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# Hirzebruch-type inequalities viewed as tools in combinatorics

Piotr Pokora

###### Abstract

The main purpose of this survey is to provide an introduction, algebro-
topological in nature, to Hirzebuch-type inequalities for plane curve
arrangements in the complex projective plane. These inequalities gain more and
more interest in many combinatorial problems related to point or line
arrangements in the plane. We would like to present a summary of the
technicalities and also some recent applications, for instance in the context
of Weak Dirac’s Conjecture. We advertise also some open problems and
questions.

Keywords Hirzebruch-Kummer covers, curve arrangements, line arrangements,
pseudoline arrangements, simplicial arrangements of lines, Dirac’s conjecture,
Beck’s Theorem of two extremes

Mathematics Subject Classification (2010) 14N10, 52C35, 32S22, 14N20

##  1 Introduction

In combinatorics, there are many interesting point-line incident problems.
Probably the most classical one is due to Sylvester [37].

######  Problem 1.1.

Prove that this is not possible to arrange any finite number of real points so
that a right line through every two of them shall pass through a third, unless
they all lie in the same right line.

This problem is also related to the famous orchard problem proposed by Jackson
as a rational amusement for winter evenings [23]. Gallai [14] proved that
Sylvester’s problem has a positive answer.

######  Theorem 1.2 (Sylvester-Gallai).

Let $\mathcal{P}\subset\mathbb{R}^{2}$ be a finite set of points. Then either

  * •

all points in $\mathcal{P}$ are collinear, or

  * •

there exists a line $\ell$ passing through exactly two points from
$\mathcal{P}$.

There are several elegant proofs of the theorem, probably the most instructive
one is given by L. M. Kelly which can be found, for instance, in [1]. Using
duality in the projective plane we can formulate Sylvester-Gallai Theorem in
the language of line arrangements and their intersection points, i.e., every
line arrangement in the real projective plane consisting of at least $3$
lines, which is not a pencil, contains at least one double intersection point.
This can be also observed using the well-known Melchior’s inequality [27]. For
an arrangement of lines $\mathcal{L}=\\{\ell_{1},...,\ell_{d}\\}$ in the
projective plane we denote by $t_{r}=t_{r}(\mathcal{L})$ the number of
$r$-fold points, i.e., points where exactly $r$-lines from the arrangement
meet.

######  Theorem 1.3 (Melchior).

Let
$\mathcal{L}=\\{\ell_{1},...,\ell_{d}\\}\subset\mathbb{P}^{2}_{\mathbb{R}}$ be
an arrangement of $d\geq 3$ lines. Assume that $\mathcal{L}$ is not a pencil,
then

$t_{2}\geq 3+\sum_{r\geq 4}(r-3)t_{r}.$

Melchior’s proof is based on a simple observation that every line arrangement
in the real projective plane provides a partition of the space into $f$
regions, $e$ edges, and $v$ vertices, and then we can use the identity
$v-e+f=\chi(\mathbb{P}^{2}_{\mathbb{R}})=1$. In fact, using the same method
one can construct a whole series of Melchior-type inequalities, which seems to
be a folklore result (this was shown for instance in a student paper [38]).

######  Theorem 1.4 (Melchior-type inequality).

Let
$\mathcal{L}=\\{\ell_{1},...,\ell_{d}\\}\subset\mathbb{P}^{2}_{\mathbb{R}}$ be
an arrangement of $d\geq 3$ lines. Assume that $\mathcal{L}$ is not a pencil
and pick $k\in\mathbb{Z}_{\geq 1}$, then

$\sum_{r=2}^{2k}(2k+1-r)t_{r}\geq 2k+1+\sum_{r>2k+1}(r-(2k+1))t_{r}.$

In particular, for $k=1$ we recover Melchior’s inequality.

It was natural to ask whether Melchior’s inequality can hold if we change the
underlying field, for instance if we consider a finite projective plane or the
complex projective plane. In both case the answer is negative.

######  Example 1.5.

Consider $\mathbb{P}^{2}_{\mathbb{Z}_{2}}$ \- the Fano plane. It is known that
there exists a unique configuration of $7$ lines and $7$ points of
multiplicity $3$, which obviously violates Melchior’s inequality.

######  Example 1.6.

Consider the following line arrangement in the complex projective plane
defined by the linear factors of the polynomial

$Q(x,y,z)=(z^{3}-y^{3})(y^{3}-z^{3})(x^{3}-z^{3}).$

It can be seen that $Q$ defines the arrangement consisting of $9$ lines and
$12$ triple intersection points, so it obviously violates Melchior’s
inequality. The arrangement defined by $Q$ is known as the dual Hesse
arrangement of lines (or CEVA’s arrangement of $9$ lines [11]).

The above (counter)examples motivated researchers to find reasonable
generalizations of Melchior’s inequality (mostly over the complex numbers)
involving the number of lines and $t_{r}$’s. It is worth mentioning that
Iitaka [22] proved (errornessly) that Melchior’s inequality holds for line
arrangements in the complex projective plane, so it shows that the problem
attracted the attention of people working in algebraic geometry. The
breakthrough came with Hirzebruch’s famous paper [20].

######  Theorem 1.7 (Hirzebruch’s inequality).

Let
$\mathcal{L}=\\{\ell_{1},...,\ell_{d}\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be
an arrangement of $d\geq 4$ lines such that $t_{d}=t_{d-1}=0$, then

$t_{2}+t_{3}\geq d+\sum_{r\geq 5}(r-4)t_{r}.$

It might be surprising to some people that Hirzebruch’s inequality is only a
by-product of his construction, the Hirzebruch-Kummer cover of the complex
projective plane branched along an arrangement of lines, which allowed him to
construct new examples of algebraic surfaces of general type, so-called ball-
quotients. We are not going into technicalities related to ball-quotient
surfaces, but for interested readers we refer to the following classical
textbook [2]. On the other side, it turned out that Hirzebruch’s inequality is
an extremely important tool in an ample variety of problems in combinatorial
geometry, for instance, as it was advertised in [33], Hirzebruch’s inequality
can be applied in the context of Sylvester-Gallai type theorems over the
complex numbers.

Our scope in this survey is to present an accessible outline of Hirzebruch’s
paper and another strong Hirzebruch-type inequalities which allowed researches
to make progress on classical conjectures in combinatorics, like the Weak
Dirac Conjecture [24, Section 6]. We hope that the survey will be useful for
these combinatorialists who want to use Hirzebruch’s ideas in their research.

Our prerequisites are not demanding, basics on differential geometry and first
lectures on algebraic geometry.

We work over the complex numbers, and we will use the natural inclusion of
$\mathbb{R}\subset\mathbb{C}$.

##  2 On Hirzebruch’s inequality for line arrangements

Before we present a condense proof of Hirzebruch’s inequality, we recall some
basics on algebraic surfaces. By an algebraic surface we mean a normal
$2$-dimensional complex projective variety – such a surface can be embedded
into $\mathbb{P}^{N}$ for some $N\in\mathbb{Z}_{>0}$. Mostly we consider only
smooth algebraic surfaces. The most important numbers that one can associate
with a smooth algebraic surface $X$ are the square of the first Chern class
which is equal to $c_{1}^{2}(X)=(\bigwedge^{2}\Omega_{X})^{2}$, were
$\Omega_{X}$ is the cotangent line bundle on $X$, and the second Chern class
$c_{2}(X)$ which coincides to the topological Euler characteristic $e(X)$. We
will need the Kodaira dimension of $X$. Let us define the $m$-th plurigenus of
$X$ as

$P_{m}(X)=\dim H^{0}(X,\omega_{X}^{\otimes m}),$

where $\omega_{X}:=\bigwedge^{2}\Omega_{X}$ is the canonical class. Now we
define the Kodaira dimension $\kappa(X)$ of $X$ to be $-\infty$ if
$P_{m}(X)=0$ for all $m>0$, otherwise $\kappa(X)$ is equal to the minimum $k$
such that the set $\\{P_{m}(X)/m^{k}:m\in\mathbb{N}\\}$ is bounded. We know
that in the case of surfaces $\kappa(X)\in\\{-\infty,0,1,2\\}$, and all those
surfaces for which $\kappa(X)=2$ are called surfaces of general type. In other
words, being a surface of general type means that $\omega_{X}$ has a large
number of sections, i.e., $\dim H^{0}(X,\omega_{X}^{\otimes m})\sim c\cdot
m^{2}$, where $c$ is a positive constant. As a simple example, the complex
projective plane has $e(\mathbb{P}^{2})=3$ and the canonical bundle is
$\mathcal{O}_{\mathbb{P}^{2}}(-3)$, which means that for all $m>0$ we have
$P_{m}(\mathbb{P}^{2})=0$, and this implies that
$\kappa(\mathbb{P}^{2})=-\infty$.

