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/lp/springer-journals/the-effective-monoids-of-some-blow-ups-of-hirzebruch-surfaces-BLe86Xsiwm
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- Springer Journals
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- Copyright © The Author(s) 2023
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Abstract
Arab. J. Math. https://doi.org/10.1007/s40065-023-00425-0 Arabian Journal of Mathematics G. Andablo-Reyes · B. L. De La Rosa-Navarro · M. Lahyane The effective monoids of some blow-ups of Hirzebruch surfaces Dedicated to Professor Brian Harbourne on the occasion of his 65th birthday Received: 16 February 2022 / Accepted: 27 February 2023 © The Author(s) 2023 Abstract 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) and (−2)-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. Mathematics Subject Classification 14J26 · 14C20 · 14C22 · 14C17 · 14Q20 1 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 [5–7,13,15,18,37]and [16]. G. Andablo-Reyes Facultad de Ciencias Físico Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Avenida Francisco J. Múgica S/N Edificio Alpha, Ciudad Universitaria, Colonia Felicitas Del Rio, C. P. 58040 Morelia, Estado de Michoacán de Ocampo, Mexico E-mail: andablo@umich.mx B. L. De La Rosa-Navarro Facultad de Ciencias, Universidad Autónoma de Baja California (UABC), Km. 103 Carretera Tijuana-Ensenada, C. P. 22860 Ensenada, Baja California, Mexico E-mail: brenda.delarosa@uabc.edu.mx M. Lahyane (B) Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Avenida Francisco J. Múgica S/N, Edi- ficio C-3, Ciudad Universitaria, Colonia Felicitas Del Rio, C. P. 58040 Morelia, Estado de Michoacán de Ocampo, Mexico E-mail: mustapha.lahyane@umich.mx 123 Arab. J. Math. 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(X) = H (X, L), L∈Pic(X) where Pic(X) is the Picard group of X,and H (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 [1,2,9,10,12,14,20–22,24,25,30–33,35,36,38,39]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). 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 associated with n is the projectivization of the locally free sheaf O 1 ⊕ O 1 (−n) of rank two on the projective line P . It is well known that the set {C , F} is a P P k k k minimal set of generators of the Néron–Severi group NS( ) of as a Z-module, where C is the class of n n n a section C of (it is unique if n is positive, and in this case, that section is usually called the exceptional n n section), and F is the class of a fibre f of . The intersection form on is given by the three equalities n n 2 2 (C ) =−n,(F) = 0, and C · F = 1; for more details, see for example [26], and [40]. Furthermore, if D n n is a prime divisor on some smooth projective surface Z , then Supp(D) denotes the support of the invertible sheaf O (D) associated with D [26]. Finally, for fixed non-negative integers r and r ,let G = C + (n + 2) f where f is afibreof ,and let 1 2 n n C C C G G G n n n f , f ,..., f , f , f ,..., f be r +r different fibres of .Now,let P , P ,..., P , Q , Q ,..., Q r 1 2 n 1 2 r 