The effective monoids of some blow-ups of Hirzebruch surfaces

We mainly give a numerical condition to ensure the finite generation of the effective monoids of some smooth projective rational surfaces. These surfaces are constructed from the blow-up of any fixed Hirzebruch surface at some special configurations of ordinary points. Under this numerical condition, we determine explicitly the list of all (-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1)$$\end{document} and (-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-2)$$\end{document}-curves. In particular, we complete a result obtained by Harbourne (Duke Math J 52(1):129–148, 1985) and another result obtained by the third author (C R Math 338(11):873–878, 2004). Moreover, the Cox rings of these surfaces are finitely generated. Our ground field is assumed to be algebraically closed of any characteristic.


Introduction
We are interested in characterizing the smooth projective rational surfaces whose effective monoids are finitely generated.This is the reason why we study the surfaces whose minimal models are the Hirzebruch ones; see [6], and also [16,40] and [34].For any smooth projective rational surface Z , the Néron-Severi group NS(Z) of Z is the quotient group of the group of divisors on Z modulo numerical equivalence, and it is a free finitely generated Z-module of finite rank ρ(Z ).A special subset of NS(Z) is the effective monoid Eff(Z) of Z , which is defined as the set of elements γ of NS(Z), such that there exists an effective divisor D on Z with γ is the class of D modulo numerical equivalence [26].It is well known that Eff(Z) has an algebraic structure of a monoid.The importance of studying the finiteness of the effective monoid of some rational surface appears clearly in the characterization of finite generation of the Cox ring of such surface; see [5][6][7]13,15,18,37] and [16].
Recall that the Cox ring, Cox(X ), of a projective variety X over an algebraically closed field k is the k-algebra given by Cox where Pic(X) is the Picard group of X , and H 0 (X, L) is the finite-dimensional k−vector space of global sections of L; for more details, see [4,27], and [28].In [7], we show the equivalence between the finite generation of the Cox ring of an anticanonical rational surface that satisfies the anticanonical orthogonal property, and the finite generation of its effective monoid.In [1,2,9,10,12,14,[20][21][22]24,25,[30][31][32][33]35,36,38,39] and [17], one may find more results about the finiteness of the effective monoids of some surfaces.Here, an anticanonical rational surface is a smooth projective rational surface whose complete anticanonical linear system is not empty [23], and we say that a surface has the anticanonical orthogonal property whenever every nef and effective divisor class (modulo numerical equivalence) on such surface that is orthogonal to an anticanonical class is the zero class.Thus, our surfaces are Harbourne-Hirschowitz ones; see [11].
In this article, we construct a family of smooth projective rational surfaces (see Sect. 2) whose sets of (−1)-curves and (−2)-curves are both finite under certain reasonable numerical condition; see Theorem 3.1.Consequently, we are able to infer the finite generation of the effective monoids of these surfaces; see Corollary 3.6.On the other hand, in Sect.4, we may observe that under the same numerical condition, these surfaces satisfy the anticanonical orthogonal property (see Lemma 4.1) and, therefore, their Cox rings are finitely generated (see Theorem 4.2).

