Towards the weighted bounded negativity conjecture for blow-ups of algebraic surfaces

In the present paper, we focus on a weighted version of the Bounded Negativity Conjecture which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a global constant. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.


Introduction
In the last years, negative curves on surfaces have been researched extensively because of their connection to many open problems. Among these, one cannot refrain from mentioning Nagata's conjecture [13] or the SHGH conjecture [6]. The present paper is devoted to yet another open question in the geometry of surfaces, namely the following folklore conjecture.
Conjecture 0.1 (Bounded Negativity Conjecture). For every smooth projective surface X over the complex numbers, there exists a nonnegative integer b ∈ Z such that C 2 ≥ −b for all integral curves C ⊂ X.
The Bounded Negativity Conjecture (BNC in short) has a long oral tradition, and it seems to date back to F. Enriques. In some cases, the conjecture is known to hold true, for instance when the anti-canonical bundle is Q-effective or when the surface is equipped with an endomorphism of degree d > 1. However, the problem acquires a completely different flavor when we start considering non-minimal surfaces, for example blow-ups of those surfaces for which the BNC is true. In those cases, it still appears that not much can be said.
After the proof that the BNC for integral curves is equivalent to the BNC for reduced divisors [1, Proposition 3.8.2], attention has been paid to configurations of curves and their so-called Hconstants [2]. This asymptotic invariant has the potential of simultaneously detecting the behavior of all blow-ups of a surface at all possible configurations of points on it.
In the present paper, we go back to focusing our attention on integral curves and bounding their negativity. In [1, Conjecture 3.7.1], the authors formulated a certain variant of the BNC, and here we propose a stronger version.
Conjecture 0.2 (Weighted BNC). For every smooth projective surface X over the complex numbers, there exists a nonnegative integer b w ∈ Z such that C 2 ≥ b w (X) · (C.H) 2 for all integral curves C ⊂ X and all big and nef line bundles H.
The importance of the BNC lies in the fact that, for instance, it implies positivity of the Seshadri constants of ample line bundles at all points of a give surface X [1, Proposition 3.6.2]. What this conjecture is asking for is a bound on the self-intersection of all integral curves on X that depends on both X and the degree of the curve C with respect of every big and nef line bundle over which the curve becomes positive. In other words, we want to bound the function "self-intersection", i.e., (−) 2 : integral curves on C −→ Z by means of a function that is quadratic in the degree with respect to every big and nef line bundle on X that does not contract C. Our paper aims at gathering evidence for the validity of this conjecture.
In particular, we provide bounds for the self-intersection numbers of irreducible and reduced curves on blow-ups of algebraic surfaces at mutually distinct points. The bounds depend on the degree of the curve with respect to a big and nef line bundle, and in fact it holds for several such bundles. The technical heart is Theorem 2.1, where we construct a line bundle on a blow-up Y of X at n distinct points that naturally arises from X. We prove this result by first showing a generalization of a result due to Sakai [16] and Orevkov-Zaidenberg [14], together with estimates on the Milnor numbers of isolated singularities. This provides a function that depends linearly on the degree with respect to a given line bundle, while the conjecture only predicts that such a function should be quadratic.

Generalization of Sakai-Orevkov-Zaidenberg
In this section, we are going to provide a generalization of the following result, proven independently by Sakai [16] and Orevkov-Zaidenberg [14]. Theorem 1.1. Let C be a reduced and irreducible curve in P 2 C of degree d having singular points p 1 , ..., p s . We denote by m p i and µ p i the corresponding multiplicity and the Milnor number of p i . If the logarithmic Kodaira dimension of P 2 C \ C is non-negative, then Our aim is to show that the above inequality holds true in a broader setting. Before we present the result, let us recall that one has the following variation on Max Noether's inequality [5,Satz 5,p. 835]. Theorem 1.2. Let X be a smooth complex projective surface and C ⊂ X an irreducible and reduced curve with singular points p 1 , ..., p s and for i ∈ {1, ..., s} we denote by µ p i the corresponding Milnor numbers. Denote by K X the canonical divisor of X, then Our approach is to mimic Sakai's idea. For an irreducible and reduced curve C, we denote by f : S → X the minimal sequence of blow-ups such that the total inverse image of C has normal crossings. Let {E 1 , ..., E n } be the set of exceptional curves for f , and we denote set D =C + i E i . For a singularity (C, p) we denote by: (1) m p is the multiplicity of (C, p); (2) r p is the number of branches of (C, p); where E p is the reduced exceptional divisor over the point p ∈ X. Let us recall that f * C =C + i m iĒi ,Ē i being the total transform of E i in S, while the reduced exceptional divisor satisfies E.C = p r p . Definition 1.3. For a singularity (C, p) we denote by (m 1 = m p , m 2 , ..., m n ) the sequence of multiplicities of all infinitely near points of p in f . We set We are now ready to show our version of the Orevkov-Sakai-Zaidenberg inequality, which we will employ in the study of the negativity of a surface carried out in Section 2.