1 2 r 1 2 r 1 2 2 1 2 be ordinary points of in general position, such that P ∈ Supp(G) ∩ Supp(f ) \Supp(C ) for every n i n i ∈{1, 2,..., r }, and Q ∈ Supp(C ) ∩ Supp(f ) \Supp(G) for every j ∈{1, 2,..., r };see Fig. 1.Next, 1 j n 2 we denote the blow-up of at the zero-dimensional subscheme ={P , P ,..., P , Q , Q ,..., Q } n 1 2 r 1 2 r 1 2 r ,r 1 2 by X ; see Fig. 2. r ,r 1 2 A minimal set of generators of NS(X ) as a Z−module is the set {C , F , −E , −E ,..., −E , −E , −E ,..., −E }, n P P P Q Q Q 1 2 r 1 2 r 1 2 Fig. 1 The configuration of the points of 123 Arab. J. Math. Fig. 2 The blow-up of at the points of where C is the class of the total transform of C , F is the class of the total transform of a fibre f of , E n n n P is the class of the exceptional divisor corresponding to the point P for every i ∈{1, 2,..., r }, and E is the i 1 Q class of the exceptional divisor corresponding to the point Q for every j ∈{1, 2,..., r }. The intersection j 2 r ,r 1 2 2 2 form on X is given by the following equalities: C =−n, F = 0, C · F = 1, C · E = 0, F · E = 0, n n ω ω E =−1for every ω ∈ , and E · E = 0for p, q ∈ , such that p = q. p q r ,r 1 2 3 The finiteness of (−1)-curves and (−2)-curves of X r ,r 1 2 In this section, we prove that there are only a finite number of (−1)-curves and (−2)-curves of X under the assumption that (n + 2) + nr + 4r − nr − r r is positive. Here, a (−1)-curve, respectively, a (−2)-curve, 2 2 1 1 2 is a smooth rational curve of self-intersection −1, respectively −2. Using the notation of the last section, we present one of the our results: r ,r 2 1 2 Theorem 3.1 If (n +2) +nr +4r −nr −r r is positive, then X has only a finite number of (−1)-curves 2 2 1 1 2 n and (−2)-curves. Proof By the forthcoming Lemma 3.2, one can assume, without loss of generality, that r and r are positive. 1 2 r ,r r 1 2 2 Let D be a (−2)-curve on X , such that the class of D does not belong to {C − E , C + (n + 2)F − n n Q n j =1 j r r ,r r r 1 1 2 1 2 E }. One may write the class of D in NS(X ) as D = aC + bF − γ E − μ E , for P n n j P Q i j i =1 j =1 =1 r ,r some integers a, b,γ ,γ ,...,γ ,μ ,μ ,...,μ . We have the equalities D =−2and D · (−K ) = 0, 1 2 1 2 r 1 2 r 1 2 r r ,r 2 1 2 r ,r and −K = C − E + G is the class of an anticanonical divisor in NS(X ), where G is the 1 2 n Q n =1 r ,r 1 2 class of the strict transform of G in NS(X ); this implies the following equalities: r r 1 2 2 2 2 2ab − a n − γ − μ =−2, (1) j =1 =1 r r 2 1 b − an − μ = 0, and 2a + b − γ = 0, =1 j =1 2 2 since D =−2, D · C − E = 0and D · G = 0. n Q =1 Using the above equalities, and the fact that γ ≥ 0and μ ≥ 0for every j ∈{1, 2,..., r } and j 1 2a + b ∈{1, 2,..., r }, it is sufficient to prove that a and b are bounded. Now, let γ¯ = γ − for all 2 j j b − an r j ∈{1, 2,..., r }, and let μ ¯ = μ − for each ∈{1, 2,..., r }. Therefore, γ¯ = 0, 1 2 j r j =1 μ ¯ = 0, and =1 r r r r 1 2 1 2 2 2 (2a + b) (b − an) 2 2 2 2 γ + μ = γ¯ + μ ¯ + + . (2) j j r r 1 2 j =1 =1 j =1 =1 123 Arab. J. Math. From Eqs. (1)and (2), we have r r 1 2 2 2 (2a + b) (b − an) 2 2 2 γ¯ + μ ¯ = 2ab − a n + 2 − − . r r 1 2 j =1 =1 Therefore, the following inequality is satisfied: 2 2 2 2abr r − a nr r + 2r r − (b + 2a) r − (b − an) r ≥ 0. 