One of the most important results in the theory of algebraic surfaces is the
Bogomolov-Miyaoka-Yau inequality (see for instance [28, 39]).

######  Theorem 2.1 (BMY).

Let $X$ be a complex smooth projective surface with $\kappa(X)\geq 0$. Then

$c_{1}^{2}(X)\leq 3c_{2}(X),$

with equality if and only if the universal cover of $X$ is the complex unit
ball $|z_{1}|^{2}+|z_{2}|^{2}<1$ (i.e. $X$ is a ball-quotient).

Before we present the main construction which allows us to prove Hirzebruch’s
inequality, we need to recall some basics on covers, for more details we refer
to [19].

######  Definition 2.2.

A branched covering $\rho:X\rightarrow Y$ is a finite surjective morphism
between normal varieties. Denote by $G$ the group of isomorphisms
$\alpha:X\rightarrow X$ so that $\rho(\alpha(x))=\rho(x)$ for all $x\in X$.
The group $G$ is called the group of covering automorphisms of $\rho$. If $G$
acts transitively on all fibers of our cover $\rho$, then the covering is
called Galois or regular. We say that a branched covering $\rho:X\rightarrow
Y$ is an abelian covering if $\rho:X\rightarrow Y$ is Galois and additionally
the group of covering automorphisms is abelian.

Now we are ready to present the main result of this section. We will provide a
detailed outline of the proof emphasizing a topological part of Hirzebruch’s
considerations. Our outline is still quite technical (and might be involving),
but we are doing this in a good faith in order to emphasize the places where
algebraic geometry methods are decisive and might be difficult to replace by
combinatorics.

######  Theorem 2.3.

Let
$\mathcal{L}=\\{\ell_{1},...,\ell_{d}\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be
an arrangement of $d\geq 6$ lines such that $t_{d}=t_{d-1}=0$. Then

$t_{2}+t_{3}\geq d+\sum_{r\geq 5}(r-4)t_{r}.$ (1)

###### Proof.

Here is the strategy. The key idea of Hirzebruch is to use abelian coverings
of the complex projective plane branched along line arrangements. This idea
leads to a of construct an interesting algebraic surfaces to whose the first
Chern class and the topological Euler characteristic can be expressed in terms
of the combinatorics of a given arrangement. Under the conditions that we have
at least $d\geq 6$ lines and $t_{d}=t_{d-1}=0$, we can deduce that our newly
constructed surface is of non-negative Kodaira dimension. Thus we can apply
the Bogomolov-Miyaoka-Yau inequality.

Starting from scratch, and following ideas from [8, Section 4], let us denote
by $s_{i}\in
H^{0}(\mathbb{P}^{2}_{\mathbb{C}},\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1))$
the defining section of $\ell_{i}$, for all $i\in\\{1,...,d\\}$. Now we
consider the following map

$f:\mathbb{P}^{2}_{{\mathbb{C}}}\ni
x\mapsto(s_{1}(x):...:s_{d}(x))\in\mathbb{P}^{d-1}_{\mathbb{C}}.$

Let us emphasize that $f$ is well-defined since by the assumption there is no
point where all lines meet, so at least one of the $s_{j}(x)$’s is non-zero.
Now we use the Kummer covering

${\rm
Km}:\mathbb{P}^{d-1}_{\mathbb{C}}\ni(x_{1}:...:x_{d})\mapsto(x_{1}^{n}:...:x_{d}^{n})\in\mathbb{P}^{d-1}_{\mathbb{C}},$

where $n\geq 2$ is the exponent. One can show that this covering is of degree
$n^{d-1}$ with the Galois group $(\mathbb{Z}/n\mathbb{Z})^{d-1}$. Obviously,
it ramifies along $x_{1}\cdot...\cdot x_{d}=0$. Our main object of interest is
the following fiber product:

$X_{n}:=\mathbb{P}^{2}_{\mathbb{C}}\times_{\mathbb{P}^{d-1}_{\mathbb{C}}}\mathbb{P}^{d-1}_{\mathbb{C}}=\\{(x,y)\in\mathbb{P}^{2}_{\mathbb{C}}\times\mathbb{P}^{d-1}_{\mathbb{C}}:f(x)={\rm
Km}(y)\\}.$

We can write $X_{n}$ even more explicitly. We know that there exists a
projective transformation on $\mathbb{P}^{2}_{\mathbb{C}}$ such that
$\ell_{1}=\\{x_{1}=0\\}$, $\ell_{2}=\\{x_{2}=0\\}$, and
$\ell_{3}=\\{x_{3}=0\\}$, so we can describe $X_{n}$ in a new coordinate
system as

$X_{n}=\\{(z_{1}:...:z_{d})\in\mathbb{P}^{d-1}_{\mathbb{C}}:z_{j}^{n}=s_{j}(z_{1}^{n},z_{2}^{n},z_{3}^{n})\,\,{\rm
for}\,\,j\in\\{4,...,d\\}\\}.$

By this explicit description, our surface $X_{n}$ is given by
$(d-3)$-homogeneous equations in $\mathbb{P}^{d-1}_{\mathbb{C}}$, which means
that $X_{n}$ is a complete intersection. In general, $X_{n}$ is never a smooth
surface (except the case when all singular points of $\mathcal{L}$ are double
intersection points), so we need to find the so-called desingularization. One
can show (using a local argument) that $X_{n}$ is singular over a point $p$ of
the arrangement $\mathcal{L}$ iff $p$ is a point of multiplicity ${\rm
mult}_{p}\geq 3$. We can resolve singularities of $X_{n}$ by one simultaneous
blow-up $\tau:Y_{n}\rightarrow X_{n}$ at all those points which correspond to
singular points of $\mathcal{L}$ with multiplicities $\geq 3$. Since $Y_{n}$
is a smooth complex projective surfaces, we can compute the Chern numbers,
namely

$c_{1}^{2}(Y_{n})/n^{d-3}=n^{2}(9-5d+3f_{1}-4f_{0})+4n(d-f_{1}+f_{0})+f_{1}-f_{0}+d+t_{2},$
$c_{2}(Y_{n})/n^{d-3}=n^{2}(3-2d+f_{1}-f_{0})+2n(d-f_{1}+f_{0})+f_{1}-t_{2}.$

In the next step, quite cumbersome, one needs to check under which conditions
on the combinatorics of $\mathcal{L}$ our surface has non-negative Kodaira
dimension – it turns out that it is enough to assume that $d\geq 6$,
$t_{d}=t_{d-1}=0$, and $n\geq 3$. This means that if $\mathcal{L}$ satisfies
the above conditions, then we can apply the Bogomolov-Miyaoka-Yau inequality:

$c_{1}^{2}(Y_{n})\leq 3c_{2}(Y_{n}).$

Let us define the following Hirzebruch polynomial

$H_{\mathcal{L}}(n)=\frac{3c_{2}(Y_{n})-c_{1}^{2}(Y_{n})}{n^{d-3}}=n^{2}(f_{0}-d)+2n(d-f_{1}+f_{0})+2f_{1}+f_{0}-d-4t_{2}.$

Since $H_{\mathcal{L}}(n)\geq 0$ for $n\geq 3$, we can compute
$H_{\mathcal{L}}(3)$, which gives us

$t_{2}+t_{3}\geq d+\sum_{r\geq 5}(r-4)t_{r},$

which is Hirzebruch’s inequality.

∎

######  Remark 2.4.

Hirzebruch’s inequality implies that every configuration of $d\geq 6$ lines
with $t_{d}=t_{d-1}=0$ contains double or triple points as the intersections.

######  Remark 2.5.

It is natural to ask whether Hirzebruch’s inequality is sharp, i.e., whether
there exists a line arrangement $\mathcal{A}$ such that
$t_{2}+t_{3}=d+\sum_{r\geq 5}(r-4)t_{r}$. There exists exactly one (!)
arrangement of lines satisfying the above equality, namely the Hesse
arrangement of lines. This arrangement consists of $d=12$ lines having
$t_{2}=12$ and $t_{4}=9$. The proof of this quite surprising result is not
elementary (in its whole generality), one need to use the theory of totally
geodesic curves in complex compact ball-quotients [2]. In the case when we
restrict our attention to real line arrangements, we refer to [5] for an
elementary proof of the fact that there are no such arrangements.

