7) $\alpha:M_{H_{1}}\;^{\oplus s_{1}}\oplus M_{H_{2}}\;^{\oplus
s_{2}}\longrightarrow M_{\mathcal{E}}\longrightarrow N_{h_{t}/X_{t}}$

for some integers $s_{1},s_{2}.$ Since rank $N_{h_{t}/X_{t}}=m+n-2,$ we can
take $s_{1}=s_{2}=m+n-2$.

Therefore, by (6), we have

$\begin{split}&2g(Y_{t})-2\\\ &\geq((a-m-1)-(m+n-2))\cdot(Y_{t}\cdot
H_{1})+((b-n-1)-(m+n-2))\cdot(Y_{t}\cdot H_{2})\\\
&=(a-(2m+n-1))\cdot(Y_{t}\cdot H_{1})+(b-(m+2n-1))\cdot(Y_{t}\cdot
H_{2})\end{split}$

which is at least $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$ when $a\geq 2m+n$
and $b\geq m+2n$. ∎

######  Remark 3.5.

The proof of Proposition 3.4 is essentially the approach taken by Ein in
[Ei88, Ei91] to prove that if $X\subseteq\mathbb{P}^{n}$ is a generic complete
intersection of type $(d_{1},\ldots,d_{k})$ satisfying $\sum_{i}d_{i}\geq
2n-k+1$, then the desingularization of every subvariety of $X$ is of general
type. In particular, such varieties $X$ are algebraically hyperbolic. The
above proof is also in the same spirit as Theorem 3.6 in [HI19], which was
proved using focal loci techniques.

The lower bounds in Proposition 3.4 are within 1 or 2 of the lower bounds
stated in Theorem 1.1(a). In the rest of this section, we present two
techniques that improve the bounds in Proposition 3.4.

In the proof of Proposition 3.4, one sees that it is possible for the
surjection (7) to be achieved with smaller $s_{1},s_{2}$, which would give us
better control of the positivity of $N_{h_{t}/X_{t}}$. This motivates the
following definition.

######  Definition 3.6.

We say that a curve $h_{t}:Y_{t}\rightarrow X_{t}$ is _of type
$(s_{1},s_{2})$_ if the induced map

$\alpha:M_{H_{1}}^{\oplus s_{1}}\oplus M_{H_{2}}^{\oplus s_{2}}\longrightarrow
N_{h_{t}/X_{t}}$

is surjective, with no summand having torsion image. In particular,
$s_{1}+s_{2}\leq m+n-2$.

Note that a curve may be of several different types, since the integers
$s_{1},s_{2}$ as defined above may not be unique.

######  Remark 3.7.

(cf [Cl03, ClR04]) Due to the genericity of the polynomials inducing the map
$\alpha$, for a fixed $0\leq u_{2}\leq s_{2}$, the rank $\gamma(i)$ of
$\alpha\left(M_{H_{1}}^{\oplus i}\oplus M_{H_{2}}^{\oplus u_{2}}\right)$ is a
strictly increasing function for $0\leq i\leq s_{1}$, while the increase
$\gamma(i+1)-\gamma(i)$ in rank is a non-decreasing function of $i$ for $0\leq
i\leq s_{1}-1.$ The analogous statement with a fixed $0\leq u_{1}\leq s_{1}$
and varying $0\leq j\leq s_{2}$ is true as well.

Now, we give an outline of the next two techniques. When $s_{1}+s_{2}$ is
large, i.e. $s_{1}+s_{2}>(m+n-2)/2$, there is a summand $M_{H_{i}}$ whose
image in a quotient of $N_{h_{t}/X_{t}}$ induced by $\alpha$ has rank one. In
this case, §3.1 shows that the curve $h_{t}:Y_{t}\rightarrow X_{t}$ lies in a
special surface in $\mathbb{P}^{m}\times\mathbb{P}^{n}$ made of “lines” (see
§3.1 for definition) passing through each point of the curve. This surface is
described in Lemma 3.8. This method produces an improvement in the estimate
for the positivity of $N_{h_{t}/X_{t}}$, which then improves the lower bounds
for $a$ and $b$ in Proposition 3.4 by 1.

When $s_{1}$ (or $s_{2}$) achieves the maximum, i.e. $s_{1}$ (or $s_{2}$)
equals $m+n-2$, $N_{h_{t}/X_{t}}$ modulo torsion is a direct sum of $m+n-2$
rank-one images of $M_{H_{1}}$ (or $M_{H_{2}}$) via $\alpha$. In this case,
§3.2 shows that the curve lies in a special locus on the hypersurface, which
turns out to be a curve of general type when $a=2m+n-2$ (or $b=m+2n-2$). This
method improves the lower bounds for $a$ and $b$ in Proposition 3.4 by 2 when
$m+n\geq 5$. When $(m,n)=(3,1)$, this method improves the lower bound for $b$
in Proposition 3.4 by 2.

###  3.1. Scroll considerations

Let us define a _$\mathbb{P}^{m}$ -line_ (resp. _$\mathbb{P}^{n}$ -line_) in
$\mathbb{P}^{m}\times\mathbb{P}^{n}$ to mean an integral curve $L$ whose
numerical class is $H_{1}^{m-1}H_{2}^{n}$ (resp. $H_{1}^{m}H_{2}^{n-1}$), and
a _line_ may refer to either a $\mathbb{P}^{m}$-line or a
$\mathbb{P}^{n}$-line. A surface in $\mathbb{P}^{m}\times\mathbb{P}^{n}$ is
called a _$\mathbb{P}^{m}$ -scroll_ (resp. _$\mathbb{P}^{n}$ -scroll_) if
there is a $\mathbb{P}^{m}$-line (resp. $\mathbb{P}^{n}$-line) through every
point of the surface.

######  Lemma 3.8 (cf Lemma 2.12 in [CR19b]).

A rank-one quotient $Q$ of $M_{H_{1}}|_{Y_{t}}$ induces a
$\mathbb{P}^{m}$-scroll containing $Y_{t}$ of
$\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1)$-degree equal to $\deg
Q+(Y_{t}\cdot H_{1}).$

###### Proof.

Recall the Euler sequence on $\mathbb{P}^{m}$:

$0\rightarrow\Omega_{\mathbb{P}^{m}}(1)\rightarrow\mathcal{O}^{m+1}_{\mathbb{P}^{m}}\rightarrow\mathcal{O}_{\mathbb{P}^{m}}(1)\rightarrow
0.$

The pullback of $\Omega_{\mathbb{P}^{m}}(1)$ via $\pi_{1}$ is isomorphic to
the Lazarsfeld-Mukai bundle $M_{H_{1}}$. By pulling back the above short exact
sequence via $\pi_{1}$ and then restricting to the curve $Y_{t}\subseteq
X_{t}$, we get

$0\rightarrow M_{H_{1}}|_{Y_{t}}\rightarrow
H^{0}(\mathbb{P}^{m}\times\mathbb{P}^{n},\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1))\otimes\mathcal{O}_{Y_{t}}\rightarrow\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1)|_{Y_{t}}\rightarrow
0.$

We define $S$ and $Q^{\prime}$ to be the sheaves that make this commutative
diagram exact:

${0}$${0}$${S}$${0}$${0}$${M_{H_{1}}|_{Y_{t}}}$${\mathcal{O}_{Y_{t}}^{m+1}}$${\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1)|_{Y_{t}}}$${0}$${Q}$${Q^{\prime}}$${0}$${0}$$\scriptstyle{(*)}$

The triangle ($*$) induces a map $\gamma:Y_{t}\rightarrow\mathbb{G}(1,m)$ that
sends a point $p\in Y_{t}$ to a line in $\mathbb{P}^{m}$ given by the rank-two
vector bundle $Q^{\prime}$, which contains $\pi_{1}(p)$. $\gamma$ lifts to a
map that sends $p\in Y_{t}$ to the $\mathbb{P}^{m}$-line
$\pi_{1}^{-1}(\gamma(p))\cap\pi_{2}^{-1}(\pi_{2}(p))$ which contains $p$,
giving a $\mathbb{P}^{m}$-scroll containing $Y_{t}$. The
$\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1)$-degree of this surface follows
from taking degrees in the above diagram. ∎

The following is the analogous result with $H_{1}$ and $H_{2}$ switched.

######  Lemma 3.9.

A rank-one quotient $Q$ of $M_{H_{2}}|_{Y_{t}}$ induces a
$\mathbb{P}^{n}$-scroll containing $Y_{t}$ of
$\pi_{2}^{*}\mathcal{O}_{\mathbb{P}^{n}}(1)$-degree equal to $\deg
Q+(Y_{t}\cdot H_{2}).$

To give an idea of their application to the proof of Theorem 1.1, suppose that
$h_{t}:Y_{t}\rightarrow X_{t}$ is a curve of type $(s_{1},s_{2})=(m+n-2,0)$.
Then the map

$M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}\Big{/}\alpha(M_{H_{1}}^{\oplus m+n-3})$

induced by (7) has a rank-one image $Q$, which by Lemma 3.8 produces a
$\mathbb{P}^{m}$-scroll containing the curve whose intersection number with
the class $H_{1}^{2}$ equals $\deg Q+(Y_{t}\cdot H_{1})$. By comparing the
numerical classes of the curve, the hypersurface and the
$\mathbb{P}^{m}$-scroll, we obtain a lower bound for $\deg Q$, which in turn
gives the following lower bound for $\deg N_{h_{t}/X_{t}}$ by (5):

$\deg N_{h_{t}/X_{t}}\geq\deg Q+\deg\alpha(M_{H_{1}}^{\oplus m+n-3})\geq\deg
Q-(m+n-3)\cdot(Y_{t}\cdot H_{1}).$

######  Remark 3.10.

The above scroll considerations were first introduced in [CR19a] to show that
a curve $C$ in a very general degree $d$ surface $X\subseteq\mathbb{P}^{3}$
satisfies the inequality

$2g(C)-2\geq\left(d-5+\frac{1}{d}\right)\cdot(C\cdot H),$

hence $X$ is algebraically hyperbolic if $d\geq 5$. Similarly, when applied to
curves $C$ contained in very general degree $d$ hypersurfaces
$X\subseteq\mathbb{P}^{n}$ of higher dimensions, the scroll method yields the
inequality

$2g(C)-2\geq\left(d-(2n-1)+\frac{1}{d}\right)\cdot(C\cdot H).$

Hence, it gives an alternate proof (see Theorem 3.21) that $X$ is
algebraically hyperbolic if $d\geq 2n-1$ and $n\geq 4$, which is a result
first obtained in [Vo96, Vo98]. Prior to [CR19a], Clemens-Ran used a similar
scroll construction in §4 of [ClR04] to discount the existence of curves of
genus $\leq 2$ in very general sextic 3-folds $X\subseteq\mathbb{P}^{4}$, in
the case where there is a surjection from $M_{H}$ onto the normal bundle of
the curve in $X$.

###  3.2. Osculating lines

A $\mathbb{P}^{m}$-line (resp. $\mathbb{P}^{n}$-line) in
$\mathbb{P}^{m}\times\mathbb{P}^{n}$ is called an _osculating
$\mathbb{P}^{m}$-line_ (resp. _osculating $\mathbb{P}^{n}$-line_) of $X_{t}$
if it intersects $X_{t}$ in exactly one point. We denote by
$\Lambda^{\mathbb{P}^{m}}_{a}|_{t}$ (resp.
$\Lambda^{\mathbb{P}^{n}}_{b}|_{t}$) the locus on $X_{t}$ swept out by
osculating $\mathbb{P}^{m}$-lines (resp. osculating $\mathbb{P}^{n}$-lines).
First, we show the following by using modifications of Lemmas 2.10–11 in
[CR19b]:

  1. (1)

When $a=2m+n-2$ and $s_{1}=m+n-2$, the curve $h_{t}:Y_{t}\rightarrow X_{t}$
lies in $\Lambda^{\mathbb{P}^{m}}_{a}|_{t}$.

  2. (2)

When $b=m+2n-2$ and $s_{2}=m+n-2$, the curve $h_{t}:Y_{t}\rightarrow X_{t}$
lies in $\Lambda^{\mathbb{P}^{n}}_{b}|_{t}$.

Then, we show that $\Lambda^{\mathbb{P}^{m}}_{a}|_{t}$ and
$\Lambda^{\mathbb{P}^{n}}_{b}|_{t}$ are curves of general type when $a,b$ are
large enough. Note that the hypothesis of the following lemma is equivalent to
the condition that the curve $h_{t}:Y_{t}\rightarrow X_{t}$ is of type
$(s_{1},s_{2})=(m+n-2,0)$ and $m+n\geq 4.$

######  Lemma 3.11 (cf Lemma 2.10 in [CR19b]).

Suppose that at a general point $(p,t)\in\mathcal{Y}$,

  1. (1)

$M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}$ has rank-one image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{1}^{\vee})$ and is not surjective,
and

  2. (2)

$M_{H_{2}}\rightarrow N_{h_{t}/X_{t}}$ has torsion image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{2}^{\vee}).$

Then there exists a $\mathbb{P}^{m}$-line
$p\in\ell\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ whose ideal
$H^{0}(\mathcal{E}\otimes\mathcal{I}_{\ell})$ is contained in
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$. Moreover, there
is only one such $\mathbb{P}^{m}$-line $\ell$, and it is the line induced by
the generic rank-one image $Q$ of $M_{H_{1}}$ in $N_{h_{t}/X_{t}}$, as in
Lemma 3.8.

###### Proof.

Consider the map $M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}$ induced by a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{1}^{\vee})$ at the point $p\in
Y_{t}$. Since the map is not surjective, its kernel is independent of the
generic polynomial by Lemma 2.2(i) in [Cl03]. We denote its kernel by
$S\subseteq H^{0}(H_{1}\otimes\mathcal{I}_{p})$ and its image by $Q\subseteq
N_{h_{t}/X_{t}}|_{p}$. By Proposition 2.1, $N_{h_{t}/X_{t}}|_{p}$ is
isomorphic to the quotient
$M_{\mathcal{E}}|_{p}\;\Big{/}\;T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}.$
Hence by the above assumptions,
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$ contains the
image of the map

$\left(S\otimes H^{0}(\mathcal{E}\otimes
H_{1}^{\vee})\right)\oplus\left(H^{0}(H_{2}\otimes\mathcal{I}_{p})\otimes
H^{0}(\mathcal{E}\otimes H_{2}^{\vee})\right)\rightarrow
H^{0}(\mathcal{E}\otimes\mathcal{I}_{p})\simeq M_{\mathcal{E}}|_{p},$

which is the ideal of the $\mathbb{P}^{m}$-line containing $p$ that is given
by Lemma 3.8.