1 2 1 2 1 2 2 1 Then, we obtain after completing the square in a 2 2 2 2 (r r − 2r + nr )b (r r − 2r + nr ) b 2r r − b r − b r 1 2 2 1 1 2 2 1 1 2 2 1 a − − − ≤ 0. 2 2 2 2 n r + nr r + 4r (n r + nr r + 4r ) n r + nr r + 4r 1 1 2 2 1 1 2 2 1 1 2 2 This implies that 2 2 2 2 (r r − 2r + nr ) b 2r r − b r − b r 1 2 2 1 1 2 2 1 + ≥ 0. 2 2 2 (n r + nr r + 4r ) n r + nr r + 4r 1 1 2 2 1 1 2 2 Therefore 2(n r + nr r + 4r ) 1 1 2 2 b − ≤ 0. (n + 2) + nr + 4r − nr − r r 2 2 1 1 2 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 2 2 2 2 (r r − 2r + nr )b (r r − 2r + nr ) b 2r r − b r − b r 1 2 2 1 1 2 2 1 1 2 2 1 a − − − ≤ 0. 2 2 2 2 n r + nr r + 4r (n r + nr r + 4r ) n r + nr r + 4r 1 1 2 2 1 1 2 2 1 1 2 2 r ,r 1 2 Therefore, X contains a finite number of (−2)-curves. r ,r 1 2 For the finiteness of the set of (−1)-curves, let N beaclass of a (−1)-curve on X , so N = aC + n n r r 1 2 bF − γ E − μ E , for some integers a, b,γ ,γ ,...,γ ,μ ,μ ,...,μ , such that N does j P Q 1 2 r 1 2 r j =1 j =1 1 2 r r 2 1 not belong to {C − E , C + (n + 2)F − E , F − E ,..., F − E , F − E ,..., F − n Q n P P P Q j i 1 r 1 j =1 i =1 r ,r E , E ,..., E , E ,..., E }. Then, N · (−K ) = 1and N =−1, so we have the following two 1 2 Q P P Q Q r 1 r 1 r 2 1 2 n cases to study: Case (1) N · (C − E ) = 1and N · G = 0, and n Q =1 Case (2) N · (C − E ) = 0, and N · G = 1. n Q =1 Assume that we are in Case (1), then r r r r 2 1 1 2 2 2 2 μ = b − an − 1, γ = b + 2a, and γ + μ = 2ab − a n + 1, =1 j =1 j =1 =1 2 2 since N · C − E = 1, N · G = 0and N =−1. n Q =1 2a + b b − an − 1 Now, let γ¯ = γ − for all j ∈{1, 2,..., r }, and let μ ¯ = μ − for each ∈ j j 1 r r 1 2 r r 1 2 {1, 2,..., r }. Therefore, γ¯ = 0, μ ¯ = 0, and 2 j j =1 =1 r r 1 2 2 2 (2a + b) (b − an − 1) 2 2 2 0 ≤ γ¯ + μ ¯ = 2ab − a n + 1 − − . r r 1 2 j =1 =1 This implies that 2 2 (2a + b) (b − an − 1) 2ab − a n + 1 − − ≥ 0. r r 1 2 123 Arab. J. Math. Then, we obtain that r r b − 2r b + nr b − nr 1 2 2 1 1 a − nr r + 4r + n r 1 2 2 1 is less than or equal to 2 2 2 r r − r b − r b + 2r b − r (r r b − 2r b + nr b − nr ) 1 2 2 1 1 1 1 2 2 1 1 + . 2 2 2 nr r + 4r + n r (nr r + 4r + n r ) 1 2 2 1 1 2 2 1 Therefore, we have the following inequality: 2 2 2 r r − r b − r b + 2r b − r (r r b − 2r b + nr b − nr ) 1 2 2 1 1 1 1 2 2 1 1 + ≥ 0, 2 2 2 nr r + 4r + n r (nr r + 4r + n r ) 1 2 2 1 1 2 2 1 and then 2(n + 4)b nr r + 4r + n r − nr − 4 1 2 2 1 1 b − − ≤ 0. 2 2 (n + 2) + nr + 4r − nr − r r (n + 2) + nr + 4r − nr − r r 2 2 1 1 2 2 2 1 1 2 Thus, completing the square in b and using our hypothesis, it follows that b is bounded. Consequently, a is r ,r 1 2 bounded too. Therefore, in Case (1), X contains a finite number of (−1)-curves. Now, assume that we are in Case (2),thatis, N · (C − E ) = 0, and N · G = 1, then n Q =1 r r r r 2 1 1 2 2 2 2 μ = b − an, γ = b + 2a − 1, and γ + μ = 2ab − a n + 1. =1 j =1 j =1 =1 2a + b − 1 b − an Let γ¯ = γ − for all j ∈{1, 2,..., r }, and let μ ¯ = μ − for each ∈{1, 2,..., r }. j j 1 2 r r 1 2 r r 1 2 Therefore, γ¯ = 0, μ ¯ = 0, and j =1 =1 r r 1 2 2 2 (2a + b − 1) (b − an) 2 2 2 0 ≤ γ¯ + μ ¯ = 2ab − a n + 1 − − . r r 1 2 j =1 =1 This implies that 2 2 (2a + b − 1) (b − an) 2ab − a n + 1 − − ≥ 0. r r 1 2 Then, after completing the square in a, we get that r r b − 2r b + nr b + 2r 1 2 2 1 2 a − nr r + 4r + n r 1 2 2 1 is less than or equal to 2 2 2 r r − r b + 2r b − r b − r (r r b + nr b + 2r − 2r b) 1 2 2 2 1 2 1 2 1 2 2 + . 