Theorem 1.4. Let C be an irreducible and reduced curve in a smooth complex projective surface X having singular points p 1 , ..., p s . We denote by m i and µ i the corresponding multiplicities and the Milnor numbers of p i 's. Assume that the logarithmic Kodaira dimension of X \ C is non-negative, then one has Proof. Since |m(K S + D)| = ∅ for a certain positive integer m, thus we can use the logarithmic Miyaoka-Sakai inequality [15] for our pair (S, D), namely First of all, we have e(S) − e(D) = e(X) − e(C). Now we would like to compute (K S + D) 2 . Following the idea of Sakai, we can see that: This leads to by the logarithmic Miyaoka-Yau inequality. The above statement is equivalent to where the last equality follows from Milnor formulae, one has As it was pointed out explicitly in [14], we have the following inequality This implies which completes the proof.
2. Bounding negativity on surfaces with κ ≥ 0 In this section, we would like to bound the negativity of curves on an algebraic surface, in the direction of the weighted bounded negativity. Let X a smooth projective surface over the complex numbers, and let σ : Y −→ X be the blow-up of X at S = {p 1 , . . . , p n }, where the p i 's are mutually distinct points of X. The following result is the technical heart of the article.
Proof. Let us assume that our curve C is not one of the exceptional divisors. The projection of C to X isC := σ(C). By pulling-back to Y , we see that σ * C = C + E, where E = n i=1 m i E i is the total exceptional divisor coming from the multiplicities ofC at the p i 's.
We can write the elements of S as follows Using Theorem 1.4, one gets Let us observe that where in the inequality above we have used that µ p (C) ≥ m p (C)−1 2 for every isolated singularity p ∈C (see for instance [10, Theorem 1.8]). From this, we deduce that At this point, we need to get rid of the multiplicities, by replacing them with a suitable intersection number. Let us choose a very ample line bundle A ∈ Pic(X), and let ϕ A : X −→ P h 0 (A)−1 be the corresponding embedding. Then, the multiplicities m p i (C) are bounded by the degree ofC in the embedding ϕ A , i.e. m p i (C) ≤ (C.A). Therefore, it follows that The line bundle K X + 3nA might not be ample, but it becomes such upon replacing A with a multiple. This means that for a suitable choice of A, the adjoint line bundle ∆ := K X + 3nA is ample, thus This concludes the proof in case C is not one of the exceptional divisor. However, if C were to be one of the exceptional divisors, the bound above would still hold true, therefore we are done.
As a consequence, we immediately get a linear bound on the self-intersection of integral curves on all surfaces Y as above having the additional requirement that their Kodaira dimension is nonnegative.
Corollary 2.2. Assume X is a surface of non-negative Kodaira dimension. Then, in the setting above, there exists a big and nef line bundle Γ that bounds negativity, i.e.
for every integral curve C ⊂ Y . In other words, if we define deg Γ C := (C.Γ), then i.e. the negativity of C is bounded by a function that depends on X, the number of points we have blown-up and the Γ-degree of C.
Proof. The line bundle ∆ in the proof of Theorem 2.1 provides us with a degree function on NS(X). As a consequence, we obtain a choice of a degree-like line bundle of Y by setting Γ := σ * ∆. The line bundle Γ will never be ample (we are pulling back along a blow-up), but it is nevertheless big and nef. Hence we can use it to provide a weighted bound for the negativity on Y .
It is interesting to observe the following facts: • if X is a minimal surface, then the bound of the negativity of Y directly arises naturally from its minimal model; • the bound on the negativity is now linear in (C.Γ), while the weighted BNC predicts the existence of a quadratic bound.