The construction of a family of smooth projective rational surfaces
First, we remind some notion about Hirzebruch surfaces over an algebraically closed field k of any characteristic.Fix a non-negative integer n.The Hirzebruch surface n associated with n is the projectivization of the locally free sheaf It is well known that the set {C n , F} is a minimal set of generators of the Néron-Severi group NS( n ) of n as a Z-module, where C n is the class of a section C n of n (it is unique if n is positive, and in this case, that section is usually called the exceptional section), and F is the class of a fibre f of n .The intersection form on n is given by the three equalities (C n ) 2 = −n, (F) 2 = 0, and C n • F = 1; for more details, see for example [26], and [40].Furthermore, if D is a prime divisor on some smooth projective surface Z , then Supp(D) denotes the support of the invertible sheaf O Z (D) associated with D [26].
Finally, for fixed non-negative integers r 1 and r 2 , let G = C n + (n + 2) f where f is a fibre of n , and let for every j ∈ {1, 2, . . ., r 2 }; see Fig. 1.Next, we denote the blow-up of n at the zero-dimensional subscheme = {P 1 , P 2 , . . ., 1 The configuration of the points of Fig. 2 The blow-up of n at the points of where C n is the class of the total transform of C n , F is the class of the total transform of a fibre f of n , E P i is the class of the exceptional divisor corresponding to the point P i for every i ∈ {1, 2, . . ., r 1 }, and E Q j is the class of the exceptional divisor corresponding to the point Q j for every j ∈ {1, 2, . . ., r 2 }.The intersection form on X r 1 ,r 2 n is given by the following equalities: , and E p • E q = 0 for p, q ∈ , such that p = q.
Proof By the forthcoming Lemma 3.2, one can assume, without loss of generality, that r 1 and r 2 are positive.Let D be a (−2)-curve on X r 1 ,r 2 n , such that the class of D does not belong to b − an − =1 μ = 0, and From Eqs. ( 1) and ( 2), we have Therefore, the following inequality is satisfied: Then, we obtain after completing the square in a a − (r 1 r 2 − 2r 2 + nr 1 )b This implies that Thus, a and b are bounded.Indeed, by our hypothesis and the last inequality, b is bounded.To see that a is bounded, we use the fact that b is bounded and the inequality above Therefore, X r 1 ,r 2 n contains a finite number of (−2)-curves.For the finiteness of the set of (−1)-curves, let N be a class of a (−1)-curve on X r 1 ,r 2 n , so so we have the following two cases to study: This implies that Then, we obtain that is less than or equal to Therefore, we have the following inequality: and then Thus, completing the square in b and using our hypothesis, it follows that b is bounded.Consequently, a is bounded too.Therefore, in Case (1), X r 1 ,r 2 n contains a finite number of (−1)-curves.Now, assume that we are in Case (2) and This implies that Then, after completing the square in a, we get that is less than or equal to Therefore, we have the following inequality: Thus, a and b are bounded.Therefore, in case Case (2), X r 1 ,r 2 n contains a finite number of (−1)-curves.
The following lemma is the special case of Theorem 3.1, when r 1 and r 2 are zero.
The following result gives the list of (−1)-curves and (−2)-curves on the surface X n+4,r 2 Remark 3.7 Let R, P 1 , P 2 , P 3 , P 4 , P 5 be ordinary points of a nodal cubic D on the projective plane P 2 k , such that R is the singular point, P 1 , P 2 and P 3 are collinear, but P i , P 4 , and P 5 are not for every i = 1, 2, 3; see Fig. 3.The surface obtained as the blow-up of P 2 k at these 6 points has 21 (−1)-curves and only one (−2)curve, instead of 27 (−1)-curves and no (−2)-curves as in the case of six points in general position of P 2 k ; see [8,Table 1,p. 34] and also Table 2.It is worth nothing that this surface is the blow-up of 1 at the points P 1 , P 2 , P 3 , P 4 , P 5 .Moreover, allowing r 2 > 0, our result completes a result obtained by Harbourne in [19] and another result obtained by the third author in [29].Also, one may observe that blow-ups of P 2 k at the node of an irreducible cubic r 2 times do not affect the finite generation of the effective monoid.

The finiteness of the Cox ring of X r 1 ,r 2 n
In this section, we prove that the surface X r 1 ,r 2 n satisfies the anticanonical orthogonal property, and we use Theorem 3.1 to prove the finite generation of the Cox ring of X r 1 ,r 2 n .

Table 1
List of (−1) and Cardinality of the set of Cardinality of the set of List of (−1) and Cardinality of the set of Cardinality of the set of Cardinality of the set of b − an − Therefore −((n + 2) 2 + nr 2 + 4r 2 − nr 1 − r 1 r 2 )b 2 ≥ 0. Therefore, using our numerical condition, we infer that b is equal to zero, and from Eqs. ( 3) and (4), we get that a is equal to zero.Thus, D = 0. Therefore, we are done.
In the following theorem, the numerical condition (n + 2) 2 + nr 2 + 4r 2 − nr 1 − r 1 r 2 > 0 gives us a family of smooth projective rational surfaces whose Cox rings are finitely generated.

3 Remark 3 . 4 2 nExample 3 . 5 .Corollary 3 . 6
With notation as above.The (−1)-curves and (−2)-curves on X n+4,r 2 n are those given in Tables1, 2, and 3.Proof It follows from the bounds given in the proof of the last theorem.It is worth noting that all the (−1)-curves (which are not exceptional) and (−2)-curves on X n+4,r come from smooth curves in n for every non-negative integers n and r 2 .Consequently, we show that the surface X 4,10 0 has no (−2)-curves, as in the case of blowing up the projective plane P 2 k at points in general position.With the notation of Theorem 3.1, the surface X 4,10 0 has not (−2)-curves.However, it has 556 (−1)-curves.Now, we handle the finite generation of the effective monoid of X r 1 ,r 2 n With notation as above, if (n + 2) 2 + nr 2 + 4r 2 − nr 1 − r 1 r 2 is a positive integer, then the effective monoid of the surface X r 1 ,r 2 n is finitely generated.ProofIt follows from Theorem 3.1 and[31].

Theorem 4 . 2
With the above notation, if (n + 2) 2 + nr 2 + 4r 2 − nr 1 − r 1 r 2 is a positive integer, then the Cox ring of the surface X r 1 ,r 2 n is finitely generated.ProofIt follows from Theorem 3.1, Lemma 4.1, and Theorem 1 of[7].Acknowledgements The authors are extremely grateful to the referees for their suggestions to improve the readability of our paper.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.