2 2 2 nr r + 4r + n r (nr r + 4r + n r ) 1 2 2 1 1 2 2 1 Therefore, we have the following inequality: 2 2 2 2(nr + n + 2r + 2n)b nr r + 4r + n r − nr − n 2 2 1 2 2 1 2 b − + ≤ 0. 2 2 (n + 2) + nr + 4r − nr − r r (n + 2) + nr + 4r − nr − r r 2 2 1 1 2 2 2 1 1 2 r ,r 1 2 contains a finite number of (−1)-curves. Thus, a and b are bounded. Therefore, in case Case (2), X The following lemma is the special case of Theorem 3.1,when r and r are zero. 1 2 Lemma 3.2 With the above notation, the surface has finitely many (−1)-curves and (−2)-curves. 123 Arab. J. Math. Fig. 3 The configuration of the ordinary points of a nodal cubic of P Proof Since K = 4, is an anticanonical rational surface (see [3, Lemma 2.1, p. 3]). Then, the result holds from [3]and [31]. n+4,r The following result gives the list of (−1)-curves and (−2)-curves on the surface X . n+4,r Corollary 3.3 With notation as above. The (−1)-curves and (−2)-curves on X are those given in Tables 1, 2,and 3. Proof It follows from the bounds given in the proof of the last theorem. n+4,r Remark 3.4 It is worth noting that all the (−1)-curves (which are not exceptional) and (−2)-curves on X come from smooth curves in for every non-negative integers n and r . n 2 4,10 Consequently, we show that the surface X has no (−2)-curves, as in the case of blowing up the projective plane P at points in general position. 4,10 Example 3.5 With the notation of Theorem 3.1, the surface X has not (−2)-curves. However, it has 556 (−1)-curves. r ,r 1 2 Now, we handle the finite generation of the effective monoid of X . Corollary 3.6 With notation as above, if (n + 2) + nr + 4r − nr − r r is a positive integer, then the 2 2 1 1 2 r ,r 1 2 effective monoid of the surface X is finitely generated. Proof It follows from Theorem 3.1 and [31]. Remark 3.7 Let R, P , P , P , P , P be ordinary points of a nodal cubic D on the projective plane P ,such 1 2 3 4 5 that R is the singular point, P , P and P are collinear, but P , P ,and P are not for every i = 1, 2, 3; see 1 2 3 i 4 5 Fig. 3. The surface obtained as the blow-up of P 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 ; see [8, Table 1, p. 34] and also Table 2. It is worth nothing that this surface is the blow-up of at the points P , P , P , P , P . Moreover, allowing r > 0, our result completes a result obtained by Harbourne in [19] 1 2 3 4 2 and another result obtained by the third author in [29]. Also, one may observe that blow-ups of P at the node of an irreducible cubic r times do not affect the finite generation of the effective monoid. r ,r 1 2 4 The finiteness of the Cox ring of X r ,r 1 2 In this section, we prove that the surface X satisfies the anticanonical orthogonal property, and we use r ,r 1 2 Theorem 3.1 to prove the finite generation of the Cox ring of X . r ,r 1 2 r ,r Lemma 4.1 With notation as above, let D be the class of a nef divisor in NS(X ), such that D · K = 0. 