######  Remark 2.6.

In the same paper [20], Hirzebruch defines the so-called characteristic
numbers of line arrangements, namely

$\gamma(\mathcal{L})=\lim_{n\rightarrow\infty}\frac{c_{1}^{2}(Y_{n})}{c_{2}(Y_{n})}=\frac{9-5d+3f_{1}-4f_{0}}{3-2d+f_{1}-f_{0}}.$

Somesse [36] proved that for complex line arrangements,

$\gamma(\mathcal{L})\leq\frac{8}{3},$

with equality if and only if $\mathcal{L}$ is the dual-Hesse arrangement of
lines. This result, in particular, implies that if $\mathcal{L}$ is an
arrangement of $d\geq 6$ lines with $t_{d}=t_{d-1}=0$, then

$2t_{2}+t_{3}\geq 3+d+\sum_{r\geq 5}(r-4)t_{r}.$

Observe that $\gamma(\mathcal{L})=3$ implies that $f_{0}=d$, and by the Erdös-
de Bruijn Theorem [9] this condition forces $\mathcal{L}$ to be a Hirzebruch
quasi-pencil, i.e., an arrangement of $d$ lines such that $t_{d-1}=1$ and
$t_{2}=d-1$. Note that for a Hirzebruch quasi-pencil,

$2d-2=2t_{2}+t_{3}\geq 3+d+\sum_{r\geq 5}(r-4)t_{r}=2d-2.$

######  Remark 2.7.

Hirzebruch’s construction provides the whole series of inequalities depending
on $n\geq 3$. In particular, for $n=5$ we obtain

$4t_{2}+3t_{3}+t_{4}\geq 4d+\sum_{r\geq 5}(2r-9)t_{r}.$

It is natural to ask whether this inequality is sharp, and it turns out that
there exists exactly one real line arrangement providing equality, the well-
known $\mathcal{A}_{1}(6)$ configuration consisting of $d=6$ lines and
$t_{3}=4$, $t_{2}=3$. For a combinatorial proof of this statement we refer to
[5]. Moreover, one can show that there is exactly one line arrangement defined
over the complex numbers providing equality, the dual-Hesse arrangement of $9$
lines and $12$ triple points.

######  Remark 2.8.

Using finer considerations on the Kodaira dimension of $Y_{n}$, we can show
that if $d\geq 6$ with $t_{d}=t_{d-1}=t_{d-2}=0$ and $n\geq 2$, then our
surface $Y_{n}$ has non-negative Kodaira dimension. The condition
$H_{\mathcal{L}}(2)\geq 0$ leads us to

$t_{2}+3t_{3}+t_{4}\geq d+\sum_{r\geq 5}(2r-9)t_{r}.$

######  Remark 2.9.

In the literature, we can find usually the following variant of Hirzebruch’s
inequality

$t_{2}+\frac{3}{4}t_{3}\geq d+\sum_{r\geq 5}(2r-9)t_{r}$ (2)

provided that $d\geq 6$ and $t_{d}=t_{d-1}=t_{d-2}=0$. In order to justify
this claim, one needs to use Miyaoka-Sakai’s improvement [21, 28, 32] of the
Bogomolov-Miyaoka-Yau inequality which tells us that if $Y_{n}$ contains
either smooth rational curves (genus = $0$) or smooth elliptic curves (genus =
$1$), then one always has $3c_{2}(Y_{n})-c_{1}^{2}(Y_{n})\geq const>0$, and
the number $const$ can be explicitly determined. This leads us to the desire
inequality.

######  Remark 2.10.

In Research Problems in Discrete Geometry by Brass, Moser, and Pach [7, p.
315; Problem 7] one of the stated research problems is to prove Hirzebruch’s
inequality (2) using only elementary methods. In the light of the above
remarks, this seems to be extremely difficult. The main ingredient of
Hirzebruch’s construction is the Bogomolov-Miyaoka-Yau inequality which is not
combinatorial in its nature. As observed in the next section, we can find even
stronger inequalities involving the number of lines and intersection points,
but these also follow from variants of the Bogomolov-Miyaoka-Yau inequality.
At this stage, at least to the author, it seems that there is no hope to find
an easy proof of (2).

######  Remark 2.11.

It is easy to observe that every configuration of $d\in\\{4,5\\}$ lines with
$t_{d}=t_{d-1}=0$ also satisfies Hirzebruch’s inequality (1).

######  Remark 2.12.

Topologically, our branched covering
$\rho:X_{n}\rightarrow\mathbb{P}^{2}_{\mathbb{C}}$ is determined by the
following defining map

$\phi:H_{1}(\mathbb{P}^{2}_{\mathbb{C}}\setminus\mathcal{L},\mathbb{Z})\rightarrow
H_{1}(\mathbb{P}^{2}_{\mathbb{C}}\setminus\mathcal{L},\mathbb{Z}/n\mathbb{Z}).$

Before we pass to (stronger) Hirzebruch-type inequalities, let us present an
interesting way to construct $K3$ surfaces with use of abelian covers branched
along $6$ general lines.

######  Example 2.13.

Consider
$\mathcal{L}=\\{\ell_{1},...,\ell_{6}\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ an
arrangement of $6$ generic lines which means that the only intersection points
of these lines are double points. We can find a projective transformation such
that $\ell_{1}=\\{x=0\\},\ell_{2}=\\{y=0\\}$, and $\ell_{3}=\\{z=0\\}$. We
denote by $\ell_{i}=a_{i}x+b_{i}y+c_{i}z$ with $i\in\\{4,5,6\\}$ the equations
of remaining $3$ lines. Now we can consider the Hirzebruch-Kummer cover
$X_{2}$ with exponent $n=2$ branched along $\ell_{1},...,\ell_{6}$. We know
that $X_{2}$ is a smooth projective surface and it can be described as

$X_{2}=\\{(z_{1},z_{2},z_{3},z_{4},z_{5},z_{6})\in\mathbb{P}^{5}_{\mathbb{C}}:z^{2}_{i}=a_{i}z_{1}^{2}+b_{i}z_{2}^{2}+c_{i}z_{3}^{2},i\in\\{4,5,6\\}\\},$

so our surface $X_{2}$ is a smooth complete intersection of $3$ quadrics in
$\mathbb{P}^{5}_{\mathbb{C}}$. This surface is well-known in algebraic
geometry, i.e., $X_{2}$ is a $K3$ surface of degree $8$. It is worth pointing
out that there is an extremely interesting link between $X_{2}$ and
$(2:1)$-cover of the complex projective plane branched along
$\ell_{1},...,\ell_{6}$ – it turns out that $X_{2}$ is the minimal
desingularization of this covering [17, p. 770].

##  3 Stronger Hirzebruch-type inequalities for complex line arrangements

Now we would like to present (stronger) Hirzebruch-type inequalities for line
arrangements in the complex projective plane. These results follow from
Langer’s version of the orbifold Miyaoka-Yau inequality for normal surfaces
with boundary divisors. Since Langer’s result is highly non-trivial (it
involves, for instance, the notion of orbifold Euler numbers, and other
technical considerations), we do not provide details – the readers can consult
[25] for details.

Let us start with the first strong Hirzebruch’s type inequality, which was
first proved by Bojanowski [6] in his Master Thesis (in Polish).

######  Theorem 3.1.

Let
$\mathcal{L}=\\{\ell_{1},...,\ell_{d}\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be
a line arrangement with $d\geq 6$ such that $t_{r}=0$ for $r>\frac{2d}{3}$.
Then

$t_{2}+\frac{3}{4}t_{3}\geq d+\sum_{r\geq
5}\bigg{(}\frac{r^{2}}{4}-r\bigg{)}t_{r}.$ (3)

One proof of this result can be deduced from [31, Theorem 2.2] with $d=1$. It
also follows from the following two Langer’s inequalities for complex line
arrangements [25, Proposition 11.3.1].

######  Theorem 3.2.

Let
$\mathcal{L}=\\{\ell_{1},...,\ell_{d}\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be
a line arrangement such that $t_{r}=0$ for $r>\frac{2d}{3}$. Then

$\sum_{r\geq 2}r^{2}t_{r}\geq\bigg{\lceil}\frac{4d^{2}}{3}\bigg{\rceil},$
$\sum_{r\geq 2}rt_{r}\geq\bigg{\lceil}\frac{d^{2}}{3}+d\bigg{\rceil}.$

It is natural to compare Bojanowski’s version of Hirzebruch’s inequality with
others, and we can easily observe the following chain of inequalities (under
the assumption that $t_{r}=0$ for $r>\frac{2d}{3}$):

$t_{2}+t_{3}\geq t_{2}+\frac{3}{4}t_{3}\geq d+\sum_{r\geq
5}\bigg{(}\frac{r^{2}}{4}-r\bigg{)}t_{r}\geq d+\sum_{r\geq 5}(2r-9)t_{r}\geq
d+\sum_{r\geq 5}(r-4)t_{r}.$

Let us now list examples of line arrangements111If you would like to learn
more about these arrangements and the geometry lurking behind them, we refer
to [2, 11] for details. for which we obtain equality in (3) – our list is
probably far away to be complete.