Uniqueness follows from a dimension count. We can view
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}\subseteq
T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$ as spaces of
polynomials. If there are two distinct $\mathbb{P}^{m}$-lines whose ideals are
contained in $T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$,
then we would have
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}=T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$,
a contradiction. ∎

The following is the analogous result with $H_{1}$ and $H_{2}$ switched.

######  Lemma 3.12.

Suppose that at a general point $(p,t)\in\mathcal{Y}$,

  1. (1)

$M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}$ has torsion image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{1}^{\vee}),$ and

  2. (2)

$M_{H_{2}}\rightarrow N_{h_{t}/X_{t}}$ has rank-one image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{2}^{\vee})$ and is not surjective.

Then there exists a $\mathbb{P}^{n}$-line
$p\in\ell\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ whose ideal
$H^{0}(\mathcal{E}\otimes\mathcal{I}_{\ell})$ is contained in
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$. Moreover, there
is only one such $\mathbb{P}^{n}$-line $\ell$, and it is the line induced by
the generic rank-one image $Q$ of $M_{H_{2}}$ in $N_{h_{t}/X_{t}}$, as in
Lemma 3.9.

The following result is a generalization of Lemma 2.11 in [CR19b].

######  Lemma 3.13 (cf Lemma 2.11 in [CR19b]).

Let $(p,t)\in\mathcal{Y}$ be a general point and let $Z:=p_{2}^{-1}(p).$ Let
$T\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ be a subvariety containing $p$
whose ideal $H^{0}(\mathcal{E}\otimes\mathcal{I}_{T})$ is contained in
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}.$ Then
$W:=h(\mathcal{Y})\cap Z$ is a union of fibers of the restriction map
$\beta:Z\rightarrow H^{0}(\mathcal{E}|_{T}).$ In particular, $W$ contains the
fiber containing $(p,t),$ which is the affine space
$(p,t)+H^{0}(\mathcal{E}\otimes\mathcal{I}_{T}).$

###### Proof.

The tangent space to a fiber of the above restriction map $\beta$ is
$H^{0}(\mathcal{E}\otimes\mathcal{I}_{T})$ at every point of the fiber, so the
fiber containing $(p,t)$ is the affine space
$(p,t)+H^{0}(\mathcal{E}\otimes\mathcal{I}_{T})$. Since
$H^{0}(\mathcal{E}\otimes\mathcal{I}_{T})\subseteq
T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)},$ Lemma 2.11 in
[CR19b] implies that $W$ is a union of fibers of the restriction map
$\beta:Z\rightarrow H^{0}(\mathcal{E}|_{T}).$ Therefore, $W$ contains the
fiber containing $(p,t),$ which is
$(p,t)+H^{0}(\mathcal{E}\otimes\mathcal{I}_{T})$. ∎

Now, we will use Lemmas 3.11 and 3.13 to prove that if $a=2m+n-2$, then a
curve of type $(s_{1},s_{2})=(m+n-2,0)$ lies in the osculation locus
$\Lambda^{\mathbb{P}^{m}}_{a}|_{t}\subseteq X_{t}$.

######  Lemma 3.14 (cf §2 in [Vo98], §3.3 in [Pa03]).

Suppose that $a=2m+n-2$, and that at a general point $(p,t)\in\mathcal{Y}$,

  1. (1)

$M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}$ has rank-one image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{1}^{\vee})$ and is not surjective,
and

  2. (2)

$M_{H_{2}}\rightarrow N_{h_{t}/X_{t}}$ has torsion image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{2}^{\vee}).$

Then the curve $h_{t}:Y_{t}\rightarrow X_{t}$ lies in the locus
$\Lambda^{\mathbb{P}^{m}}_{a}|_{t}\subseteq X_{t}$.

###### Proof.

Let $\ell$ denote the $\mathbb{P}^{m}$-line associated to $(p,t)$ obtained
from Lemma 3.11. We can assume without loss of generality that

$p=\\{X_{1}=\ldots=X_{m}=Y_{1}=\ldots=Y_{n}=0\\},$

and

$\ell=\\{X_{2}=\ldots=X_{m}=Y_{1}=\ldots=Y_{n}=0\\}.$

Consider the restriction map

$\beta:T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}\rightarrow
H^{0}(\mathcal{O}_{\ell}(a)(-p)),$

whose kernel is $H^{0}(\mathcal{E}\otimes\mathcal{I}_{\ell})$. Denote by $F$
the polynomial corresponding to $t$.

Since $\mathcal{Y}$ is $G$-invariant,
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$ contains the
elements of $T_{(\mathbb{P}^{m}\times\mathbb{P}^{n})\times V}|_{(p,t)}$ that
are tangent to the $G$-orbit of $(p,t)$ and project to 0 in
$T_{\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{p}$. In particular,
$T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}$ contains the
elements

$F,\;X_{1}\dfrac{\partial F}{\partial X_{0}},\;\ldots\;,X_{1}\dfrac{\partial
F}{\partial X_{m}}.$

By Lemma 3.13, we may assume that $F$ is generic in the space
$(p,t)+H^{0}(\mathcal{E}\otimes\mathcal{I}_{\ell}),$ so we can assume that the
$m-1$ elements

$X_{1}\frac{\partial F}{\partial X_{2}}|_{\ell},\ldots,X_{1}\frac{\partial
F}{\partial X_{m}}|_{\ell}$

in Im $\beta$ are independent modulo the subspace

$K:=\langle\;F|_{\ell},X_{1}\frac{\partial F}{\partial
X_{0}}|_{\ell},X_{1}\frac{\partial F}{\partial
X_{1}}|_{\ell}\;\rangle\subseteq H^{0}(\mathcal{O}_{\ell}(a)(-p)).$

By a dimension count, we have

$\begin{split}\text{dim Im
}\beta&=\dim\left(T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}\right)-\dim
H^{0}(\mathcal{E}\otimes\mathcal{I}_{\ell})\\\
&=\dim\left(T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}\right)-\dim
H^{0}(\mathcal{E}\otimes\mathcal{I}_{\ell})-\dim\left(h^{*}T_{\mathcal{X}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}\Big{/}T_{\mathcal{Y}/\mathbb{P}^{m}\times\mathbb{P}^{n}}|_{(p,t)}\right)\\\
&=a-(m+n-2)\\\ &=m\end{split}$

when $a=2m+n-2.$ Hence, $\dim K\leq\dim\text{Im }\beta-(m-1)=m-(m-1)=1$, which
implies that $F|_{\ell}=a\cdot X_{1}^{a}Y_{0}^{b}.$ Therefore, the curve
$h_{t}:Y_{t}\rightarrow X_{t}$ lies in the locus
$\Lambda^{\mathbb{P}^{m}}_{a}|_{t}\subseteq X_{t}.$ ∎

The following is the analogous result when $b=m+2n-2$ and the curve is of type
$(s_{1},s_{2})=(0,m+n-2)$ instead.