1 2 If (n + 2) + nr + 4r − nr − r r is a positive integer, then D = 0. 2 2 1 1 2 123 Arab. J. Math. 4,r Table 1 List of (−1) and (−2)-curves of X 4,1 4,1 (−1)-curves of X Cardinality of the set of (−1)-curves on X 0 0 F − E , i ∈{1, 2, 3, 4} 4 F − E 1 E , i ∈{1, 2, 3, 4} 4 E 1 C − E , i ∈{1, 2, 3, 4} 4 0 P C − E 1 0 Q C + F − E − E , i , i ∈{1, 2, 3, 4} 6 0 P Q 1 2 i 1 =1 C + F − E , i , i , i ∈{1, 2, 3, 4} 4 0 P 1 2 3 =1 i C + 2F − E − E 1 0 P Q i =1 i 1 2C + F − E − E 1 0 P Q i =1 i 1 4,1 4,1 (−2)-curves of X Cardinality of the set of (−2)-curves on X 0 0 4,1 There are no (−2)-curves on X Zero 4,r 4,r 2 2 (−1)-curves of X with r = 1 Cardinality of the set of (−1)-curves on X with r = 1 2 2 0 0 F − E , i ∈{1, 2, 3, 4} 4 F − E , j ∈{1, 2,..., r } r Q 2 2 E , i ∈{1, 2, 3, 4} 4 E , j ∈{1, 2,..., r } r Q 2 2 C − E , i ∈{1, 2, 3, 4} 4 0 P C + F − E − E , i , i ∈{1, 2, 3, 4}, j ∈{1, 2,..., r } 6r 0 P Q 1 2 2 2 =1 i j C + F − E , i , i , i ∈{1, 2, 3, 4} 4 0 P 1 2 3 =1 i 3 2 2 C + 2F − E − E , i , i , i ∈{1, 2, 3, 4}, j , j ∈{1, 2,..., r } 4 0 P Q 1 2 3 1 2 2 i j =1 =1 C + 2F − E − E , j ∈{1, 2,..., r } r 0 P Q 2 2 i =1 i j 4 3 2 C + 3F − E − E , j , j , j ∈{1, 2,..., r } 0 P Q 1 2 3 2 i =1 i =1 j 2C + F − E − E , j ∈{1, 2,..., r } r 0 P Q 2 2 i j i =1 4,2 4,2 (−2)-curves of X Cardinality of the set of (−2)-curves on X 0 0 C − E − E 1 0 Q Q 1 2 4,r 4,r 2 2 (−2)-curves of X with r = 1, 2 Cardinality of the set of (−2)-curves on X with r = 1, 2 2 2 0 0 4,r There are no (−2)-curves on X with r = 1,2Zero 0 Arab. J. Math. 5,r Table 2 List of (−1) and (−2)-curves of X 5,0 5,0 (−1)-curves of X Cardinality of the set of (−1)-curves on X 1 1 C 1 F − E , i ∈{1, 2,..., 5} 5 E , i ∈{1, 2,..., 5} 5 C + F − E , i , i ∈{1, 2,... , 5} 10 1 P 1 2 =1 i C + 2F − E , i , i , i , i ∈{1, 2,... , 5} 5 1 P 1 2 3 4 =1 i 2C + 2F − E 1 1 P i =1 i 5,0 5,0 (−2)-curves of X Cardinality of the set of (−2)-curves on X 1 1 5,0 There are no (−2)-curves on X Zero 5,r 5,r 2 2 (−1)-curves of X with r > 0 Cardinality of the set of (−1)-curves on X with r > 0 2 2 1 1 F − E , i ∈{1, 2,..., 5} 5 F − E , j ∈{1, 2,..., r } r Q 2 2 E , i ∈{1, 2,..., 5} 5 E , j ∈{1, 2,..., r } r Q 2 2 C + F − E , i , i ∈{1, 2,... , 5} 10 1 P 1 2 =1 i C + 2F − E − E , i , i , i ∈{1, 2,..., 5}, j ∈{1, 2,..., r } 10r 1 P Q 1 2 3 2 2 =1 i j 4 2 C + 3F − E − E , i , i , i , i ∈{1, 2,... , 5}, j , j ∈{1, 2,..., r } 5 1 P Q 1 2 3 4 1 2 2 =1 i =1 j 5 3 C + 4F − E − E , j , j , j ∈{1, 2,..., r } 1 P Q 1 2 3 2 i =1 i =1 j C + 2F − E , i , i , i , i ∈{1, 2,... , 5} 5 1 P 1 2 3 4 =1 i C + 3F − E − E , j ∈{1, 2,..., r } r 1 P Q 2 2 i =1 i j 2C + 2F − E 1 1 P i =1 5,1 5,1 (−2)-curves of X Cardinality of the set of (−2)-curves on X 1 1 C − E 1 1 Q 5,r 5,r 2 2 (−2)-curves of X with r > 1 Cardinality of the set of (−2)-curves on X with r > 1 2 2 1 1 Tere are no (−2)-curves Zero