The effect of points fattening on del Pezzo surfaces

In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $\mathbb{S}_r$ arising by blowing-up of $\mathbb{P}^2$ at $r$ general points, with $ r \in \{1, \dots, 8 \}$. Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide complete answer for del Pezzo surfaces.


Introduction
In this paper, we follow the approach to fat point schemes initiated by Bocci and Chiantini in [3]. The initial degree α(I) of a homogeneous ideal I ⊂ C[P n ] is the least degree t such that the homogeneous component I t in degree t is non-zero. Although this notion was known since 1981 (see [4]), Bocci and Chiantini used this invariant for the first time in order to study fat points subschemes in the projective plane.
This definition can be extended to symbolic powers I (m) of I, namely α(I (m) ) is the least degree t such that the homogeneous component (I (m) ) t in degree t is non-zero. The sequence α(I), α(I (2) ), α(I (3) ), . . .
is called the initial sequence of I. Let Z ⊂ P 2 (C) be the set of points and I be its radical ideal. By Nagata-Zariski theorem ( [7], Theorem 3.14) the ideal of scheme mZ is the m−th symbolic power of I. Bocci and Chiantini proved, among others, that sets of points Z in P 2 (C) such that α(I (2) ) − α(I) = 1, are either contained in a single line or form the so-called star-configuration. Results of Bocci and Chiantini have been generalized in [6] by Dumnicki, Szemberg and Tutaj-Gasińska. They were studying configurations of points in P 2 (C) with α(I (m+1) ) − α(I (m) ) = 1 for some m ≥ 2 and obtained their full characterization (see [6], Theorem 3.4). These considerations were extended for another types of spaces. Except for spaces P n the problem of points fattening was considered among others by me in [1] for the space P 1 × P 1 and by Di Rocco, Lundman and Szemberg in [5] for Hirzebruch surfaces (with appropriately modified definition of the initial degree).
The aim of this paper is to make similar classification with respect to points fattening on del Pezzo surfaces. In this papper a del Pezzo surface (over an arbitrary field) is a smooth surface X with the ample anticanonical bundle −K X .
In fact considerations on points fattening effect was initiated on del Pezzo surface P 2 (C) and this path of research was continued to another one, namely P 1 × P 1 . Over C there are exactly 10 del Pezzo surfaces: P 2 , P 1 × P 1 and 8 surfaces S r arising by blowing-up of P 2 in r general points, where 1 ≤ r ≤ 8. In this paper, we complete a picture for the last 8 del Pezzo surfaces. More precisely, for each of the surfaces S r we establish maximal integer m, such that α(I) = α(I (2) ) = · · · = α(I (m) ) = 1 holds and we describe subschemes realizing this sequence of equalities. We focus mainly on the smallest possible value of α(I), namely 1. In the case of the surfaces S 1 and S 2 we additionally give characterization of subschemes satisfying a more general condition, namely α(I (m) ) = α(I (m+1) ) · · · = α(I (m+a) ) for some integers m and a. We conclude our paper presenting a Chudnovsky-type inequlity.

Basic notions and auxiliary facts
The original definition of the initial degree given in [3] was extended in [5] for arbitrary smooth projective variety with an ample class. Definition 2.1. (Initial degree) Let X be a smooth projective variety with an ample line bundle L on X and let Z be a reduced subscheme of X defined by the ideal sheaf I Z ⊂ O Z . For a positive integer m the initial degree (with respect to L) of the subscheme mZ is the integer Analogously, the initial sequence (with respect to L) of a subscheme Z is the sequence The initial sequence is sequence of positive integers with the following properties: 1) The initial sequence is weakly growing, i.e., α(mZ) ≤ α(nZ) for n ≥ m.