  1. 1.

_Icosahedron arrangement_ consisting of $15$ lines and
$t_{2}=15,t_{3}=10,t_{5}=6$;

  2. 2.

_CEVA’s arrangements_ consisting of $3n$ lines with $n\geq 3$, and
$t_{3}=n^{2},t_{n}=3$;

  3. 3.

_extended CEVA’s arrangements_ consisting of $3n+3$ lines with $n\geq 3$, and
$t_{2}=3n$, $t_{3}=n^{2}$, $t_{n+2}=3$;

  4. 4.

_Hesse arrangement_ consisting of $12$ lines and $t_{4}=9,t_{2}=12$;

  5. 5.

the merger of CEVA’s arrangement of $9$ lines and the Hesse arrangement
consisting of $d=12+9$ lines with $t_{2}=36$, $t_{4}=9$, $t_{5}=12$;

  6. 6.

_Klein’s arrangement_ consisting of $21$ lines and $t_{3}=28,t_{4}=21$;

  7. 7.

_Wiman’s arrangement_ consisting of $45$ lines and
$t_{3}=120,t_{4}=45,t_{5}=36$.

As we can observe, there exists an infinite series of line arrangements such
that equality in (3) holds – for instance CEVA’s line arrangements. Moreover,
note that there exists an interesting line combinatorics (which we understand
here as a vector of the form $(d,t_{2},...,t_{d-1})$) $\mathcal{C}$
constructed in [2, p. 116] consisting of $d=12m+3$ lines and
$t_{2}=12m^{2}+15m+3$, $t_{6}=4m^{2}+m$ with $m\in\mathbb{Z}_{\geq 3}$. It can
be shown that this combinatorics cannot be realized over the real numbers
(i.e., there does not exist any line arrangement defined over the real numbers
possessing the mentioned combinatorics), so this leads to the first open
problem of this survey.

######  Problem 3.3.

Is it possible to construct arrangements of $d=12m+3$ lines in the complex
projective plane such that $t_{2}=12m^{2}+15m+3$, $t_{6}=4m^{2}+m$ where
$m\in\mathbb{Z}_{\geq 3}$?

Simple calculations reveal that combinatorics $\mathcal{C}$ satisfies the
equality in $(\ref{langorb})$, and if one can show that there exists
$m_{0}\in\mathbb{Z}_{\geq 3}$ for which we can realize $\mathcal{C}$ over the
complex numbers, then $\mathcal{C}$ leads to a new example of complex and
compact $2$-dimensional ball-quotient (and in fact this is the main reason why
this problem is really attractive).

Now we are in a good position to present (probably) the strongest known
Hirzebruch-type inequality for complex line arrangements. The inequality in
question is the main result of Bojanowski’s thesis [6, Theorem 2.3].

######  Theorem 3.4.

Let $\mathcal{L}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be an arrangement of $d$
lines. Pick a natural number $n\in[3,...,d)$ and assume that $t_{r}=0$ for
$r>d-n+1$. Then

$t_{2}+\frac{3}{4}t_{3}\geq
d+\sum_{r=5}^{s-1}\bigg{(}\frac{r^{2}}{4}-r\bigg{)}+\sum_{r=s}^{d-n}((n-1)r-n^{2})t_{r}+\sum_{r>d-n}((n-2)r-(n-1)^{2})t_{r},$

where $s=\min\\{2n,d-n\\}$.

##  4 Applications

In this section, we focus on applications of Hirzebruch-type inequalities in
the context of interesting combinatorial problems in incident point-line
theory. We are going to present only three aspects, in order to avoid too many
repetitions, for more applications which use Langer’s inequalities and in some
sense Hirzebruch-type inequalities, we refer for instance to a recent paper by
Frank de Zeeuw [10].

###  4.1 Weak Dirac’s Conjecture

Let us denote by $\mathcal{P}\subset\mathbb{P}^{2}_{\mathbb{C}}$ a finite set
of mutually distinct $n$ points and let $\mathcal{L}(\mathcal{P})$ be the set
of lines determined by $\mathcal{P}$, where a line which passes through at
least two points from $\mathcal{P}$ is said to be determined by $\mathcal{P}$.
In 1961, P. Erdös proposed the following Weak Dirac’s Conjecture [12].

######  Conjecture 4.1 (WDC).

Every set $\mathcal{P}$ of $n$ non-collinear points in the plane (presumably
over the real numbers) contains a point which is incident to at least
$\lceil\frac{n}{c}\rceil$ lines from $\mathcal{L}(\mathcal{P})$ for a certain
constant $c>0$.

Weak Dirac’s Conjecture was solved independently by Beck [3] and Szemerédi-
Trotter [34], but they do not specified the actual value of $c$. In 2012,
Payne and Wood showed the WDC with $c=37$ [29], and one of the main
ingredients of their proof is Hirzebruch’s inequality. For many years people
believed that the conjecture should hold with $c=2$, but it turned out that
there are some counterexamples, see for instance [18]. On the other side, as
we can read in [24, Chapter 6], it was more plausible to believe that $c=3$,
and it turned out that this prediction is correct [40].

######  Theorem 4.2 (Zeye Han).

The Weak Dirac Conjecture holds with $c=3$.

###### Proof.

We will follow Zeye Han’s approach [40]. First of all, note that if
$\mathcal{P}$ is a finite set of non-collinear points and it contains at least
$\lceil\frac{n}{3}\rceil+1$ points which lie on a line $\ell$, then we are
done – it is enough to consider a point $p\in\mathcal{P}\setminus\ell$ which
is incident, by definition/construction, to at least
$\lceil\frac{n}{3}\rceil+1$ lines, so we may assume that $\mathcal{P}$ does
not contain $\lceil\frac{n}{3}\rceil+1$ collinear points. Let us denote by
$l_{r}$ the number of $r$-rich lines, i.e., those lines containing exactly $r$
points from $\mathcal{P}$. According to the dual version of Bojanowski’s
inequality, we have

$l_{2}+\frac{3}{4}l_{3}\geq n+\sum_{r\geq
5}\bigg{(}\frac{r^{2}}{4}-r\bigg{)}l_{r}\Longleftrightarrow
l_{2}+\frac{3}{4}l_{3}\geq n+\sum_{r\geq 5}\frac{{r\choose
2}}{2}l_{r}-\frac{3}{4}\sum_{r\geq 5}rl_{r}.$

Using the combinatorial count ${n\choose 2}=\sum_{r\geq 2}{r\choose 2}t_{r}$
we obtain

$l_{2}+\frac{3}{4}l_{3}\geq n+\frac{{n\choose
2}}{2}-\sum_{r=2}^{4}\frac{{r\choose 2}}{2}l_{r}-\frac{3}{4}\sum_{r\geq
5}rl_{r}\Leftrightarrow\sum_{r\geq 2}rl_{r}\geq\frac{n(n+3)}{3},$

and this leads to

$\sum_{p\in\mathcal{P}}{\rm mult}_{p}\geq\frac{n(n+3)}{3}.$

Using the Pigeonhole Principle, there exists a point from $\mathcal{P}$ which
is incident to at least $\lceil\frac{n}{3}\rceil+1$ lines from
$\mathcal{L}(\mathcal{P})$, and it completes the proof. ∎

It is worth mentioning that Han’s result can be also obtained using Langer’s
inequality (cf. [10, Corollary 1.2]).

###  4.2 Beck’s Theorem of two extremes

In this subsection, we would like to report on some progress towards better
estimations in Beck’s Theorem [3, Theorem 3.1].

######  Theorem 4.3 (Beck).

For a finite set $\mathcal{P}$ of $n$ points in $\mathbb{R}^{2}$ one of the
following is true:

  * •

there exists a line that contains $c_{1}n$ points from $\mathcal{P}$ for some
positive $c_{1}$;

  * •

there are at least $c_{2}n$ lines spanned by $\mathcal{P}$.

Beck in his paper gave $c_{1}=\frac{1}{100}$ and $c_{2}$ was unspecified,
Payne and Wood in [29, Theorem 5] provided $c_{1}=c_{2}=\frac{1}{100}$. With
use of Langer’s inequality, Frank de Zeeuw observed [10, Theorem 2.1] that one
can significantly improve estimations on $c_{1}$ and $c_{2}$.

######  Theorem 4.4.

Let $\mathcal{P}$ be a finite set of $n$ points in $\mathbb{R}^{2}$, then one
of the following is true:

  * •

there is a line that contains more than $\frac{6+\sqrt{3}}{9}n$ points of
$\mathcal{P}$;

  * •

there are at least $\frac{n^{2}}{9}$ lines spanned by $\mathcal{P}$.