Arab. J. Math. n+4,r Table 3 List of (−1) and (−2)-curves of X with n ≥ 2and r ≥ 0 n 2 n+4,r n+4,r 2 2 (−1)-curves of X with n ≥ 2and r ≥ 0 Cardinality of the set of (−1)-curves on X with n ≥ 2and r ≥ 0 n 2 n 2 F − E , i ∈{1, 2,..., n + 4} n + 4 F − E , j ∈{1, 2,..., r } r Q 2 2 E , i ∈{1, 2,..., n + 4} n + 4 E , j ∈{1, 2,..., r } r Q 2 2 n + 4 n+1 C + nF − E , i ,..., i ∈{1, 2,..., n + 4} n P 1 n+1 =1 i n + 1 i ,..., i ∈{1, 2,..., n + 4} n + 4 n+2 1 n+2 C + (n + 1)F − E − E , r n P Q 2 =1 i j j ∈{1, 2,..., r } n + 1 i ,..., i ∈{1, 2,..., n + 4} r n+3 2 1 n+3 2 C + (n + 2)F − E − E , (n + 4) n P Q =1 i =1 j j , j ∈{1, 2,..., r } 2 1 2 2 n+4 3 2 C + (n + 3)F − E − E , j , j , j ∈{1, 2,..., r } n P Q 1 2 3 2 i =1 j i =1 n+3 C + (n + 1)F − E , i ,..., i ∈{1, 2,..., n + 4} n + 4 n P 1 n+3 =1 i n+4 C + (n + 2)F − E − E , j ∈{1, 2,..., r } r n P Q 2 2 i =1 i j 6,0 6,0 (−2)-curves of X Cardinality of the set of (−2)-curves on X 2 2 C 1 6,r 6,r 2 2 (−2)-curves of X with r > 0 Cardinality of the set of (−2)-curves on X with r > 0 2 2 2 2 6,r There are no (−2)-curves on X with r >0Zero n+4,0 n+4,0 (−2)-curves of X with n > 2 Cardinality of (−2)-curves on X with n > 2 n n n + 4 n+2 C + nF − E , i ,..., i ∈{1, 2,..., n + 4} n P 1 n+2 =1 i n + 2 n+4,r n+4,r 2 2 (−2)-curves of X with n > 2and r > 0 Cardinality of the set of (−2)-curves on X with n > 2and r > 0 n 2 n 2 i ,..., i ∈{1, 2,..., n + 4} n+3 1 n+3 C + (n + 1)F − E − E , r (n + 4) n P Q 2 =1 i j j ∈{1, 2,..., r } 2 Arab. J. Math. r ,r r r 1 2 1 2 Proof Let D be a class of a nef divisor in NS(X ), so D = aC + bF − γ E − μ E , n n j P Q j =1 =1 for some integers a, b,γ ,γ ,...,γ ,μ ,μ ,...,μ . Therefore, D ≥ 0, D · (C − E ) = 0, and 1 2 r 1 2 r n Q 1 2 =1 D · G = 0. From these, we have the following: r r 1 2 2 2 2 2ab − a n − γ − μ ≥ 0, (3) j =1 =1 2a + b − γ = 0, and (4) j =1 b − an − μ = 0. =1 2a + b b − an Now, let γ¯ = γ − for all j ∈{1, 2,..., r }, and let μ ¯ = μ − for each ∈{1, 2,..., r }. j j 1 2 r r 1 2 Therefore r r 1 2 2 2 (2a + b) (b − an) 2 2 2 0 ≤ γ¯ + μ ¯ ≤ 2ab − a n − − . r r 1 2 j =1 =1 This implies that 2 2 (2a + b) (b − an) 2ab − a n − − ≥ 0. r r 1 2 Consequently, after completing the square in a, we obtain that 2 2 2 2 (r r − 2r + nr )b (r r − 2r + nr ) b (r + r )b 1 2 2 1 1 2 2 1 1 2 a − ≤ − . 2 2 2 2 nr r + 4r + n r (nr r + 4r + n r ) nr r + 4r + n r 1 2 2 1 1 2 2 1 1 2 2 1 Therefore 2 2 −((n + 2) + nr + 4r − nr − r r )b ≥ 0. 2 2 1 1 2 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) + nr + 4r − nr − r r > 0 gives us a family 2 2 1 1 2 of smooth projective rational surfaces whose Cox rings are finitely generated. Theorem 4.2 With the above notation, if (n + 2) + nr + 4r − nr − r r is a positive integer, then the Cox 2 2 1 1 2 r ,r 1 2 ring of the surface X is finitely generated. Proof It 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/. 123 Arab. J. Math. Funding Gloria Andablo-Reyes and Mustapha Lahyane acknowledge a partial support from Coordinación de la Investigación Científica de la Universidad Michoacana de San Nicolás de Hidalgo (UMSNH) during 2022. Brenda Leticia De La Rosa-Navarro was supported by Programa para el Desarrollo Profesional Docente, para el Tipo Superior under Grant No. UABC-PTC-558. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Declarations Conflict of interest The authors declare that there are no conflicts of interest. References 1. Campillo, A.; Piltant, O.; Reguera-López, A.J.: Cones of curves and of line bundles at “infinity”. J. Algebra 293, 503–542 (2005) 2. Campillo, A.; Piltant, O.; Reguera-López, A.J.: Cones of curves and of line bundles on surfaces associated with curves having one place at infinity. Proc. Lond. Math. Soc. 84(3), 559–580 (2020) 3. Cerda Rodríguez, J.A.; Faila, G.; Lahyane, M.; Osuna Castro, O.: Fixed loci of anticanonical complete linear systems of anticanonical rational surfaces. Balkan J. Geom. Appl. 17(1), 1–8 (2012) 4. Cox, D.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995) 5. De La Rosa Navarro, B.L.; Frías Medina, J.B.; Lahyane, M.; Moreno Mejía, I.; Osuna Castro, O.: A geometric criterion for the finite generation of the Cox ring of projective surfaces. Rev. Mat. Iberoam. 31(4), 1131–1140 (2015) 6. De La Rosa Navarro, B.L.; Frías Medina, J.B.; Lahyane, M.: Rational surfaces with finitely generated Cox rings and very high Picard numbers. Rev. R. Acad. Cienc. Exact. Fís. Nat. Ser. A Math. RACSAM 111(2), 297–306 (2017) 7. De La Rosa Navarro, B.L.; Frías Medina, J.B.; Lahyane, M.; Moreno Mejía, I.; Osuna Castro, O.: Erratum to "A geometric criterion for the finite generation of the Cox ring of projective surfaces”. Rev. Mat. Iberoam. 33(1), 375–376 (2017) 8. Demazure, M.: Surfaces de Del Pezzo II–V. In: Demazure, M., Pinkham, H., Teissier, B. (eds.) Séminaire sur les Singularités des Surfaces, pp. 23–69. Springer, Heidelberg (1980) 9. Failla, G.; Lahyane, M.; Molica, Bisci G.: Rational surfaces of Kodaira type IV. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10(3), 741–750 (2007) 10. Failla, G.; Lahyane, M.; Molica Bisci, G.: The finite generation of the monoid of effective divisor classes on Platonic rational surfaces. In: Chéniot, D., Dutertre, N., Murolo, C., Trotman, D., Pichon, A. (eds.) Singularity Theory: Dedicated to Jean–Paul Brasselet on his 60th Birthday, pp. 565–576. World Sci. Publ, Hackensack (2007) 11. Frías-Medina, J.B.; Lahyane, M.: Harbourne–Hirschowitz surfaces whose anticanonical divisors consist only of three irre- ducible components. Int. J. Math. 29(12), 1850072-1–1850072-19 (2018) 12. Galindo, C.; Monserrat, F.: On the cone of curves and of line bundles of a rational surface. Int. J. Math. 15(4), 393–407 (2004) 13. Galindo, C.; Monserrat, F.: The total coordinate ring of a smooth projective surface. J. Algebra 284, 91–101 (2005) 14. Galindo, C.; Monserrat, F.: The cone of curves associated to a plane configuration. Comment. Math. Helv. 80(1), 75–93 (2005) 15. Galindo, C.; Monserrat, F.: The cone of curves and the Cox ring of rational surfaces given by divisorial valuations. Adv. Math. 290, 1040–1061 (2016) 16. Galindo, C.; Monserrat, F.; Moreno-Ávila, C.J.: Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces. Rev. Mat. Complut. 