######  Lemma 3.15.

Suppose that $b=m+2n-2$, and that at a general point $(p,t)\in\mathcal{Y}$,

  1. (1)

$M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}$ has torsion image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{1}^{\vee}),$ and

  2. (2)

$M_{H_{2}}\rightarrow N_{h_{t}/X_{t}}$ has rank-one image for a generic
polynomial in $H^{0}(\mathcal{E}\otimes H_{2}^{\vee})$ and is not surjective.

Then this curve lies in the locus $\Lambda^{\mathbb{P}^{n}}_{b}|_{t}\subseteq
X_{t}$.

######  Remark 3.16.

When $n=1$, Lemma 3.15 follows directly from Lemmas 2.10 and 2.11 in [CR19b]
by the argument used there toward classifying very general surfaces in
$\mathbb{P}^{2}\times\mathbb{P}^{1}$. (See [CR19b], Proof of Theorem 3.4, Case
4.) This argument goes as follows. By Lemmas 3.12 and 3.13 above, $W$ is a
union of fibers of the restriction map $\beta:Z\rightarrow
H^{0}(\mathcal{E}|_{\ell})$ where $\ell$ is the $\mathbb{P}^{n}$-line
$\pi_{1}^{-1}\pi_{1}(p)$, so $X_{t}\cap\ell$ is a subscheme $B$ of $b$ (not
necessarily distinct) points on $\mathbb{P}^{1}$. If $B$ contained $\geq 2$
distinct points, then its $GL(2)$-orbit would have codimension $\leq b-2$,
which equals $m-2$ when $b=m+2n-2=m$. Since $\mathcal{Y}$ is $G$-invariant,
$W$ contains the fibers of $\beta$ over $B$, so we would have
$\operatorname{codim}(\mathcal{Y}\subseteq\mathcal{X})\leq m-2.$ However, this
is a contradiction since
$\operatorname{codim}(\mathcal{Y}\subseteq\mathcal{X})=m+n-2=m-1$ by (2).
Hence, $B$ has to be one unique point with multiplicity $b$.

For the rest of this section, we investigate the geometry of the osculation
loci $\Lambda^{\mathbb{P}^{m}}_{a}|_{t}$ and
$\Lambda^{\mathbb{P}^{n}}_{b}|_{t}$. Let us define the incidence variety

$\Delta^{\mathbb{P}^{m}}_{a}=\\{(L,p,t)\mid\pi_{1}^{-1}(L)\cap\pi_{2}^{-1}(\pi_{2}(p))\cap
X_{t}=a\cdot[p]\\}\subseteq\mathbb{G}(1,m)\times(\mathbb{P}^{m}\times\mathbb{P}^{n})\times
V,$

and denote its fiber over $t$ by $\Delta^{\mathbb{P}^{m}}_{a}|_{t}.$

######  Lemma 3.17 (cf Theorem 5.2 in [CR20]).

For a very general $t\in U$, $\Delta^{\mathbb{P}^{m}}_{a}|_{t}$ is a smooth,
irreducible variety of dimension $2m-1+n-a$. Moreover, it is of general type
when $a\geq\max\\{\sqrt{2m}+1,3\\}$ and $b\geq n+2$.

###### Proof.

Since $\Delta^{\mathbb{P}^{m}}_{a}$ is a vector bundle over the space of pairs
$(L,p)\in\mathbb{G}(1,m)\times(\mathbb{P}^{m}\times\mathbb{P}^{n})$ such that
$\pi_{1}(p)\in L$, it is a smooth and irreducible variety.

Now we prove that the canonical bundle $K_{\Delta^{\mathbb{P}^{m}}_{a}}$ is
very ample when $a,b$ are large enough. For convenience, let us denote
$\mathbb{G}:=\mathbb{G}(1,m)\times(\mathbb{P}^{m}\times\mathbb{P}^{n})\times
V$, and let us denote the various projection maps by
$\rho_{1}:\mathbb{G}\rightarrow\mathbb{G}(1,m)$,
$\rho_{2}:\mathbb{G}\rightarrow\mathbb{P}^{m}\times\mathbb{P}^{n}$,
$\rho_{2,m}:\mathbb{G}\rightarrow\mathbb{P}^{m}$,
$\rho_{2,n}:\mathbb{G}\rightarrow\mathbb{P}^{n}$, and
$\rho_{3}:\mathbb{G}\rightarrow V$. We will construct
$\Delta^{\mathbb{P}^{m}}_{a}$ as the vanishing locus of some vector bundle on
$\mathbb{G}$. Let $\sigma,H_{1},H_{2}$ be the pullback of the hyperplane
divisors via $\rho_{1},\rho_{2,m},\rho_{2,n}$ respectively. Then the canonical
bundle of $\mathbb{G}$ is of class

$K_{\mathbb{G}}=(-m-1)\sigma+(-m-1)H_{1}+(-n-1)H_{2}=:(-m-1,-m-1,-n-1).$

First consider the natural map

$e:\rho_{3}^{*}\mathcal{O}_{V}(-1)\rightarrow\rho_{2}^{*}\mathcal{E}$

of vector bundles on $\mathbb{G}$. Its zero scheme is the locus of points
$(L,p,t)$ with $p\in X_{t}.$ Then consider the natural map

$f:\rho_{2,m}^{*}\mathcal{O}_{\mathbb{P}^{m}}(-1)\hookrightarrow\rho_{1}^{*}\mathcal{O}^{m+1}_{\mathbb{G}(1,m)}\twoheadrightarrow\rho_{1}^{*}Q,$

where the surjection comes from the following tautological sequence on
$\mathbb{G}(1,m)$

$0\rightarrow S\rightarrow\mathcal{O}^{m+1}_{\mathbb{G}(1,m)}\rightarrow
Q\rightarrow 0.$

The zero scheme of $f$ is the locus of points $(L,p,t)$ with $\pi_{1}(p)\in
L.$ Since $\Delta^{\mathbb{P}^{m}}_{1}$ is the common zero scheme of $e,f$, we
can compute the class of its canonical bundle as follows:

(8)
$\begin{split}K_{\Delta^{\mathbb{P}^{m}}_{1}}&=K_{\mathbb{G}}+c_{1}(\rho_{2}^{*}\mathcal{E})+c_{1}(\rho_{1}^{*}Q\otimes\rho_{2,m}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1))\\\
&=K_{\mathbb{G}}+c_{1}(\rho_{2}^{*}\mathcal{E})+c_{1}(\rho_{1}^{*}Q)+(m-1)\cdot
c_{1}(\rho_{2,m}^{*}\mathcal{O}_{\mathbb{P}^{m}}(1)))\\\
&=(-m-1,-m-1,-n-1)+(0,a,b)+(1,0,0)+(0,m-1,0)\\\ &=(-m,a-2,b-n-1).\end{split}$

Note that on $\Delta^{\mathbb{P}^{m}}_{1}$, the natural map

$\rho_{2,m}^{*}\mathcal{O}_{\mathbb{P}^{m}}(-1)\hookrightarrow\rho_{1}^{*}\mathcal{O}^{m+1}_{\mathbb{G}(1,m)}$

from above factors as

${\rho_{2,m}^{*}\mathcal{O}_{\mathbb{P}^{m}}(-1)}$${\rho_{1}^{*}\mathcal{O}^{m+1}_{\mathbb{G}(1,m)}}$${\rho_{1}^{*}S}$$\scriptstyle{\mu}$

.