###  4.3 Simplicial line arrangements

Let $\mathcal{A}=\\{H_{1},...,H_{d}\\}\subset\mathbb{R}^{n}$ be a central
arrangement of $d$ (linear) hyperplanes. We say that $\mathcal{A}$ is
simplicial if every connected component of
$\mathbb{R}^{n}\setminus\bigcup_{i=1}^{d}H_{i}$ is an open simplicial cone.
Using a natural projectivization we can think about rank $n=3$ simplicial
hyperplane arrangements as line arrangements in $\mathbb{P}^{2}_{\mathbb{R}}$.
Let us recall some numerical properties of simplicial line arrangements, and
we assume from now on that our arrangements are _irreducible_ :

  * •

$t_{2}=3+\sum_{r\geq 4}(r-3)t_{r}$;

  * •

$t_{r}=0$ for $r>d/2$, which means that we can freely use Langer’s
inequalities and Bojanowski’s inequality (3).

Now we present some very recent and interesting results from the PhD thesis of
Geis [16]. We start with an interesting observation which gives us a bound on
multiplicities of singular points of a certain class of simplicial line
arrangements [16, Remark 2.13 iv].

######  Proposition 4.5.

Let $\mathcal{L}$ be a simplicial line arrangement in
$\mathbb{P}^{2}_{\mathbb{R}}$ such that $t_{2}\geq t_{3}$ and $t_{i}=0$ for
$i\not\in\\{2,3,x\\}$. Then $x\leq 8$.

###### Proof.

Since $t_{2}\geq t_{3}$ and by Bojanowski’s inequality (3.1), one has the
following chain of inequalities:

$\frac{7}{4}t_{2}\geq t_{2}+\frac{3}{4}t_{3}\geq
d+\frac{x(x-4)}{4}t_{x}=d+\frac{x(x-4)(t_{2}-3)}{4(x-3)},$

where the last equality follows from Melchior’s inequality for simplicial line
arrangements. Assume now that $x\geq 9$, which implies that

$0\leq\bigg{(}\frac{x(x-4)}{4(x-3)}-\frac{7}{4}\bigg{)}t_{2}\leq\frac{3x(x-4)}{4(x-3)}-d.$

This allows us to deduce that

$d\leq\frac{3x(x-4)}{4(x-3)}\leq\frac{3x}{4}\leq\frac{3d}{8}<d,$

a contradiction. ∎

Next, we present an application of one of Langer’s inequalities providing a
quadratic lower bound on ${\rm max}(t_{2},t_{3})$ for simplicial arrangements
[16, Theorem 5.2].

######  Theorem 4.6.

Let $\mathcal{L}$ be a simplicial line arrangement in
$\mathbb{P}^{2}_{\mathbb{R}}$. Then

${\rm max}(t_{2},t_{3})>\bigg{\lceil}\frac{d^{2}+3d}{27}\bigg{\rceil}.$

In order to give you some feeling about this result, let us recall that Erdös
and Purdy [13] proved that if $\mathcal{L}$ is an arrangement of $d\geq 25$
lines in the real projective plane such that $t_{d}=0$, then

${\rm max}(t_{2},t_{3})\geq d-1.$

Moreover, they also proved that if $t_{2}<d-1$, then $t_{3}\geq cd^{2}$ for
some positive constant $c$.

Before we finish this section, it is worth presenting a Melchior-type
inequality for simplicial line arrangements also showed by Geis [16, Lemma 5.2
c] – the key advantage of this result is that it provides constraints on the
number of triple points.

######  Proposition 4.7.

Let $\mathcal{L}$ be an simplicial line arrangement in
$\mathbb{P}^{2}_{\mathbb{R}}$. Then

$t_{3}\geq 4+\sum_{r\geq 5}(r-4)t_{r}.$

It is worth mentioning that the proof provided by Geis does not use any
Hirzebruch-type inequalities.

Concluding this section, if we combine Melchior’s inequality with Bojanowski’s
inequality, we obtain the following chain of inequalities

$t_{3}+\frac{4}{3}t_{4}+t_{5}\geq\frac{4}{3}(d-3)+\frac{1}{3}\sum_{r\geq
6}\bigg{(}r^{2}-8r+12\bigg{)}t_{r}\geq\frac{4}{3}(d-3),$

which seems to be an interesting observation. Of course this inequality is
sharp.

##  5 Generalizations of Hirzebruch’s inequalities for plane curve
arrangements

In this section, we present some natural generalizations of Hirzebruch’s
inequality for line arrangements in the context of higher degree plane curves.
We start with the following definition.

######  Definition 5.1.

Let $\mathcal{C}=\\{C_{1},...,C_{k}\\}$ be an arrangement of irreducible
curves in the complex projective plane. We say that $\mathcal{C}$ is a
$d$-arrangement if the following conditions hold:

  1. 1.

all irreducible components $C_{i}$ are smooth and of the same degree $d\geq
1$,

  2. 2.

all intersection points are ordinary singularities (i.e., these look locally
like intersections of lines),

  3. 3.

there is no point where all curves meet simultaneously.

As we can observed, $d$-arrangements are higher degree generalizations of line
arrangements, for instance $2$-arrangements will be called conic arrangements,
even if in general conic arrangements might have non-ordinary intersection
points. The first result presents a Hirzebruch-type inequality for
$d$-arrangements [30, Theorem 2.3].

######  Theorem 5.2.

Let $\mathcal{C}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be a $d$-arrangement of
$k\geq 4$ curves with $d\geq 2$. Then

$(5d^{2}-6d)k+t_{2}+\frac{3}{4}t_{3}\geq\sum_{r\geq 5}(r-4)t_{r}.$

It is natural to ask whether one can find an improvement of the above
inequality, for instance in order to have the so-called quadratic right-hand
side. It turns out that this can be achieved with help of Langer’s ideas
around his version of the orbifold Miyaoka-Yau inequality [31, Theorem 2.2].

######  Theorem 5.3.

Let $\mathcal{C}=\\{C_{1},...,C_{k}\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be a
$d$-arrangement of $k\geq 3$ curves with $d\geq 2$. Then

$t_{2}+\frac{3}{4}t_{3}+d^{2}k(dk-k-1)\geq\sum_{r\geq
5}\bigg{(}\frac{r^{2}}{4}-r\bigg{)}t_{r}.$

It is also an interesting question whether one can extend Hirzebruch-type
inequalities in the context of arrangements admitting different degrees of
irreducible curves. Probably the first result in this spirit is devoted to
conic-line arrangements in the complex projective plane having only ordinary
singularities [31, Theroem 2.1].

######  Theorem 5.4.

Let $\mathcal{LC}=\\{L_{1},...,L_{l},C_{1},...,C_{k}\\}$ be an arrangement of
$l$ lines and $k$ conics such that $t_{r}=0$ for $r>\frac{2(l+2k)}{3}$, and we
assume that all intersection points of the arrangement are ordinary
singularities. Then

$t_{2}+\frac{3}{4}t_{3}+(4k+2l-4)k\geq l+\sum_{r\geq
5}\bigg{(}\frac{r^{2}}{4}-r\bigg{)}t_{r}.$

Finally we consider an interesting topological generalization of line
arrangements in the real projective plane – pseudolines arrangements.

######  Definition 5.5.

An arrangement $\mathcal{C}\subset\mathbb{P}^{2}_{\mathbb{R}}$ of $d\geq 3$
smooth closed curves is an arrangement of pseudolines if:

  * •

all intersection points are transversal, i.e., locally can be described as
$x_{1}x_{2}=0$,

  * •

every pair of pseudolines intersect at exactly one point,

  * •

there is no point where all curves meet.

For such arrangements, topological in nature, Shnurnikov proved the following
inequality [35].

######  Theorem 5.6 (Shnurnikov).

Let $\mathcal{C}$ be a pseudoline arrangement of $d\geq 5$ curves such that
$t_{d}=t_{d-1}=t_{d-2}=t_{d-3}=0$. Then

$t_{2}+\frac{3}{2}t_{3}\geq 8+\sum_{r\geq 4}(2r-7.5)t_{r}.$

There exists exactly one combinatorial type of pseudoline arrangements for
which we obtain equality in Shnurnikov’s inequality, namely $d=7$ with
$t_{4}=2$ and $t_{2}=9$.

Let us emphasize that pseudoline arrangements can be viewed algebraically as
rank $3$ simple oriented matroids [4].