Properties in Fact 2.2 are generally known facts, thus we take them for granted. The choice of a line bundle L strictly depends on the variety X. In the projective plane the α-invariant was taken with respect to the line bundle O P 2 (1). This line bundle is 1 3 of the anticanonical bundle −K P 2 = O P 2 (3). Similarly on P 1 × P 1 it is natural to work with the α-invariant taken with respect to the line bundle O P 1 ×P 1 (1,1). In this case the line bundle is half of the anti-canonical divisor as on The most natural choice of the line bundle on del Pezzo surfaces S r seems to be the ancticanonical bundle which is not divisible in the Picard group P ic(S r ).
The fattening effect can be also considered more generally for graded linear series.
be a graded linear system. We define Then we have the following property.
Proof. It follows immediately from the fact, that W • has more sections than V • . .
Remark 2.5. Note that if Z ⊆ S t \{E s+1 , . . . , E t }, then α W• (mZ) is α(mZ) counted on the surface S t , whereas α V• (mZ) can be regarded either as α(mZ) on S t or α(mZ) on S s . This allows us to compare α's computed on various del Pezzo surfaces, provided that it makes sense to consider the underlying set Z on both surfaces.
The study of sequences of minimal growth can be paralleled by the investigation of ratios of the type α(mZ) m .
There are some estimates for such quotients. The first result of this kind, α(mZ) m ≥ α(Z) + 1 2 appears in a work of Chudnovsky and it concerns finite sets of points Z in the projective plane (see [9], Proposition 3.1 and [8]). It was generalized to other spaces, i.e. P 3 , P 1 × P 1 and Hirzebruch surfaces (see [5], [6], [10] and [1]). All these inequalities are called by the common name Chudnowsky-type results. In Section 4 we present our result of this kind for del Pezzo surfaces S r with r ≤ 6.
Our paper is concluded by a comparison of points fattening effect for surface S 1 considered as a del Pezzo surface and on the other hand as a Hirzebruch surface.

The points fattening effect on S r
In this section we present some results concerning the fattening effect on S r . Let us recall that S r arises as the blowing-up of the complex projective plane in fixed r general points P 1 , ..., P r . We denote by f r : S r → P 2 the blow up, where E 1 , . . . , E r are the exceptional divisors. If r is fixed, then we write simply f instead of f r .
In further considerations we will use the following observations about blow-ups.
Remark 3.1. If F is a plane curve of degree 3k in P 2 passing through the points P 1 , . . . , P r , so that mult Definition 3.2. (Adapted transform) We keep the notation as in Remark 3.1.
The adapted transform of F is the divisor where F denotes the proper transform of F .
if Q / ∈ E 1 ∪ . . . ∪ E r . Furthermore, if equality holds in (2), then f r (D) is a union of lines through f r (Q).
First we consider the situation, when Q / ∈ E 1 ∪. . .∪E r . Since f r is an isomorphism except the points {P 1 , . . . , P r }, then mult fr(Q) (f r (D)) = m. The multiplicity of the singular point of the plane curve can be at most the degree of this curve, thus f r (D) may have at most 3k−tuple points, what finishes the proof of statement (2).
We assume now, that Q ∈ E i for some i ∈ {1, . . . , r}. Let us denote by F = f r (D).
But mult Q ( F ) ≤ mult P i (F ), thus we finally obtain the statement (1).
A natural consequence of Lemma 3.3 is the following property for surfaces S r .
for some positive integers m and t ≥ 3, then Z ⊂ E 1 ∪ · · · ∪ E r .
We begin with surfaces S 1 and S 2 .

Surfaces S 1 and S 2
Theorem 3.5. Let Z ⊂ S 1 be a finite set of points. Then the following conditions are equivalent Proof. The implication from i) to ii) is obvious. It is enough to consider the nonreduced curve F = 3L ⊂ P 2 for some line passing through the point P 1 . Indeed, it gives rise to In order to prove the reverse implication, let Z = {Q 1 , . . . , Q s } and we assume that Let us consider possible types of cubic curves in the projective plane and their adapted transforms. The curve F has to pass through the point P 1 and in order to get the highest possible multiplicities along the exceptional divisor E 1 it should have the highest possible multiplicity at P 1 . We have the following types of cubic curves in P 2 : a) an irreducible cubic (possibly singular); b) a union of an irreducible conic and a line; c) a union of three lines (possibly not distinct).