Let $R^{\vee}$ be the cokernel of $\mu$. We have the short exact sequence

$0\rightarrow\rho_{2,m}^{*}\mathcal{O}_{\mathbb{P}^{m}}(-1)\xrightarrow{\mu}\rho_{1}^{*}S\rightarrow
R^{\vee}\rightarrow 0,$

on $\Delta^{\mathbb{P}^{m}}_{1}$, which induces a filtration $F^{\bullet}$ on
$\rho_{1}^{*}Sym^{a}S^{\vee}$ with

$F^{i}/F^{i+1}=\rho_{2,m}^{*}\mathcal{O}_{\mathbb{P}^{m}}(a-i)\otimes R^{i}.$

Note that

(9) $c_{1}(F^{i}/F^{i+1})=(0,a-i,0)+i\cdot(1,-1,0)=(i,a-2i,0).$

Since $\Delta^{\mathbb{P}^{m}}_{a}$ is the zero scheme of the (well-defined)
natural map

$\mathcal{O}_{V}(-1)\rightarrow F^{1}/F^{a},$

we can compute its canonical bundle using adjunction again as follows:

$\displaystyle K_{\Delta^{\mathbb{P}^{m}}_{a}}$
$\displaystyle=K_{\Delta^{\mathbb{P}^{m}}_{1}}+c_{1}(F^{1}/F^{a})$
$\displaystyle=K_{\Delta^{\mathbb{P}^{m}}_{1}}+\sum_{i=1}^{a-1}c_{1}(F^{i}/F^{i+1})$
$\displaystyle=(-m,a-2,b-n-1)+\sum_{i=1}^{a-1}(i,a-2i,0)\qquad\text{by
(\ref{eqn-delta1}) and (\ref{eqn-filtration})}$
$\displaystyle=\left(-m+\frac{a(a-1)}{2},a-2,b-n-1\right).$

Hence $K_{\Delta^{\mathbb{P}^{m}}_{a}}$ is very ample when
$a\geq\max\\{\sqrt{2m}+1,3\\}$ and $b\geq n+2$.

Finally we verify the dimension of $\Delta^{\mathbb{P}^{m}}_{a}|_{t}.$ Since a
very general fiber of the projection map $(L,p,t)\mapsto(L,p)$ has codimension
$a$ in $V$, we have

$\dim\Delta^{\mathbb{P}^{m}}_{a}=\dim\mathbb{G}(1,m)+n+1+(\dim
V-a)=2m-1+n-a+\dim V,$

and so $\dim\Delta^{\mathbb{P}^{m}}_{a}|_{t}=2m-1+n-a$. ∎

######  Lemma 3.18 (cf Proposition 5.3 in [CR20]).

For a very general $t\in U,$ the projection map
$\Delta^{\mathbb{P}^{m}}_{a}|_{t}\rightarrow\Lambda^{\mathbb{P}^{m}}_{a}|_{t}$
is an isomorphism when $a\geq 2m+n-2.$

###### Proof.

Let us define

$\mathcal{D}=\\{(p,t)\mid p\in X_{t}\text{ and }\exists L_{1}\neq
L_{2}\in\mathbb{G}(1,m)\text{ s.t. }L_{1}\cap
L_{2}=\pi_{1}(p)\\}\subseteq\mathcal{X}_{V}.$

From the proof of Lemma 4.1, $X_{t}$ does not contain $\mathbb{P}^{m}$-lines
when $a\geq 2m+n-2$, so it is enough to prove that codim
$(\mathcal{D}\subseteq\mathcal{X}_{V})\geq m+n.$

Define the incidence variety

$I=\\{(p,t,L_{1},L_{2})\mid p\in X_{t},L_{1}\neq L_{2}\text{ and }L_{1}\cap
L_{2}=\pi_{1}(p)\\}.$

The space of tuples $(p,L_{1},L_{2})$ such that $L_{1}\cap L_{2}=\pi_{1}(p)$
has dimension equal to

$2(m-1)+(1+n)+(m-1)=3m+n-2.$

Since a fiber of $I$ over such a tuple has dimension $\dim V-2a$, we have

$\dim I=\dim V+3m+n-2-2a,$

so $\mathcal{D}$ being the image of $I$ must have dimension $\leq\dim
V+3m+n-2-2a$. Therefore,

$\begin{split}\text{codim
}(\mathcal{D}\subseteq\mathcal{X}_{V})&\geq\dim\mathcal{X}_{V}-(N+3m+n-2-2a)\\\
&=(\dim V+m+n-1)-(\dim V+3m+n-2-2a)\\\ &=2a-2m+1,\end{split}$

which is $\geq m+n$ when $a\geq\frac{3m+n-1}{2}.$ ∎

The following are the analogous results to Lemmas 3.17 and 3.18 for
$\Delta^{\mathbb{P}^{n}}_{b}|_{t}$ and $\Lambda^{\mathbb{P}^{n}}_{b}|_{t}.$

######  Lemma 3.19.

For a very general $t\in U$, $\Delta^{\mathbb{P}^{n}}_{b}|_{t}$ is a smooth,
irreducible variety of dimension $m-1+2n-b$. Moreover, it is of general type
when $a\geq m+2$ and $b\geq\max\\{\sqrt{2n}+1,3\\}$.

######  Lemma 3.20.

For a very general $t\in U,$ the projection map
$\Delta^{\mathbb{P}^{n}}_{b}|_{t}\rightarrow\Lambda^{\mathbb{P}^{n}}_{b}|_{t}$
is an isomorphism when $b\geq m+2n-2.$

One can use the analogues of the above tools in the $\mathbb{P}^{n}$ setting
to show that a very general hypersurface $X_{d}\subseteq\mathbb{P}^{n}$ of
degree $d\geq 2n-2$ is algebraically hyperbolic when $n\geq 5$. Our proof is
built upon the proofs of [Pa04] and [ClR04] for the $n\geq 6$ case. The key to
our improved result is the use of Coskun and Riedl’s scroll argument from
[CR19a], which also gives an alternate proof of Voisin’s result on the
algebraic hyperbolicity of very general hypersurfaces of degree $d\geq 2n-1$
when $n\geq 4$ [Vo96, Vo98].

######  Theorem 3.21.

A very general hypersurface $X\subseteq\mathbb{P}^{n}$ of degree $d$ is
algebraically hyperbolic when _(1)_ $n\geq 4$ and $d\geq 2n-1$; or _(2)_
$n\geq 5$ and $d\geq 2n-2$.

