OPTIMAL BOUND FOR THE NUMBER OF ( − 1 )-CURVES ON EXTREMAL RATIONAL

We give an optimal bound for the number of (−1)-curves on an extremal rational surface X under the assumption that −KX is numerically effective and having self-intersection zero. We also prove that a nonelliptic extremal rational surface has at most nine (−1)-curves.


Introduction.
Let X be a smooth projective rational surface defined over the field of complex numbers.From now on we assume that −K X is numerically effective (in short NEF, i.e., the intersection number of the divisor K X with any effective divisor on X is less than or equal to zero, where K X is a canonical divisor on X) and of selfintersection zero.
It is easy to see that X is obtained by blowing up 9 points (possibly infinitely near) of the projective plane.
Nagata [4] proved that if the 9 points are in general positions, then X has an infinite number of (−1)-curves (i.e., smooth rational curves of self-intersection −1).
Miranda and Persson [3] studied the case when the position of the 9 points give a rational elliptic surface with a section.They classified all such surfaces which have a finite number of (−1)-curves and called them extremal Jacobian elliptic rational surfaces.For each case, they gave the number of (−1)-curves.
We use the following notations: (i) ∼ is the linear equivalence of divisors on X; is the group of divisors on X; (iv) NS(X) is the quotient group Div(X)/∼ of Div(X) by ∼ (the linear, algebraic, and numerical equivalences are the same on Div(X) since X is a rational surface); (v) D • D denotes the intersection number of the divisor D with the divisor D , in particular the self-intersection of D is Following [3], we define a smooth rational projective surface having a finite number of (−1)-curves on it as an extremal rational surface.The extremal rational surfaces are classified by the following theorem which can be found in [1, Theorem 3.1, page 65].
Theorem 1.1.Let X be a smooth projective rational surface having −K X NEF and of self-intersection zero.Then the following statements are equivalent: (1) X is extremal; (2) X satisfies the following two conditions: (a) the rank of the matrix (C i •C j ) i,j=1,...,r is equal to 8, where {C i : i = 1,...,r } is the finite set of (−2)-curves on X; a (−2)-curve is a smooth rational curve of self-intersection −2; (b) there exist r strictly positive rational numbers a i , i = 1,...,r , such that From this theorem we deduce the following lemma.
Lemma 1.2.Let X be an extremal surface.With the same notation as Theorem 1.1, if all of the a i , i = 1,...,r , are strictly positive integers, then a (−1)-curve on X meets only one (−2)-curve C i in one point and necessarily the coefficient a i of C i must be equal to one.
Proof.Let E be a (−1)-curve on X.We have On the other hand, for every j ∈ {1, 2,...,r }, the intersection number of E with C j is a nonnegative integer.Therefore, there exists i ∈ {1, 2,...,r } such that a i E • C i = 1 and for every j ∈ {1, 2,...,r }, j ≠ i, E • C j = 0. Hence the lemma follows.
In this note, we give an optimal bound for the number of (−1)-curves on an extremal rational surface.Keeping the same notations as in Theorem 1.1, our result is as follows.
Theorem 1.3.Let X be an extremal rational surface.The number of (−1)-curves on X is bounded by the integer where [[ ]] denotes the greatest integer function.This bound is optimal.
2. The proof.Let X be a smooth projective rational surface such that K 2 X = 0, where K X is a canonical divisor of X.We assume that −K X is NEF, that is, K X •D ≤ 0 for every effective divisor D on X.
For each (r + 2)-tuple (p, q; n 1 ,...,n r ) of integers, where r is a strictly positive integer, we consider the set Ᏹ r , where {C i : i = 1,...,r } is the finite set of (−2)curves on X.We think of Ᏹ n 1 ,...,nr p,q as a set of elements of NS(X) with imposed intersection with the set of (−2)-curves like a linear system with imposed base points.We prove that if the set of (−2)-curves on X is maximal in a sense that will be explained in Proposition 2.1, then for each nonzero integer q, the set Ᏹ n 1 ,...,nr p,q has at most one element.
Proposition 2.1.Let X be a smooth projective rational surface having an anticanonical divisor −K X of self-intersection zero.If the set of (−2)-curves on X spans the orthogonal complement of K X , then for each (r + 2)-tuple (p, q; n 1 ,...,n r ) of integers, with q nonzero, the set Ᏹ n 1 ,...,nr p,q has at most one element.
Proof.If the set Ᏹ n 1 ,...,nr p,q is not empty, consider two elements [D] and [D ].First, we have D −D belongs to the orthogonal complement of r ) and the fact that the set of (−2)-curves on X spans the orthogonal complement of K X , we conclude that (D − D ) 2 = 0. Hence there exists a rational number m such that D = D + mK X .Furthermore D 2 = D 2 .Since q ≠ 0, we have m = 0 and hence D is linearly equivalent to D , that is, An immediate consequence is the following corollary.
Corollary 2.2.Let X be a smooth projective rational surface having an anticanonical divisor −K X of self-intersection zero.If the set of (−2)-curves on X spans the orthogonal complement of K X , then for two different (−1)-curves E and E on X, there exists i ∈ {1,...,r } such that The suggested bound is optimal for certain extremal rational surfaces (see Remark 2.3).Remark 2.3.It is interesting to know that for which extremal rational surfaces, the set of (−1)-curves is in one-to-one correspondence with i=r i=1 ([0,[[1/a i ]]]∩N) {(0,..., 0)}.For example, in the case of an extremal elliptic Jacobian rational surface [3, Table 5.1, page 544], the only such surfaces for which there is a bijection are (i) the surface X 22 which has the set {II, II * } as set of singular fibers; (ii) the surface X 211 which has the set {II * ,I 1 ,I 1 } as set of singular fibers.More generally, for a given extremal surface X, we ask: which r -tuple (n 1 ,...,n r ) of Remark 2.4.Let X be an extremal rational surface which is not elliptic, then we have the following facts: (1) the set of (−2)-curves on X is connected and hence has one of the three types of configurations Ã8 , D8 , or Ẽ8 .In all cases there are only nine (−2)-curves on the surface; r of integers belongs to the set i=r i=1 ([0,[[1/a i ]]] ∩ N) {(0,...,0)} which has exactly −1 + i=r i=1 (1 + [[1/a i ]]) elements.Consider the map φ defined from the set of (−1)curves on X to i=r i=1 ([0,[[1/a i ]]] ∩ N) {(0,...,0)}, it is given by φ(E) = (E • C i ) i=1,...,r for every (−1)-curve E on X. Corollary 2.2 confirm that φ is injective.Therefore, the first result of Theorem 1.3 holds.