In the case a) the divisor A(F ) on S 1 has points of multiplicity at most two. In the case b) the highest possible multiplicity of a point on E 1 is three, this happens in the case when the line is tangent to the conic at point P 1 .
Let us pass to the case c). We know that the adapted transform of a curve F consisting of some triple line L has a quintuple point. Except this arrangement of three lines we never get the quintuple points, which completes the proof.
Remark 3.6. In fact one can weaken condition ii) in Theorem 3.5. Assuming Let us denote by F = f (D) having deg(F ) = 3k. Since F is of degree 3k its multiplicity at P 1 is at most 3k. Hence the multiplicity of E 1 in D is at most 2k. This contributes to the multiplicity of D at every point Q 1 , . . . , Q t . The remaining multiplicity at these points must come from components of F passing through P 1 at directions corresponding to Q 1 , . . . , Q t . We have On the other hand, since α(mZ) = k, it must be since otherwise one could find 3(k − 1) lines through P 1 . Their images in P 2 would show α(mZ) ≤ k − 1 contradicting the assumption. Combining (3) and (4) we get that 3k − 3 + 2kt − 2t + 4t < 3k + 2kt and thus t < 3 2 , what finally means that Z is a single point.
On S 1 , there also exist infinitely many sets satisfying a weaker condition, namely and these sets are not necessarily the same as in Theorem 3.5.
Theorem 3.7. Let Z ⊂ S 1 be a finite set of points and let m be a positive integer. Then the following conditions are equivalent: Proof. The sets in ii) satisfy the condition for example with m = 1 and m = 4 respectively. We will prove the opposite implication. Suppose now that Z = {Q 1 , . . . , Q t } is a set such that α(mZ) = · · · = α((m + 3)Z) = k for some integers k and t and let D ∈ |3kH − kE 1 | be a divisor such that mult Q i (D) ≥ m + 3 for any point Q i ∈ Z. Let us denote by F = f (D), with deg(F ) = 3k. In fact we can repeat reasoning used in the proof of Remark 3.6, but this time with the following estimates By (5) combined with (6) we get what gives t < 3. Proof. Suppose to the contrary that Z = {Q 1 , . . . , Q t } is such that α(mZ) = · · · = α((m + 5)Z) = k for some integers k and t, and let D ∈ |3kH − kE 1 | be a divisor such that mult Q i (D) ≥ m + 3 for Q i ∈ Z. Let F = f (D), with deg(F ) = 3k. In the spirit of the proof of Remark 3.6, we get estimates Combining (7) and (8) we obtain t < 1, but t is a positive integer, a contradiction.
One may also consider sets with three initial values equal to 1. The list of possible types gets much longer but the arguments used to obtain their classification are similar to those used above. We refer the interested reader to [11] (Subchapter 6.1).
In further considerations lines joining points P 1 , . . . , P r plsy an important role. From now on we denote by L ij the line passing through the points P i and P j for fixed distinct i, j ∈ {1, . . . , r}. We are ready to formulate analogous results concerning the fattening effect on S 2 .
Theorem 3.13. Let Z ⊂ S 2 be a finite set of points such that Z ⊂ E i for i ∈ {1, 2}. Then Z satisfies the condition α(mZ) = · · · = α((m + 3)Z) < α((m + 4)Z), for some positive integer m if and only if Z has the following form: either All theorems from 3.9 to 3.13 can be proved analogously as in the case of surface S 1 , or the reader can find alternative proofs in [11]. In the subchapter 2.4 of [11] reader can also find a description of sets satisfying the condition Question about the maximal integer a satisfying condition α(mZ) = α((m + 1)Z) · · · = α((m + a)Z) is not trivial and it is still an open problem for some of studied before surfaces. For example, fattening effect for the Hirzebruch surfaces is described in [5] only with respect to the condition α(Z) = · · · = α(mZ) = 1.