##  6 Speculations

In this short section, I consider possible combinatorial approaches towards
Hirzebruch-type inequalities. It is a notoriously difficult question whether
we can show any Hirzebruch-type inequality using only elementary combinatorial
methods. At this moment, unfortunately, it seems to be out of reach. However,
we can translate this problem using different languages. One of the most
promising is the language of tropical geometry, we refer to [15] for a short
introduction to the subject, or to the very recent textbook [26]. Let
$\mathcal{C}$ be an arrangement of smooth curves in the complex projective
plane, and let $\overline{\mathcal{C}}$ denote its tropicalization. Of course
it might happen that our curves are intersecting along segments (even not
bounded segments), but instead of that we can use the notion of stable
intersections in order to avoid such situations. This idea leads to a tropical
model of curve arrangements in the complex projective plane (as a one of
possibilities). Now we would like to formulate some problems.

######  Problem 6.1.

Is it possible to show a Hirzebruch-type inequality using the language of
tropical geometry, or its tropical variation?

######  Problem 6.2.

Is it possible to find tropical analogues of the Bogomolov-Miyaoka-Yau
inequality?

## Acknowledgement

The idea to write up this survey was born after email correspondence between
the author and Frank de Zeeuw, and this was a very stimulating exchange. I
would like to thank David R. Wood for suggestions and remarks that allowed to
improve this survey, to Terence Tao for pointing out [33], to Renzo Cavalieri
for his remarks about the tropical world, to Sławomir Rams for his remarks,
and to Igor Dolgachev for reading the manuscript. Finally, I would like to
thank David Geis for allowing me to reproduce some of the results from his PhD
thesis, and for useful remarks.

During the project the author was supported by the Fundation for Polish
Science (FNP) Scholarship Start No. 076/2018.

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Address: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich
8, PL-00-656 Warszawa, Poland.
E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>


# Stochastic Chaos and Predictability in Laboratory Earthquakes

Adriano Gualandi<EMAIL_ADDRESS>Istituto Nazionale di Geofisica e
Vulcanologia, Osservatorio Nazionale Terremoti, Rome, Italy  Davide Faranda
Laboratoire des Sciences du Climat et de l’Environnement, UMR 8212 CEA-CNRS-
UVSQ, Université Paris-Saclay, IPSL, 91191 Gif-sur-Yvette, France  London
Mathematical Laboratory, London, UK  LMD/IPSL, Ecole Normale Superieure, PSL
research University, Paris, France  Chris Marone  La Sapienza University of
Rome, Department of Earth Sciences, Rome, Italy  Pennsylvania State
University, Geosciences, USA  Massimo Cocco  Istituto Nazionale di Geofisica e
Vulcanologia, Sezione Roma 1, Rome, Italy  Gianmarco Mengaldo  National
University of Singapore, Department of Mechanical Engineering, College of
Design and Engineering, Singapore

###### Abstract

Laboratory earthquakes exhibit characteristics of a low-dimensional random
attractor with a dimension similar to that of natural slow earthquakes. A
model of stochastic differential equations based on rate- and state-dependent
friction explains the laboratory observations. We study the transition from
stable sliding to stick-slip events and find that aperiodic behavior can be
explained by small perturbations ($<1$‰) in the stress state. Friction’s
nonlinear nature amplifies small scale perturbations, reducing the
predictability of the otherwise periodic macroscopic dynamics.

††preprint: APS/123-QED

Friction is a complex nonlinear phenomenon that can be modeled through many
degrees of freedom (dofs) [1]. Nonetheless, laboratory studies of friction in
geomaterials led to the discovery of phenomenological laws to describe the
behavior of sliding surfaces with a limited number of dofs. Rate- and state-
dependent (RS) friction is today a standard framework to describe friction in
a wide range of systems [2, 3, 4, 5]. Within such formalism, the friction
($\mu$) depends on the sliding rate ($v$) and an internal state variable
($\theta$) that describes memory effects. Stick-slip behavior, similar to that
for the seismic cycle for earthquake faulting, is characterized by inter-
seismic periods of load increase followed by failure events with fast or slow
unloading. Repeating failure events are explained by fault healing and changes
in frictional state $\theta$ [2, 6]. Stick-slip laboratory experiments have
been used extensively to mimic the seismic cycle, for example, via a double
direct shear apparatus [7, 8]. Our goal is to see if slow and fast labquakes
live in the same phase space, and if they have the same number of dofs
observed for slow earthquakes in nature [9]. We use data from 14 stick-slip
friction experiments conducted at different imposed normal stress
($\sigma_{n}$) conditions [10]. In the experiments, two layers of quartz
powder are put under $\sigma_{n}$ and then sheared using an acrylic piston to
modulate the elastic stiffness around a value of $k=14.8$ MPa/$\mu$m [7] (Fig.
1a). The applied normal stress is used as a control parameter to
systematically traverse the critical stability condition for which the
stiffness of the loading apparatus ($k$) is equal to the critical stiffness of
the system ($k_{c}\propto\frac{1}{\sigma_{n}}$) [6, 11]. We observe both slow
and fast stick-slip motion, with slow/fast events occurring at low/high
$\sigma_{n}$ values. We deduce from the observations the number of dofs
governing the system at this transition regime and set up a model with the
appropriate number of phase space dimensions to match laboratory data and
explain the observed range of labquake behaviors.

Each experiment includes several labquake cycles. We use data in a
$200~{}\textrm{s}$ time window (Fig. 1 and Tab. S1) in order to have a
sufficient number of cycles to perform dynamical systems analysis. We do not
extend further this window to limit friction evolution effects associated with
shear fabric development and wear. Throughout each experiment, the loading
velocity $v_{0}$ and the applied normal stress $\sigma_{n}$ are kept constant
to some precision using servo-control. Their mean values and standard
deviations are reported in Tab. S1. The nominal loading velocity was
$v_{0}=10$ $\mu$m/s for all experiments. Data are sampled every $\Delta
t=0.01\textrm{ s}$.

Figure 1: (a) Sketch of the biaxial apparatus (BRAVA) used for the experiments
[7]. (b) Frictional shear stress time series ($\tau_{f}$). The system shows
stick-slip behavior at different values of applied normal stress
$\sigma_{n0}$. Red/green dots represent local max/min values of $\tau_{f}$.
Orange dots are the closest points during a slip event to the average value of
the shear stress $\langle\tau_{f}\rangle$. (c) Zoom of the black dashed box of
panel (b). Interevent times are taken between the orange dots to reduce
sensitivity to measurement noise associated with using max/min values and
define the returning of the trajectory to a well specified region of the phase
space (i.e., the hyperplane $\tau_{f}=\langle\tau_{f}\rangle$). Note that the
dynamics is richer when $\frac{k}{k_{c}}\sim 1$ (intermediate values of
$\sigma_{n}$), without the presence of a single characteristic labquake.

To infer the dimensionality of the system we use the scalar time series of the
frictional shear stress ($\tau_{f}$, Fig. 1b-c) since it is the measure with
the smallest noise contribution (Tab. S2). We calculate the Lyapunov dimension
($D_{KY}$) via the Kaplan-Yorke conjecture [12], using the Lyapunov spectrum
calculated with the method of [13] (Fig. 2a-b). We also estimate the
information dimension ($D_{1}$) as the average instantaneous dimension,
exploiting recent results in extreme value theory applied to dynamical systems
[14, 15]. With both techniques we find in most cases relatively small
dimensions ($<5$) (Fig. 2b), suggesting that a reduced order model may suffice
to explain the observations. Details on the two approaches are provided in
section S1 of the Supplementary Material. Remarkably, similar low
dimensionality was observed for slow slip events in nature [9], suggesting
that it might be a common feature of frictional faulting across multiple
spatio-temporal scales.

We can now seek a low-dimensional model to explain the observations. For a
spring-slider model obeying RS friction, the number of variables needed to
explain the dynamics (i.e., the number of dofs) is equal to 2 plus the number
of internal state variables. We notice that the dimension of the system tends
to decrease with higher $\sigma_{n}$, i.e. we observe smaller dimension for
faster ruptures (Fig. 2b). This observation points towards the fact that
inertial effects can be neglected for the conditions of our experiments. In
fact, if inertial effects were important, we would expect fast labquakes to
have a higher dimension than slow events. We thus deem reasonable to introduce
a radiation damping approximation, where the inertial term is replaced by a
viscous term to represent energy outflow as seismic waves [16]. This
approximation reduces the number of dofs of the system by one. For a non-
inertial spring-slider system with RS friction and a single state variable it
is impossible to get a bifurcation in the slip behavior: the phase space is
2-dimensional, and a closed orbit cannot split into two slipping modes without
trajectory intersections occurring in the phase space. An extra variable is
needed.