###### Proof.

Suppose $X$ is parametrized by $t\in U$ and let $h_{t}:Y_{t}\rightarrow X_{t}$
be a curve in $X.$ Then, there is a surjection

$\alpha:M_{H}\;^{\oplus s}\rightarrow N_{h_{t}/X_{t}}$

for some integer $s$. Take $s$ to be the minimum. Since rank
$N_{h_{t}/X_{t}}=n-2$, we have $s\leq n-2.$

Suppose that $n\geq 4$. If $s\leq n-3,$ then

$\deg
N_{h_{t}/X_{t}}\geq(n-3)\cdot\deg\alpha\left(M_{H}\right)\geq(n-3)\cdot-\deg
H|_{Y_{t}}.$

Therefore,

$\begin{split}2g(Y_{t})-2&=(K_{X_{t}}\cdot Y_{t})+\deg N_{h_{t}/X_{t}}\\\
&\geq((d-n-1)-(n-3))\cdot(Y_{t}\cdot H)\\\ &=(d-(2n-2))\cdot(Y_{t}\cdot
H)\end{split}$

which is at least $(Y_{t}\cdot H)$ when $d\geq 2n-1$. If $s=n-2,$ then the map

$M_{H}\rightarrow N_{h_{t}/X_{t}}/\alpha(M_{H}\;^{\oplus n-3})$

given by multiplication by a generic polynomial in
$H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(d-1))$ has a rank-one image
$Q$, which induces a surface scroll $\Sigma\subseteq\mathbb{P}^{n}$ of degree
$\deg Q+(Y_{t}\cdot H)$. Since the curve $Y_{t}$ is contained in the curve
$\Sigma\cap X_{t}$ whose degree is $d\cdot(\deg Q+(Y_{t}\cdot H)),$ we must
have

$\deg Q\geq\left(\frac{1}{d}-1\right)\cdot(Y_{t}\cdot H).$

Therefore,

$\begin{split}2g(Y_{t})-2&=(K_{X_{t}}\cdot Y_{t})+\deg N_{h_{t}/X_{t}}\\\
&\geq((d-n-1)-(n-3))\cdot(Y_{t}\cdot H)+\deg Q\\\
&\geq(d+\frac{1}{d}-(2n-1))\cdot(Y_{t}\cdot H)\end{split}$

which is at least $\frac{1}{d}(Y_{t}\cdot H)$ when $d\geq 2n-1$. This
concludes the proof for $n\geq 4$ and $d\geq 2n-1$.

Now suppose that $n\geq 5$ and $d=2n-2.$ If $s\leq n-4$, then as in the case
$s\leq n-3$ from above, we have $\deg N_{h_{t}/X_{t}}\geq(n-4)\cdot-\deg
H|_{Y_{t}}$. Therefore,

$2g(Y_{t})-2\geq((d-n-1)-(n-4))\cdot(Y_{t}\cdot H)=(d-(2n-3))\cdot(Y_{t}\cdot
H)=(Y_{t}\cdot H).$

If $s=n-3$, then as in the case $s=n-2$ from above, the map

$M_{H}\rightarrow N_{h_{t}/X_{t}}/\alpha(M_{H}\;^{\oplus n-4})$

has a rank-one image $Q$, since we assume that $n\geq 5$. Then it follows from
the scroll method as applied above that $2g(Y_{t})-2\geq\frac{1}{d}(Y_{t}\cdot
H)$. If $s=n-2$, then the map

$M_{H}\rightarrow N_{h_{t}/X_{t}}$

has rank-one image and is not surjective. By the analogues of Lemmas 3.11,
3.13 and 3.14 for $\mathbb{P}^{n}$, the curve lies in the 1-osculation locus
of $X_{t}$, which is a curve of general type by §4.1 in [Pa03]. ∎

######  Remark 3.22.

Consider a curve in a very general sextic threefold, i.e. when $n=4$ and
$d=2n-2$. If the curve is of type $s=2$, then it satisfies the inequality (1)
by the same argument as the case $n\geq 5$, $d=2n-2$ and $s=n-2$ in Theorem
3.21. If the curve is of type $s=1$, i.e. the map $M_{H}\rightarrow
N_{h_{t}/X_{t}}$ induced by a generic degree 5 polynomial has a rank-two
image, then the scroll method as above does not apply (see also Remark 3.10).

##  4\. Proof of Theorem 1.1

###  4.1. Proof of Theorem 1.1(2).

We verify that very general hypersurfaces of low bidegrees $(a,b)$ contain
lines or elliptic curves. Hence, they are not algebraically hyperbolic.

######  Lemma 4.1.

Suppose that $a\leq 2m+n-3$ or $b\leq m+2n-3$. Then a very general
hypersurface $X\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ of bidegree
$(a,b)$ contains a line. In particular, it is not algebraically hyperbolic.

###### Proof.

View $X$ as a very general $\mathbb{P}^{n}$-family of degree $a$ hypersurfaces
in $\mathbb{P}^{m}$ via the projection map
$\pi_{2}:X\rightarrow\mathbb{P}^{n}$. The space of degree $a$ hypersurfaces in
$\mathbb{P}^{m}$ containing a line has codimension $(a+1)-2(m-1)$ in
$H^{0}(\mathbb{P}^{m},\mathcal{O}_{\mathbb{P}^{m}}(a))$, so it has non-trivial
intersection with $X$ when $a\leq 2m+n-3$. Therefore $X$ contains a
$\mathbb{P}^{m}$-line when $a\leq 2m+n-3$. The same argument shows that $X$
contains a $\mathbb{P}^{n}$-line when $b\leq m+2n-3$. ∎

######  Lemma 4.2.

Let $(m,n)=(2,2)$. Suppose that $a=4$ or $b=4$. Then a very general
hypersurface $X\subseteq\mathbb{P}^{2}\times\mathbb{P}^{2}$ of bidegree
$(a,b)$ contains an elliptic curve. In particular, it is not algebraically
hyperbolic.

###### Proof.

As above, we can view $X$ as a very general net of degree 4 curves in
$\mathbb{P}^{2}$. By the degree-genus formula for plane curves, a general
member of this net has geometric genus 3. We prove that there exists a curve
in this net that has two nodes. Such a curve would be an elliptic curve since
its geometric genus is $3-2=1$. Requiring a quartic plane curve to contain two
particular distinct nodes in $\mathbb{P}^{2}$ imposes $2\times 3=6$ conditions
on the equation of the curve. Since the two points can vary over a $(2\times
2=4)$-dimensional family, the space of quartic plane curves containing any two
distinct nodes has codimension $6-4=2$ in
$H^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(4))$. Hence, this family
intersects the two-dimensional family that sweeps out the very general
hypersurface $X$. ∎

###  4.2. Proof of Theorem 1.1(1)

We divide up the proof by $(m,n)$. Then, for each possible type
$(s_{1},s_{2}),$ we produce a uniform $\varepsilon>0$ such that all curves
$h_{t}:Y_{t}\rightarrow X_{t}$ of type $(s_{1},s_{2})$ (see Definition 3.6)
lying in a very general hypersurface
$X_{t}\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ satisfies

(10) $2g(Y_{t})-2\geq\varepsilon\cdot((Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2}))$

provided that the bidegree $(a,b)$ of $X_{t}$ lies in the stated range. We
assume without loss of generality that $s_{1}\geq s_{2}.$

####  4.2.1. Proof of Theorem 1.1(1)(a).

Let $m+n\geq 5$.