The available tools are not useful in the case of greater r. In particular, when considered sets have some points on the exceptional divisors. From that reason for remaining surfaces S r we only establish maximal m, for which α(Z) = · · · = α(mZ) = 1 hold and we describe sets Z with that property.

Surfaces
The natural sequence of inclusions between linear systems suggests, that the sequence of equalities α(Z) = α(2Z) = · · · = 1 should become shorter with r growing. In the case of r = 1 and r = 2 we had α(5Z) = 1 and we proved moreover, that there it is not possible to obtain more than five consecutive initial values equal.
We present now such a characterization for remaining surfaces S r .
Theorem 3.14. Let Z ⊂ S 3 be a finite set of points. The following conditions are equivalent: i) α(Z) = · · · = α(4Z) = 1; where C is an irreducible conic curve passing through the points P 1 , P 2 , P 3 , P 4 and L is the line tangent to C at the point P i for i ∈ {1, 2, 3, 4}.
where C is a conic passing through the points P 1 , . . . , P 5 and L i is a line tangent to C at the point P i , b) Q ∈ L ij ∩ L kl for pairwise distinct i, j, k, l ∈ {1, 2, 3, 4, 5}.
where C j is a conic curve determined by five points of P 1 , . . . , P 6 excluding P j and L ij is of course the line passing through the points P i and P j , but simultaneously L ij is the tangent line to the curve C j at the point P i .
Let us note that S 6 is the first example of surfaces S r , where the existence of a set Z satisfying the condition depends on the geometry of points P 1 , . . . , P 6 . For given six points in general position in the projective plane, there always exists a cubic curve consisting of three lines, passing through these points. But these lines do not have to intersect at one point (see Figure 1). It is a rather strong requirement.  Similarly when the cubic splits into a conic and a line. Each five points determine a conic curve in a unique way. But the line joining the sixth point with one of the previous five does not necessary need to be tangent to this conic (see Figure 2). It is also a situation, which may happen or not, and it depends of the arrangement of the starting six points (although they are always in general position). It is a quite interesting phenomenon. Especially that for remaining two surfaces S r the condition α(Z) = α(2Z) = α(3Z) = 1 in never satisfied.
where C ij is the irreducible conic passing through the five points of P 1 , . . . , P 7 , distinct from P i and P j .
Proofs of Theorems 3.14 to 3.18 are based on a review of plane cubics passing through the points P 1 , . . . , P r (analogously as in the proof of Theorem 3.5). Thus we skip details here. We refer the curious reader to [11].
The line bundle has the least number of sections of all line bundles L r considered so far, namely h 0 (L 8 ) = 1. For that reason we expected, that here. Thus the fact, that there exists a set Z, where α(Z) = α(2Z) = 1 was surprising. Proof. If Z = {Q} and f (Q) is double point on a cubic, distinct of any P i , then it is obvious, that Q has multiplicity 2 on S 8 and of course α(Z) = α(2Z) = 1. Let us focus on the opposite implication. Let D ∈ |L 8 | be such that mult Q (D) ≥ 2 for any point Q ∈ Z. The curve F = f (D) has degree 3 and passes through eight distinct points P 1 , . . . , P 8 in general position. Then F is irreducible. Irreducible cubic has at most one singular point and it can not be any of P ′ i s (general points). Thus it must be f (Q).
To finish the proof we need to show, that there always exists a singular cubic curve passing through 8 given general points.
Let us notice that cubics passing through 8 fixed points form a pencil, if no four points lie on a line and no seven lie on a conic. Since P 1 , . . . , P 8 are in general position, the family of cubics passing through these points is a pencil. We denote it by S. Every two cubics in S meet in nine points, thus the set {P 1 , . . . , P 8 } determines a new point. This point is determined uniquely (CayleyBacharach theorem, see [12], Theorem 1). Let us denote it by P 9 .