Figure 2: (a) Lyapunov spectrum estimated from $\tau_{f}$ laboratory data.
Legend is shared with panel (c). (b) Lyapunov dimension ($D_{KY})$ and
information dimension ($D_{1}$) estimated from $\tau_{f}$ time series. (c) and
(d): bifurcation diagrams using the interevent times of the previous
($T_{pre}$) and next ($T_{next}$) events. Figure 3: Time evolution of shear
stress ($\tau_{f}$, top row), normal stress ($\sigma_{n}$, middle row), and
layer thickness ($H$, bottom row) for experiments b726 (first column), b698
(second column) and i417 (third column). Red line in the second row shows
cumulative sum of the de-meaned $\sigma_{n}$ and clearly reveals perturbations
associated with labquakes. Fourth column shows zoom (blue dashed lines in
column 3) of experiment i417. Note drop in $\sigma_{n}$ of $\sim 0.3$ MPa and
in $H$ of $\sim 2~{}\mu$m.

In the classic spring-slider analysis, $\sigma_{n}$ is assumed to be constant,
but experimental data shows otherwise (Fig. 3). We notice a clear variation in
$\sigma_{n}$ when $\tau_{f}$ drops, followed by a relaxation to recover the
imposed target value of $\sigma_{n0}$. Also the layer thickness ($H$) varies
systematically over the labquake cycle, showing both a relaxation after a
shear stress drop and a clear reduction (i.e., compaction) with time (Fig. 3).
The variations in $H$ are a function of relaxation of shear-induced dilation
[17], layer thinning [18], and the drop in $\sigma_{n}$ that occurs as the
servo-control system responds to the labquake. The exact partition of layer
compaction between these causes is unknown. Thus, we account for normal stress
variations and dilation using the formalism of [19] for porous materials,
where the porosity ($\phi$) and the pore pressure ($p$) are introduced as
variables in the dynamics and linked to $\sigma_{n}$ via Terzaghi’s principle
[20]. The only difference that we introduce is the possibility for the
characteristic distance over which the porosity relaxes ($L_{\phi}$) to differ
from the RS friction characteristic slip distance ($L$). The model is
summarized by the following set of non-dimensional ordinary differential
equations (ODE) (see section S2.1 for its derivation):

$\displaystyle\dot{x}$
$\displaystyle=\frac{e^{x}\left[\left(\beta_{1}-1\right)x\left(1+\lambda
u\right)+y-u\right]+\kappa\left(\frac{v_{0}}{v_{{}*{}}}-e^{x}\right)-\dot{u}\frac{1+\lambda
y}{1+\lambda u}}{1+\lambda u+\nu e^{x}}$ (1a) $\displaystyle\dot{y}$
$\displaystyle=\kappa(\frac{v_{0}}{v_{{}*{}}}-e^{x})-\nu e^{x}\dot{x}$ (1b)
$\displaystyle\dot{z}$ $\displaystyle=-\rho e^{x}(\beta_{2}x+z)$ (1c)
$\displaystyle\dot{u}$ $\displaystyle=-\alpha-\gamma u+\dot{z}$ (1d)

The state vector $\pmb{\zeta}$, i.e. that vector that fully characterizes the
state of the system, is made of the non-dimensional variables
$[x,y,z,u]=[\ln\left(\frac{v}{v_{{}*{}}}\right),\frac{\tau_{f}-\tau_{0}}{a\sigma_{n0}},\frac{1}{\lambda\beta\sigma_{n0}}\left(\phi-\phi_{0}\right),-\frac{1}{\lambda}\frac{p}{\sigma_{n0}}]$,
where $v_{{}*{}}$ is a reference sliding velocity,
$\tau_{0}=\mu_{0}\sigma_{n0}$ is the product of the reference friction
coefficient $\mu_{0}$ and the reference normal stress $\sigma_{n0}$, $\lambda$
is equal to $\frac{a}{\mu_{0}}$ with $a$ being the RS direct effect parameter
(i.e., the instantaneous response of the friction coefficient to a sudden step
in the sliding velocity), $\beta$ is the product of the elastic component of
porosity and the combined compressibility of the fluid (air) in the pores and
the elastic pores, and $\phi_{0}$ is a reference porosity. The other
parameters appearing in system (1) are the ratio between the RS evolutionary
and direct effect parameters $\beta_{1}=\frac{b}{a}$, the non-dimensional
spring stiffness $\kappa$, viscous parameter $\nu$, dilatancy coefficient
$\beta_{2}$, pore pressure in the surrounding $\alpha$, diffusivity $\gamma$,
and the ratio $\rho=\frac{L}{L_{\phi}}$.

We adopt values for the frictional parameters derived from [7] who used the
same material as for our tests. We adjust the values of
$\beta_{2},\gamma,\rho$ and $\nu$ in order to reproduce the period doubling as
observed in our experiments (Fig. 1, section S2.2 and Tab. S4). For the used
parameters, the solution of the ODE system does not show chaotic behavior, and
the shear stress drops repeat periodically with either a single characteristic
labquake or a characteristic sequence of slow and fast labquakes in a row
(Fig. S2). In other words, with the adopted parameters the system exhibits a
cyclic attractor and a limit cycle. We notice that a quasi-static spring-
slider system with two RS state variables is a particular case of the set of
ODE previously derived (see section S2.3). Such a system shows the typical
behavior of the route to chaos when sufficiently reducing the stiffness of the
system $\kappa$ (i.e., when increasing the reference normal stress
$\sigma_{n0}$) [21]. The range of values for which deterministic chaos is
observed is quite narrow ($0.06840\lesssim\kappa\lesssim 0.06864$), and
further reducing $\kappa$ leads to unstable behavior and divergence of the
solution [21]. We find chaotic behavior for the system of eqs. (1) for only a
narrow range of $\sigma_{n0}$ values. The window over which such chaotic
behavior is observed depends on the selected parameters. For the adopted
parameters, such range is smaller than the minimum step in $\sigma_{n0}$, so
we do not further investigate this matter here, but we do not exclude that
part of the observed complexity may be due to this transition to deterministic
chaos. We note that the radiation damping term allows one to explore a larger
range of values with respect to the quasi-static case, and the modifications
introduced in system (1) allow us to reproduce the major feature of the data
consisting in period doubling followed by a return to a single characteristic
labquake cycle as $\sigma_{n0}$ increases (Fig. S2). To obtain these results
we had to use a radiation damping term 35 times larger than what is typically
adopted to simulate earthquakes. The classically adopted value assumes an
infinite fault plane in an elastic full space and radiation due only to
outgoing planar shear waves [16]. These assumptions may be false in laboratory
samples of few cm, and the results suggest that in the laboratory a larger
fraction of energy is dissipated with respect to this idealized condition.

Period doubling in the $\tau_{f}$ time series was obtained by [10] using a
boundary integral software for seismic cycle simulations. Such model contains
many more dofs with respect to the simple spring-slider model that we have
implemented, and many more than those deducible from the observed time series.
Nonetheless, our model better describes the actual shape of the frictional
shear stress time series (compare Figs. S2a and S3 with Fig. 1b), reproduces
the bifurcation at the same normal stress values actually imposed in the
laboratory, and takes into account variations in normal stress (even if not
perfectly, see Fig. S4). We thus prefer to use the low-dimensional model to
describe the observations.

We further notice that, differently from the ODE simulations, the laboratory
data are not exactly periodic. This can be seen either directly from the time
series of $\tau_{f}$, or from Fig. 2 where we plot for each event the time to
the next event ($T_{next}$) vs the time since the previous event ($T_{pre}$)
(panel c) and the ratio $\frac{T_{pre}}{T_{next}}$ as a function of
$\sigma_{n0}$ (panel d). In those plots we clearly see a spread of points
around the values that would represent periodic cycles of loading and failure.
Another way to measure the aperiodicity of the cycles is via the Coefficient
of Variation (CV), defined as the ratio between the standard deviation and the
mean of the observed interevent times. The observed CV are reported in Fig.
S5a. Experiments close to the transition regime (i.e., $\frac{k}{k_{c}}\sim
1$) show high CVs, indicating a higher complexity than experiments more
distant from such a critical state. Nonetheless, also experiments with a
single characteristic labquake do not have a null CV. We notice that
$\tau_{f}$ and $\sigma_{n}$ are subject to noise fluctuations (Fig. 3). These
fluctuations can be described by a Wiener process $W_{T}$ of intensity
$\varepsilon_{\tau_{f}}$ and $\varepsilon_{\sigma_{n}}$, respectively. We
estimate these intensities using the standard deviation of the time series
relative to the experiment at $\sigma_{n0}=13.6$ MPa. In particular, we use
$\varepsilon_{\tau_{f}}=\textrm{std}(\tau_{f}-\tilde{\tau}_{f})$ and
$\varepsilon_{\sigma_{n}}=\textrm{std}(\sigma_{n})$, where $\tilde{\tau}_{f}$
is a cubic spline approximation used to smooth the time series and filter out
the low frequencies. We obtain values of $\varepsilon_{\tau_{f}}\simeq 0.004$
MPa and $\varepsilon_{\sigma_{n}}\simeq 0.006$ MPa. Adding only observational
noise to the ODE simulations does not explain the observed CV variability
(Fig. S6). Introducing such noises in terms of $y$ and $u$ leads to a set of
stochastic differential equations (SDE) with eqs. (1b) and (1d) being modified
into (section S2.1):