Case 1: Suppose that $s_{1},s_{2}\leq m+n-4$. Then, by (6), we have

$\begin{split}&2g(Y_{t})-2\\\ &\geq((a-m-1)-(m+n-4))\cdot(Y_{t}\cdot
H_{1})+((b-n-1)-(m+n-4))\cdot(Y_{t}\cdot H_{2})\\\
&=(a-(2m+n-3))\cdot(Y_{t}\cdot H_{1})+(b-(m+2n-3))\cdot(Y_{t}\cdot
H_{2}),\end{split}$

which is at least $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$ when $a\geq 2m+n-2$
and $b\geq m+2n-2.$

Case 2: Suppose that $s_{1}=m+n-3$, so $s_{2}\leq 1$. Consider the map

$M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}\Big{/}\alpha(M_{H_{1}}^{\oplus m+n-4})$

induced by $\alpha$. Since $m+n\geq 5$, this map must have a rank-one image
$Q$, so we can apply the scroll considerations of Lemma 3.8. Let
$\Sigma\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ denote the
$\mathbb{P}^{m}$-scroll given by $Q$. The $H_{1}$-degree of $\Sigma$ is
$\Sigma\cdot H_{1}^{2}$, which is equal to $\deg Q+(Y_{t}\cdot H_{1})$ by
Lemma 3.8. Since $\Sigma$ is a $\mathbb{P}^{m}$-scroll, we have $\Sigma\cdot
H_{2}=(Y_{t}\cdot H_{2})\cdot H_{1}^{m-1}H_{2}^{n}.$ Hence, $\Sigma$ is of
numerical class

$(Y_{t}\cdot H_{2})\cdot H_{1}^{m-1}H_{2}^{n-1}+\left(\deg Q+(Y_{t}\cdot
H_{1})\right)\cdot H_{1}^{m-2}H_{2}^{n}.$

$Y_{t}$ is contained in the curve $\Sigma\cap X_{t}$ whose class is

$(a\cdot(H_{2}\cdot Y_{t}))\cdot H_{1}^{m}H_{2}^{n-1}+(b\cdot(H_{2}\cdot
Y_{t})+a\cdot(\deg Q+(H_{1}\cdot Y_{t})))\cdot H_{1}^{m-1}H_{2}^{n}.$

Since $\Sigma\cap X_{t}-Y_{t}$ is effective, we have $H_{1}\cdot(\Sigma\cap
X_{t}-Y_{t})\geq 0$. Hence, we obtain the inequality

$b\cdot(H_{2}\cdot Y_{t})+a\cdot(\deg Q+(H_{1}\cdot Y_{t}))-(H_{1}\cdot
Y_{t})\geq 0,$

which can be rearranged into

$\deg Q\geq\left(\frac{1}{a}-1\right)\cdot(H_{1}\cdot
Y_{t})+\left(-\frac{b}{a}\right)\cdot(H_{2}\cdot Y_{t}).$

Therefore, by Lemma 3.3,

$\begin{split}&2g(Y_{t})-2\\\ &\geq((a-m-1)-(m+n-4))\cdot(Y_{t}\cdot
H_{1})+((b-n-1)-1)\cdot(Y_{t}\cdot H_{2})+\deg Q\\\
&\geq\left(a+\frac{1}{a}-(2m+n-2)\right)\cdot(H_{1}\cdot
Y_{t})+\left(b-\frac{b}{a}-n-2\right)\cdot(H_{2}\cdot Y_{t}),\end{split}$

which is at least some multiple of $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$
when $a\geq 2m+n-2$ and $b>(n+2)\cdot\left(\dfrac{a}{a-1}\right)$. One can
check that $b\geq m+2n-2>(n+2)\cdot\left(\dfrac{a}{a-1}\right)$ when $m+n\geq
5$ and $m\geq n.$

Case 3: Suppose that $s_{1}=m+n-2$, so $s_{2}=0.$ Then, the map

$M_{H_{1}}\rightarrow N_{h_{t}/X_{t}}$

induced by $\alpha$ has a rank-one image $Q$, so we can apply the scroll
considerations as in Case 2 to obtain

$\deg Q\geq\left(\frac{1}{a}-1\right)\cdot(H_{1}\cdot
Y_{t})+\left(-\frac{b}{a}\right)\cdot(H_{2}\cdot Y_{t}).$

Therefore, by Lemma 3.3,

$\begin{split}&2g(Y_{t})-2\\\ &\geq((a-m-1)-(m+n-3))\cdot(Y_{t}\cdot
H_{1})+(b-n-1)\cdot(Y_{t}\cdot H_{2})+\deg Q\\\
&\geq\left(a+\frac{1}{a}-(2m+n-1)\right)\cdot(H_{1}\cdot
Y_{t})+\left(b-\frac{b}{a}-n-1\right)\cdot(H_{2}\cdot Y_{t}),\end{split}$

which is at least some multiple of $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$
when $a\geq 2m+n-1$ and $b>(n+1)\cdot\left(\dfrac{a}{a-1}\right)$. One can
check that $b\geq m+2n-2>(n+1)\cdot\left(\dfrac{a}{a-1}\right)$ when $m+n\geq
5$ and $m\geq n.$ Hence, we focus on the case when $a=2m+n-2$ and $b\geq
m+2n-2.$ Suppose that the hypothesis of Lemma 3.11 is satisfied. Then, by
Lemma 3.14, the curve $h_{t}:Y_{t}\rightarrow X_{t}$ lies in the subvariety
$\Lambda^{\mathbb{P}^{m}}_{a}|_{t}\subseteq X_{t}$. Lemmas 3.17 and 3.18 imply
that $\Lambda^{\mathbb{P}^{m}}_{a}|_{t}$ is a smooth, irreducible curve of
general type. Therefore, there is some $\varepsilon>0$ such that (10) is
satisfied. ∎

####  4.2.2. Proof of Theorem 1.1(1)(b).

Let $(m,n)=(2,2)$.

Case 1: Suppose that $s_{1},s_{2}\leq 1.$ We proceed as in Case 1 of §4.2.1
and obtain the inequality

$2g(Y_{t})-2\geq(a-4)\cdot(Y_{t}\cdot H_{1})+(b-4)\cdot(Y_{t}\cdot H_{2}),$

which is at least $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$ when $a,b\geq 5.$

Case 2: Suppose that $s_{1}=2$, so $s_{2}=0$. We proceed as in Case 2 of
§4.2.1. Let $Q$ be the rank-one image of $\alpha:M_{H_{1}}\rightarrow
N_{h_{t}/X_{t}},$ and let $\Sigma\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$
denote the $\mathbb{P}^{m}$-scroll given by $Q$. By Lemma 3.8 and the same
calculations as above, we have

$\deg Q\geq\left(\frac{1}{a}-1\right)\cdot(H_{1}\cdot
Y_{t})+\left(-\frac{b}{a}\right)\cdot(H_{2}\cdot Y_{t}),$

and so

$\begin{split}2g(Y_{t})-2&\geq((a-3)-1)\cdot(Y_{t}\cdot
H_{1})+(b-3)\cdot(Y_{t}\cdot H_{2})+\deg Q\\\
&\geq\left(a+\frac{1}{a}-5\right)\cdot(H_{1}\cdot
Y_{t})+\left(b-\frac{b}{a}-3\right)\cdot(H_{2}\cdot Y_{t}),\end{split}$

which is at least some multiple of $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$
when $a,b\geq 5.$ ∎

####  4.2.3. Proof of Theorem 1.1(1)(c).

Let $(m,n)=(3,1)$.