Let S be a blow-up of P 2 in all nine points P 1 , . . . , P 9 . Then S is the total space of the pencil S and we have the morphism ϕ : S → P 1 , whose fibers are the elements of S. Let e(·) be the topological Euler characteristic. Thus we have e(S) = e(P 2 ) + 9 = 12.
Suppose now, to the contrary, that ϕ has only smooth fibers. Then from the topological point of view we have where E is an elliptic curve. We have then e(S) = e(P 1 ) · e(E) = 2 · 0 = 0.
Thus S must contain singular fibers. Since the points P 1 , . . . , P 8 are in general position, these singular fibers are irreducible cubics, which ends the proof.

The Chudovsky-type result for surfaces S r
We conclude our considerations by a lower bound on the growth rate of the initial sequence for surfaces S r . We present a general estimate for sets Z satisfying the condition α(Z) ≥ 2. The assumption of very-ampleness of line bundle L r is significant, thus our result concerns surfaces S r with r ≤ 6. Proof. We recall first the notation L r = 3H − E 1 − · · · − E r . We assume that α(Z) ≥ 2. Let us denote α(Z) by α. We have h 0 (3mH −mE 1 −· · ·−mE r ) = 3m + 2 2 −r· m + 1 2 = (9 − r)m 2 + (9 − r)m + 2 2 .
We choose a minimal subset W ⊆ Z, such that α(W ) = α, i.e., there is no element in (α − 1) · L r . The minimality of W is taken with respect to the inclusion (thus there can be several sets satisfying this condition). It follows that the points in W impose independent conditions on the space of sections in |(α − 1) · L r |. Then We claim that |α · L r ⊗ I W | has no additional base points on S r , i.e. is not contained in W . Let W = {Q 1 , . . . , Q t }. For any Q i , there exists a curve C i ∈ (α − 1) · L r , such that C i does not vanish at Q i and it does vanish at all points in W \ {Q i }. Let s i denote the section in H 0 (S r , (α − 1) · L r ) corresponding to C i . Then the sections s 1 , . . . , s t form a basis of H 0 (S r , (α − 1) · L r ). Suppose that R i ∈ S r \ W is a base point of |α · L r ⊗ I W |. There exists a section s i ∈ {s 1 , . . . , s t } not vanishing at R. Indeed, otherwise R would be a common zero of |(α − 1) · L r )|, which is not possible by the choice of W . Since L r is very ample, the system |L r ⊗ I Q i | is then base point free away from Q i . Hence there exists a section s ∈ H 0 (S r , L r ⊗ I Q i ) not vanishing at R. Then, in particular, |α · L r ⊗ I W | has no base component. Thus is a section not vanishing at R. Let A ∈ |α(Z)L r | and B ∈ |α(mZ)L r |. Using Bezout theorem we obtain This ends the proof. Firstly let us observe that if Z is set from Theorem 3.5, then its initial sequence is of the form 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, . . .
The divisor F = 3kL for the line L passing through the P 1 and corresponding to the point Q ∈ E 1 , gives rise to D = 3k L + 2kE 1 ∈ |3kH − kE 1 | on the blow up S 1 and mult Q (D) = 5k for any Q ∈ Z. Hence α(5kZ) ≤ k for any positive integer k.
By Lemma 2.3 we conclude, that estimate α(mZ) m ≥ 1 5 concerns any initial sequence {α(mZ)} with α(Z) = 1 for all surfaces S r . In the case of surfaces S 1 and S 2 we were able to show that this estimate is optimal (in the sense that 1 5 is borderline value). Probably this estimate is not sharp for r ≥ 3.
The surface S 1 was considered with respect to the fattening effect in [5] as a Hirzebruch surface. An interesting phenomenon is that from the point of view of Hirzebruch surfaces the most natural choice of the reference line bundle for S 1 is while if we consider it as a del Pezzo surface, we work with the anticanonical line bundle, i.e., L 1 = 3H − E 1 .