3. Bounding negativity on blow-ups of P 2 In this section, we will study the problem of bounding negativity for blow-ups of P 2 . We present here two different approaches to find bounds for the intersection numbers for curves on blow-ups of the complex projective plane. We start with the first approach using Orevkov-Sakai-Zaidenberg's inequality.
Theorem 3.1. Let σ : Y −→ P 2 be the blow-up of P 2 at S = {p 1 , . . . , p n }, where the p i 's are distinct points of P 2 , and let C be an irreducible and reduced curve on Y . Then, where L is the pull-back of a line in P 2 .
Proof. In this case, there do exist curves for which the logarithmic Kodaira dimension of the complement is −∞. As it was shown by Wakabayashi [17], if D ⊂ P 2 C is an irreducible and reduced curve of degree d ≥ 4 having s ≥ 1 singular points, which is not a rational cuspidal curve with one cusp, then the logarithmic Kodaira dimension of P 2 C \ D is non-negative. Therefore, we can apply Theorem 1.4 to bound the self-intersection of these curves. In fact, it was pointed by Sakai [16] that the inequality in Theorem 1.1 holds for all irreducible and reduced curves D ⊂ P 2 of degree d ≥ 3 -it is enough to verify the remaining cases by simple computations.
Let C ⊂ Y be an irreducible an reduced curve, and let us denote byC its image under σ. If C.H ≥ 3, H being the class of a line in P 2 , then we can repeat the proof of Theorem 2.1 to obtain where L = σ * H. We are left to deal with curves C ⊂ Y whose imageC is either a line or a conic. For such curves, we have that However, due to the restriction on the degree,C is necessarily smooth and m p i (C) = 1 for all i = 1, . . . , s. Therefore, we find that C 2 ≥ 1 − n, and thus we have proven the result.
Our second approach to the problem allows us to improve our previous bound from Theorem 3.1, and this is a consequence of a classical result in the theory of algebraic curves [18,Theorem 7.22].
In the setting of Theorem 3.1, by using the inequality µ p ≥ (m p (C) − 1) 2 for p ∈ Sing(C), the Plücker-Tessier formula implies that (again, we use the notation in the proof of Theorem 2.1): which in turn shows that and we got a better constant than in the statement of Theorem 3.1.
We would like to conclude by making the following remark, which considers the case of a blow-up of P 2 at a set S of points in very general position. Assume that S = {p 1 , ..., p n } are points in very general position and we consider the blowing-up π : X → P 2 along S. Let C ⊂ X be an irreducible and reduced curve, and denote byC ⊂ P 2 its image. Then by [19, Lemma 1], one has: C 2 ≥ − min{m q 1 , ....m qs , m q ′ 1 , . . . , m q ′ t } ≥ −d, which means that in generic case the better bound C 2 ≥ −d holds for every irreducible and reduced curve C ⊂ X. Notice that this bound does not depend on the number of points that we have blown up the surface.

Bounding negativity on blow-ups of Hirzebruch surfaces
We denote by F m the m th Hirzebruch surface, and let us consider the case m = 1 only, so that F m is a minimal surface (F 1 is P 2 blown-up at one point). If F is the class of a fiber, and H is the tautological section of F m , then We would like to mimic the argument for blow-ups of P 2 . Let σ : Y −→ F m be the blow-up of F m at a set S = {p 1 , . . . , p n } of distinct points. Suppose that C ⊂ Y is a curve with the property that κ(Y \ C) ≥ 0, and letC be its image under σ. By the proof of Theorem 2.1, we get Now, the line bundle A := H + F is very ample by [4, Exercise IV.18 (2)], and it embeds F m into P m+3 as a surface of degree m + 2. Therefore, The line bundle ∆ := K X + 3nA is always very ample on F m , thus yielding a big and nef line bundle Γ := σ * ∆ on Y that bounds the negativity on Y : It is natural to ask for which classes of curves we can apply our lower-bound, and the answer is provided by the following Wakabayashi-type result [12,Theorem 1.4].
Theorem 4.1. On a Hirzebruch surface F m , let C be an irreducible curve of genus g and type (a, b) with b > 2, a > 2 − 1 2 bm, and a ≥ 0. Then • If g > 0, then the logarithmic Kodaira dimension of F m \ C is equal to 2.
• If g = 0 and C has at least three cusps, then the logarithmic Kodaira dimension of F m \ C is equal to 2. • If g = 0 and C at least two cusps, then the logarithmic Kodaira dimension of F m \ C is at least equal to 0.