Case 1: Suppose that $(s_{1},s_{2})=(1,0).$ We proceed as in Case 1 of §4.2.1
and obtain the inequality

$2g(Y_{t})-2\geq(a-5)\cdot(Y_{t}\cdot H_{1})+(b-2)\cdot(Y_{t}\cdot H_{2}),$

which is at least $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$ when $a\geq 6$ and
$b\geq 3.$

Case 2: Suppose that $(s_{1},s_{2})=(1,1).$ We proceed as in Cases 2 and 3 of
§4.2.1. For $i=1,2$, let $Q_{i}$ be the rank-one image of
$M_{H_{i}}\rightarrow N_{h_{t}/X_{t}},$ and let
$\Sigma_{i}\subseteq\mathbb{P}^{m}\times\mathbb{P}^{n}$ denote the scroll
given by $Q_{i}$. By Lemma 3.8 and the same calculations as above, we have

(11) $\deg Q_{1}\geq\left(\frac{1}{a}-1\right)\cdot(H_{1}\cdot
Y_{t})+\left(-\frac{b}{a}\right)\cdot(H_{2}\cdot Y_{t}),$ (12) $\deg
Q_{2}\geq\left(-\frac{a}{b}\right)\cdot(H_{1}\cdot
Y_{t})+\left(\frac{1}{b}-1\right)\cdot(H_{2}\cdot Y_{t}).$

Using (11), we have

$\begin{split}2g(Y_{t})-2&\geq(a-4)\cdot(Y_{t}\cdot
H_{1})+((b-2)-1)\cdot(Y_{t}\cdot H_{2})+\deg Q_{1}\\\
&\geq\left(a+\frac{1}{a}-5\right)\cdot(H_{1}\cdot
Y_{t})+\left(b-\frac{b}{a}-3\right)\cdot(H_{2}\cdot Y_{t}),\end{split}$

which is at least some multiple of $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$
when $a\geq 5$ and $b\geq 4.$

Using (12), we have

$\begin{split}2g(Y_{t})-2&\geq((a-4)-1)\cdot(Y_{t}\cdot
H_{1})+(b-2)\cdot(Y_{t}\cdot H_{2})+\deg Q_{2}\\\
&\geq\left(a-\frac{a}{b}-5\right)\cdot(H_{1}\cdot
Y_{t})+\left(b+\frac{1}{b}-3\right)\cdot(H_{2}\cdot Y_{t}),\end{split}$

which is at least some multiple of $(Y_{t}\cdot H_{1})+(Y_{t}\cdot H_{2})$
when $a\geq 8$ and $b\geq 3.$

Case 3: Suppose that $(s_{1},s_{2})=(2,0).$ Then the same proof as Case 4 of
§4.2.1 applies to show that there is some $\varepsilon>0$ such that (10) is
satisfied when $a\geq 5$ and $b\geq 3.$∎

######  Remark 4.3.

§4.2.3 shows where our proof techniques could not resolve the remaining open
cases in $\mathbb{P}^{3}\times\mathbb{P}^{1}$. Specifically, our techniques
could not produce an $\varepsilon>0$ such that curves of some types
$(s_{1},s_{2})$ satisfy (10), hence we could not confirm algebraic
hyperbolicity. We list such types of curves for each open case:

  1. (1)

For $(a,b)=(5,3)$, curves of types $(s_{1},s_{2})=(1,0)$ or $(1,1)$;

  2. (2)

For $(a,b)=(5,b)$ with $b\geq 4$, curves of type $(s_{1},s_{2})=(1,0)$;

  3. (3)

For $(a,b)=(6,3)$ or $(7,3)$, curves of type $(s_{1},s_{2})=(1,1)$.

## References

  * [Br78] R. Brody. Compact manifolds in hyperbolicity. Trans. Amer. Math. Soc. 235, 213–219 (1978)
  * [CL99] L. Chiantini and A. F. Lopez. Focal loci of families and the genus of curves on surfaces. Proc. Amer. Math. Soc. 127, no. 12, 3451–3459 (1999)
  * [Cl86] H. Clemens, Curves on generic hypersurfaces, Ann. Sci. École Norm. Sup. (4) 19, no. 4, 629–636 (1986)
  * [Cl03] H. Clemens, Lower Bounds on Genera of Subvarieties of Generic Hypersurfaces, Communications in Algebra, 31:8, no. 4, 3673–3711 (2003)
  * [ClR04] H. Clemens and Z. Ran. Twisted genus bounds for subvarieties of generic hypersurfaces. Amer. J. Math. 126, no. 1, 89–120 (2004)
  * [Co05] I. Coskun. The arithmetic and the geometry of Kobayashi hyperbolicity. Snowbird lectures in algebraic geometry. Contemp. Math., 388, 77–88 (2005)
  * [CR19a] I. Coskun and E. Riedl. Algebraic hyperbolicity of the very general quintic surface in $\mathbb{P}^{3}$. Adv. Math. 350, 1314–1323 (2019)
  * [CR19b] I. Coskun and E. Riedl. Algebraic hyperbolicity of very general surfaces. arXiv:1912.07689
  * [CR20] I. Coskun and E. Riedl. Clustered families and applications to Lang-type conjectures. arXiv:2010.11301
  * [De95] J.P. Demailly. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In Algebraic geometry Santa Cruz 1995, volume 62 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 285–360 (1997)
  * [De18] J.P. Demailly. Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. Jpn. J. Math. 15, 1–120 (2020)
  * [Ei88] L. Ein. Subvarieties of generic complete intersections. Invent. Math. 94, 163–169 (1988)
  * [Ei91] L. Ein. Subvarieties of generic complete intersections II. Math. Ann. 289, 465–471 (1991)
  * [HI19] C. Haase and N. Ilten, Algebraic hyperbolicity for surfaces in toric threefolds. J. Algebraic Geom. 30, 573-602 (2021)
  * [Pa03] G. Pacienza. Rational curves on general projective hypersurfaces. J. Algebraic Geom. 12, no. 2, 245–267 (2003)
  * [Pa04] G. Pacienza. Subvarieties of general type on a general projective hypersurface. Trans. Amer. Math. Soc. 356, no. 7, 2649–2661 (2004)
  * [Ro21] S. Robins. Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3. Beitr Algebra Geom (2022). https://doi.org/10.1007/s13366-021-00612-0
  * [Vo96] C. Voisin. On a conjecture of Clemens on rational curves on hypersurfaces. J. Differential Geom. 44, 200–214 (1996)
  * [Vo98] C. Voisin. A correction: “On a conjecture of Clemens on rational curves on hypersurfaces”. J. Differential Geom. 49, 601–611 (1998)
  * [Vo03] C. Voisin. On some problems of Kobayashi and Lang; algebraic approaches. Current Developments in Mathematics, vol. 2003, no. 1, 53–-125 (2003)
  * [Xu94] G. Xu. Subvarieties of general hypersurfaces in projective space. J. Differential Geom. 39, 139–172 (1994