33(2), 349–372 (2020) 17. Galindo, C.; Monserrat, F.; Moyano-Fernández, J.J.; Nickel, M.: Newton–Okounkov bodies of exceptional curve valuations. Rev. Mat. Iberoam. 36(7), 2147–2182 (2020) 18. Giuffrida, S.; Maggioni, R.: The global ring of a smooth projective surface. Matematiche (Catania) 55(1), 133–159 (2000) 19. Harbourne, B.: Blowings-up of P and their blowings-down. Duke Math. J. 52(1), 129–148 (1985) 20. Harbourne, B.: Complete linear systems on rational surfaces. Trans. Am. Math. Soc. 289(1), 231–236 (1985) 21. Harbourne, B.: The geometry of rational surfaces and Hilbert functions of points in the plane. In: Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conference Proceedings, vol. 6, pp. 95–111. American Mathematical Society, Providence (1986) 22. Harbourne, B.: Rational surfaces with K > 0. Proc.Am. Math.Soc. 124(3), 727–733 (1996) 23. Harbourne, B.: Anticanonical rational surfaces. Trans. Am. Math. Soc. 349(3), 1191–1208 (1997) 24. Harbourne, B.: Free resolutions of fat point ideals on P . J. Pure Appl. Algebra 125, 213–234 (1998) 25. Harbourne, B.; Miranda, R.: Exceptional curves on rational numerically elliptic surfaces. J. Algebra 128, 405–433 (1990) 26. Hartshorne, R.: Algebraic Geometry. Springer, New York (1977) 27. Hu, Y.; Keel, S.: Mori dream spaces and GIT. Michig. Math. J. 48, 331–348 (2000) 28. Laface, A.; Velasco, M.: A survey on Cox rings. Geom. Dedicata. 139, 269–287 (2009) 29. Lahyane, M.: Exceptional curves on rational surfaces having K ≥ 0. C. R. Math. 338(11), 873–878 (2004) 30. Lahyane, M.: On the finite generation of the effective monoid of rational surfaces. J. Pure Appl. Algebra 214(7), 1217–1240 (2010) 31. Lahyane, M.; Harbourne, B.: Irreducibility of −1-classes on anticanonical rational surfaces and finite generation of the effective monoid. Pac. J. Math. 218(1), 101–114 (2005) 32. Miranda, R.; Persson, U.: On extremal rational elliptic surfaces. Math. Z. 193(4), 537–558 (1986) 33. Monserrat, F.: Curves having one place at infinity and linear systems on rational surfaces. J. Pure Appl. Algebra 211(3), 685–701 (2007) 123 Arab. J. Math. 34. Moreno-Ávila, C.J.: Global geometry of surfaces defined by non-positive and negative at infinity valuations. Ph.D. Thesis, University of Jaume I (2021) 35. Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. (2) 116(1), 133–176 (1982) 36. Nagata, M.: On rational surfaces, II. Mem. Coll. Sci. Univ. Kyoto Ser. A 33(2), 271–293 (1960) 37. Ottem, J.C.: On the Cox ring of P blown up in points on a line. Math. Scand. 109(1), 22–30 (2011) 38. Rosoff, J.: On the Semi-group of Effective Divisor Classes of an Algebraic Variety: The Question of Finite Generation. Ph.D. Thesis, University of California, Berkeley (1978) 39. Rosoff, J.: Effective divisor classes and blowings-up of P . Pac. J. Math. 89(2), 419–429 (1980) 40. Rosoff, J.: Effective divisor classes on a ruled surface. Pac. J. Math. 202(1), 119–124 (2002) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Journal
Arabian Journal of Mathematics – Springer Journals
Published: Mar 13, 2023
Keywords: 14J26; 14C20; 14C22; 14C17; 14Q20