$\displaystyle\textrm{d}y$
$\displaystyle=\left[\kappa\left(\frac{v_{0}}{v_{{}*{}}}-e^{x}\right)-\nu
e^{x}\dot{x}\right]\textrm{d}T+\varepsilon_{y}\textrm{d}W_{T}$ (2a)
$\displaystyle\textrm{d}u$ $\displaystyle=\left[-\alpha-\gamma
u+\dot{z}\right]\textrm{d}T+\varepsilon_{u}\textrm{d}W_{T}$ (2b)

The simulated time series are shown in Fig. S7. Of course, also $v$ and $\phi$
(i.e., $x$ and $z$) may be subject to noise that can perturb the dynamics of
the system. Contrary to $\tau_{f}$ and $\sigma_{n}$, we do not directly
measure them. Before introducing further complications, we consider the noises
on $\sigma_{n}$ and $\tau_{f}$ as the most relevant ones. To mimic the
laboratory time series, we finally add measurement noise of
$\varepsilon_{\tau_{f}}$ to the SDE generated $\tau_{f}$ time series (Fig.
4a). The Lyapunov spectrum, the system dimension, and the interevent times are
shown in Figs. 4b-e. We find a Lyapunov spectrum similar to the one derived
from the data, $D_{KY}$ between 3 and 4, and both $D_{KY}$ and $D_{1}$
decreasing with increasing $\sigma_{n0}$ (see Fig. 2b for comparison).
Furthermore, in the bifurcation diagrams we see a spread similar to the one
observed in the laboratory time series. This result suggests that the
aperiodicities in the returning time of laboratory earthquakes (of any type,
either slow or fast or mixed) could be the result of a stochastic noise
component that enters the dynamics and gets amplified due to the
nonlinearities of the equations. As a consequence, with the current available
set up, stick-slip cycles near the critical transition regime obtained from
laboratory experiments can be reproducible only statistically and not
deterministically, in the sense that, with the same rocks and conditions
applied, we can expect to reproduce similar average interevent times and
standard deviations, but not the same exact sequence of ruptures. Similar
observations were made for a different frictional system of labquakes [22],
and this type of feedback between the apparatus’ vibrations and the system
dynamics has been documented for turbulent flows [14]. These behaviors may be
a more general characteristic of nonlinear frictional systems.

Figure 4: (a) Frictional shear stress $\tau_{f}$ simulated using the system of
SDE with stochastic terms added to $y$ and $u$, and addition of observational
noise $\varepsilon_{\tau_{f}}$ on $\tau_{f}$. Imposed $\sigma_{n0}$ to mimic
experiments b726, b698, b728, b721 and i417. (b)-(e): As Fig. 2, but for SDE
simulated time series with stochastic terms added to $y$ and $u$ and the
addition of observational noise $\varepsilon_{\tau_{f}}$ to $\tau_{f}$.

Our results point towards the existence of a random attractor (Fig. 5) to
describe the seismic cycle: minimal variations of the order of less than 1‰ on
the shear and normal stresses applied to the fault (intended as the
experimental frictional interface) influence the large scale dynamics and the
recurrence time of a rupture, inducing CV of a few percent points ($>3\%$,
Fig. S5a). The implications on natural faults are quite relevant. At
seismogenic depths ($\sim 10$ km), variations of less than 1‰ of the
lithostatic stress would correspond to hundreds of kPa or less. Possible
causes that can generate stress perturbations of 10-100 kPa are: other
tectonic sources (with both static and dynamic stress variations [23]),
magmatic intrusions in volcano-tectonic environments [24], surface atmospheric
loading [25, 26], anthropogenic activity, and solid tides [27]. The laboratory
experiments here considered were conducted to explore the transition regime
$\frac{k}{k_{c}}\sim 1$, which corresponds to regions of transition between
linearly stable to unstable behavior. Such a transition zone might be similar
to the one where episodic tremors are observed in nature, since we found that
labquakes at the transition regime have dimension similar to the one of slow
earthquakes in Cascadia [9]. Given that $k_{c}$ depends on $\sigma_{n}$ [6],
variations $\varepsilon_{\sigma_{n}}$ can influence the stability of the
system, making it sensitive to small stress perturbations. This suggests that
we should not treat faults as isolated systems, especially when modeling the
limits of the seismogenic zone in nature.

Figure 5: Phase space subregion $xyu$ of the ODE (top) and SDE (bottom)
simulations for the experiments b726, b698, b728, b721 and i417.

The lack of an appropriate deterministic description of all the aforementioned
stress perturbations may explain why earthquake forecast is a difficult task
and statistical methods are used [28, 29]. Nonetheless, the addition of the
available physical information is a fundamental aspect to improve our
forecasts. For example, a commonly used statistical distribution to describe
the recurrence of a characteristic earthquake is the one derived from the
study of a Brownian Relaxation Oscillator (BRO). This is a time and slip
predictable stick-slip system subject to a stochastic perturbation of its
“load state”. The distribution that describes the expected waiting time of the
next event, given the (physical) knowledge of the current “load state” of the
system, is the so-called Brownian Passage Time (BPT) distribution [30]. Our
results relying on laboratory experiments may explain why a mix of physical
knowledge and statistical methods like those obtained with a BPT model are
often used for earthquakes forecast [31]. Compared to the BRO, here we have
introduced the stochastic perturbations under a more rigorous physical
description of friction, i.e. under the RS framework using two length-scale
parameters. Despite a low average dimension of the attractor, the maximum
instantaneous dimension deduced from the data ranges from $\sim 12$ (for
experiment b727, $\sigma_{n0}=24.017$ MPa) to $\sim 47$ (for experiment b695,
$\sigma_{n0}=17.909$ MPa). This suggests that extra dofs are needed to fully
characterize the dynamics in some regions of the phase space. Supplying a
stochastic term to the dynamics is an admittance of our ignorance on how to
monitor and describe these extra dofs [32]. The maximum instantaneous
dimension increases from $\sim 2$ to $\sim 38$ when perturbing the dynamics
with a stochastic term, and it goes up to $\sim 83$ when introducing also
observational noise. The model we propose is one of the possible candidates
and it is based on the modelling of small scale fluctuations via stochastic
terms. This choice is not unique and deterministic descriptions could hold as
well. Whether solids or fluids should be phenomenologically modelled via
stochastic or deterministic equations is still an open problem [33, 34] which
affects, for example, the quality of weather forecasts and climate predictions
[35]. Furthermore, another limitation may come from the fact that
instabilities are introduced by inertial effects, that certainly play a
significant role for natural earthquakes. Despite these limitations, the
proposed description constitutes a step forward towards a model that can
explain major features of the laboratory shear stress time series and
reconcile the number of dofs that can be deduced from the observations.
Furthermore, this model can explain fluctuations of the labquakes stress drop
caused by the inclusion of stochastic perturbations to the dynamics, thus
contributing to the debate on stress drop scaling with earthquake size [36].
Thanks to the ODE and SDE models, we can estimate the effect of the stochastic
term to the predictability of the dynamics. We calculate the Lyapunov time
($t_{Lyap}$) using the maximum Lyapunov exponent of the spectrum obtained from
clean time series, i.e. without additional observational noise (Figs. S5b-c).
The limited $t_{Lyap}$ for ODE models is due to the finite $\Delta t$. For SDE
models $t_{Lyap}<1$ s. While there exist model-free techniques capable of
predicting the behavior of chaotic systems up to $8t_{Lyap}$ into the future
[37], as future steps we envision the application of data assimilation
approaches that can use the model here presented to advance the state of the
system. Similar to weather forecast, we think that ensemble forecast is the
most reasonable way to assess the future state of the system because of the
unavoidable stochastic terms that affect the dynamics. We do not exclude that
this model may also be used to improve laboratory earthquakes forecast with
machine learning approaches, generating at will synthetic data that resembles
the major characteristics of the observed time series and reducing the
shortcoming of limited training datasets.

###### Acknowledgements.

We thank Carolina Giorgetti, Luca Dal Zilio, Giacomo Pozzi, Lauro Chiaraluce
and Marco Maria Scuderi for insightful discussions.

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