Hirzebruch-type inequalities viewed as tools in combinatorics

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.


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.
• all points in P are collinear, or • there exists a line ℓ passing through exactly two points from 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 L = {ℓ 1 , ..., ℓ d } in the projective plane we denote by t r = t r (L) the number of r-fold points, i.e., points where exactly r-lines from the arrangement meet. Theorem 1.3 (Melchior). Let L = {ℓ 1 , ..., ℓ d } ⊂ P 2 R be an arrangement of d ≥ 3 lines. Assume that L is not a pencil, then 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 = χ(P 2 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 L = {ℓ 1 , ..., ℓ d } ⊂ P 2 R be an arrangement of d ≥ 3 lines. Assume that L is not a pencil and pick k ∈ Z ≥1 , then 2k r=2 (2k + 1 − r)t r ≥ 2k + 1 + 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.
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].
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 ballquotient 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 R ⊂ C.

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 P N for some N ∈ 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 2 1 (X) = ( 2 Ω X ) 2 , were Ω 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 where ω X := 2 Ω X is the canonical class. Now we define the Kodaira dimension κ(X) of X to be −∞ if P m (X) = 0 for all m > 0, otherwise κ(X) is equal to the minimum k such that the set {P m (X)/m k : m ∈ N} is bounded. We know that in the case of surfaces κ(X) ∈ {−∞, 0, 1, 2}, and all those surfaces for which κ(X) = 2 are called surfaces of general type. In other words, being a surface of general type means that ω X has a large number of sections, i.e., dim H 0 (X, ω ⊗m X ) ∼ c · m 2 , where c is a positive constant. As a simple example, the complex projective plane has e(P 2 ) = 3 and the canonical bundle is O P 2 (−3), which means that for all m > 0 we have P m (P 2 ) = 0, and this implies that κ(P 2 ) = −∞.
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 κ(X) ≥ 0. Then 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 ρ : X → Y is a finite surjective morphism between normal varieties. Denote by G the group of isomorphisms α : X → X so that ρ(α(x)) = ρ(x) for all x ∈ X. The group G is called the group of covering automorphisms of ρ. If G acts transitively on all fibers of our cover ρ, then the covering is called Galois or regular. We say that a branched covering ρ : X → Y is an abelian covering if ρ : X → 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. 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 ≥ 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 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 where n ≥ 2 is the exponent. One can show that this covering is of degree n d−1 with the Galois group (Z/nZ) d−1 . Obviously, it ramifies along x 1 · ... · x d = 0. Our main object of interest is the following fiber product: We can write X n even more explicitly. We know that there exists a projective transformation on P 2 C such that ℓ 1 = {x 1 = 0}, ℓ 2 = {x 2 = 0}, and ℓ 3 = {x 3 = 0}, so we can describe X n in a new coordinate system as By this explicit description, our surface X n is given by (d − 3)-homogeneous equations in P d−1 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 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 L iff p is a point of multiplicity mult p ≥ 3. We can resolve singularities of X n by one simultaneous blow-up τ : Y n → X n at all those points which correspond to singular points of L with multiplicities ≥ 3. Since Y n is a smooth complex projective surfaces, we can compute the Chern numbers, namely In the next step, quite cumbersome, one needs to check under which conditions on the combinatorics of L our surface has non-negative Kodaira dimension -it turns out that it is enough to assume that d ≥ 6, t d = t d−1 = 0, and n ≥ 3. This means that if L satisfies the above conditions, then we can apply the Bogomolov-Miyaoka-Yau inequality: . Let us define the following Hirzebruch polynomial Since H L (n) ≥ 0 for n ≥ 3, we can compute H L (3), which gives us which is Hirzebruch's inequality. Remark 2.5. It is natural to ask whether Hirzebruch's inequality is sharp, i.e., whether there exists a line arrangement A such that t 2 + t 3 = d + r≥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 Somesse [36] proved that for complex line arrangements, with equality if and only if L is the dual-Hesse arrangement of lines. This result, in particular, implies that if L is an arrangement of d ≥ 6 lines with t d = t d−1 = 0, then Observe that γ(L) = 3 implies that f 0 = d, and by the Erdös-de Bruijn Theorem [9] this condition forces 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, Remark 2.7. Hirzebruch's construction provides the whole series of inequalities depending on n ≥ 3. In particular, for n = 5 we obtain 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 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 ≥ 6 with t d = t d−1 = t d−2 = 0 and n ≥ 2, then our surface Y n has non-negative Kodaira dimension. The condition H L (2) ≥ 0 leads us to Remark 2.9. In the literature, we can find usually the following variant of Hirzebruch's inequality provided that d ≥ 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 2 1 (Y n ) ≥ 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 ∈ {4, 5} lines with t d = t d−1 = 0 also satisfies Hirzebruch's inequality (1).
Remark 2.12. Topologically, our branched covering ρ : X n → P 2 C is determined by the following defining map φ : 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 L = {ℓ 1 , ..., ℓ 6 } ⊂ P 2 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 ℓ 1 = {x = 0}, ℓ 2 = {y = 0}, and ℓ 3 = {z = 0}. We denote by ℓ i = a i x + b i y + c i z with i ∈ {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 ℓ 1 , ..., ℓ 6 . We know that X 2 is a smooth projective surface and it can be described as so our surface X 2 is a smooth complete intersection of 3 quadrics in P 5 C . This surface is wellknown 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 ℓ 1 , ..., ℓ 6 -it turns out that X 2 is the minimal desingularization of this covering [17, p. 770].

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).
C be a line arrangement with d ≥ 6 such that t r = 0 for r > 2d 3 . Then 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].
C be a line arrangement such that t r = 0 for r > 2d 3 . Then r≥2 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 > 2d 3 ): Let us now list examples of line arrangements 1 for which we obtain equality in (3) -our list is probably far away to be complete.
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 )) 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 ∈ Z ≥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 ∈ Z ≥3 ? Simple calculations reveal that combinatorics C satisfies the equality in (3), and if one can show that there exists m 0 ∈ Z ≥3 for which we can realize C over the complex numbers, then 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].
C be an arrangement of d lines. Pick a natural number n ∈ [3, ..., d) and assume that t r = 0 for r > d − n + 1. Then where s = min{2n, d − n}.

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].

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

Simplicial line arrangements
Let A = {H 1 , ..., H d } ⊂ R n be a central arrangement of d (linear) hyperplanes. We say that A is simplicial if every connected component of R n \ d i=1 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 P 2 R . Let us recall some numerical properties of simplicial line arrangements, and we assume from now on that our arrangements are irreducible: • 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 L be a simplicial line arrangement in P 2 R such that t 2 ≥ t 3 and t i = 0 for i ∈ {2, 3, x}. Then x ≤ 8.
Proof. Since t 2 ≥ t 3 and by Bojanowski's inequality (3.1), one has the following chain of inequalities: where the last equality follows from Melchior's inequality for simplicial line arrangements. Assume now that x ≥ 9, which implies that This allows us to deduce that Next, we present an application of one of Langer's inequalities providing a quadratic lower bound on max(t 2 , t 3 ) for simplicial arrangements [16,Theorem 5.2]. In order to give you some feeling about this result, let us recall that Erdös and Purdy [13] proved that if L is an arrangement of d ≥ 25 lines in the real projective plane such that t d = 0, then max(t 2 , t 3 ) ≥ d − 1.
Moreover, they also proved that if t 2 < d − 1, then t 3 ≥ 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.
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].

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 C be an arrangement of smooth curves in the complex projective plane, and let 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?