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 \(\textrm{NS}(Z)\) of Z is the quotient group of the group of divisors on Z modulo numerical equivalence, and it is a free finitely generated \({\mathbb {Z}}\)-module of finite rank \(\rho (Z)\). A special subset of \(\textrm{NS}(Z)\) is the effective monoid \(\textrm{Eff}(Z)\) of Z, which is defined as the set of elements \(\gamma \) of \(\textrm{NS}(Z)\), such that there exists an effective divisor D on Z with \(\gamma \) is the class of D modulo numerical equivalence [26]. It is well known that \(\textrm{Eff}(Z)\) 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, \(\textrm{Cox}(X)\), of a projective variety X over an algebraically closed field k is the k-algebra given by

$$\begin{aligned} \textrm{Cox}(X)=\bigoplus _{{\mathcal {L}}\in \textrm{Pic}(X)} \textrm{H}^0(X,{\mathcal {L}}), \end{aligned}$$

where \(\textrm{Pic}(X)\) is the Picard group of X, and \(\textrm{H}^0(X,{\mathcal {L}})\) is the finite-dimensional \(k-\)vector space of global sections of \({\mathcal {L}}\); 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).

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 \(\Sigma _n\) associated with n is the projectivization of the locally free sheaf \({\mathcal {O}}_{{\mathbb {P}}^1_k}\oplus {\mathcal {O}}_{{\mathbb {P}}^1_k}(-n)\) of rank two on the projective line \({\mathbb {P}}^1_k.\) It is well known that the set \(\{{\mathfrak {C}}_n,{\mathfrak {F}}\}\) is a minimal set of generators of the Néron–Severi group \(\textrm{NS}(\Sigma _n)\) of \(\Sigma _n\) as a \({\mathbb {Z}}\)-module, where \({\mathfrak {C}}_n\) is the class of a section \(C_n\) of \(\Sigma _n\) (it is unique if n is positive, and in this case, that section is usually called the exceptional section), and \({\mathfrak {F}}\) is the class of a fibre f of \(\Sigma _n.\) The intersection form on \(\Sigma _n\) is given by the three equalities \(({\mathfrak {C}}_n)^2=-n, \, ({\mathfrak {F}})^2=0,\) and \({\mathfrak {C}}_n\cdot {\mathfrak {F}}=1\); for more details, see for example [26], and [40]. Furthermore, if D is a prime divisor on some smooth projective surface Z,  then \(\mathrm{{Supp}}(D)\) denotes the support of the invertible sheaf \({\mathcal {O}}_Z(D)\) associated with D [26].

Fig. 1
figure 1

The configuration of the points of \(\Omega \)

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Fig. 2
figure 2

The blow-up of \(\Sigma _n\) at the points of \(\Omega \)

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Finally, for fixed non-negative integers \(r_1\) and \(r_2\), let \(G=C_n+(n+2)f\) where f is a fibre of \(\Sigma _n\), and let \(f_1^{G},f_2^{G},\ldots ,f_{r_1}^{G},f_1^{C_n},f_2^{C_n},\dots ,f_{r_2}^{C_n}\) be \(r_1+r_2\) different fibres of \(\Sigma _n\). Now, let \(P_1,P_2,\dots ,P_{r_1}, Q_1,Q_2,\ldots ,Q_{r_2}\) be ordinary points of \(\Sigma _n\) in general position, such that \(P_{i} \in \big (\textrm{Supp}(G) \cap Supp(f_i^{G})\big ) {\setminus } \textrm{Supp}(C_n)\) for every \(i \in \{1,2,\ldots ,r_1\},\) and \(Q_j \in \big (\textrm{Supp}(C_n) \cap Supp(f_i^{C_n})\big ) {\setminus } \textrm{Supp}(G) \) for every \(j \in \{1,2,\ldots ,r_2\}\); see Fig. 1. Next, we denote the blow-up of \(\Sigma _n\) at the zero-dimensional subscheme \( \Omega = \{P_1,P_2,\ldots ,P_{r_1}, Q_1,Q_2, \ldots ,Q_{r_2}\}\) by \(X_n^{r_1,r_2};\) see Fig. 2.

A minimal set of generators of \(\textrm{NS}(X_n^{r_1,r_2})\) as a \({\mathbb {Z}}-\)module is the set

$$\begin{aligned} \{{\mathcal {C}}_n,{\mathcal {F}}, -{\mathcal {E}}_{P_1},-{\mathcal {E}}_{P_2},\ldots ,-{\mathcal {E}}_{P_{r_1}},-{\mathcal {E}}_{Q_1},-{\mathcal {E}}_{Q_2},\ldots ,-{\mathcal {E}}_{Q_{r_2}}\}, \end{aligned}$$

where \({\mathcal {C}}_n\) is the class of the total transform of \(C_n,\) \({\mathcal {F}}\) is the class of the total transform of a fibre f of \(\Sigma _n,\) \({\mathcal {E}}_{P_i}\) is the class of the exceptional divisor corresponding to the point \(P_i\) for every \(i \in \{1,2,\ldots ,r_1\},\) and \({\mathcal {E}}_{Q_j}\) is the class of the exceptional divisor corresponding to the point \(Q_{j}\) for every \( j \in \{1,2,\ldots ,r_2\}.\) The intersection form on \(X_n^{r_1,r_2}\) is given by the following equalities: \({\mathcal {C}}_n^2=-n, \, {\mathcal {F}}^2=0,\) \({\mathcal {C}}_n\cdot {\mathcal {F}}=1,\) \({\mathcal {C}}_n\cdot {\mathcal {E}}_{\omega }=0,\) \({\mathcal {F}}\cdot {\mathcal {E}}_{\omega }=0,\) \({\mathcal {E}}_{\omega }^2=-1\) for every \(\omega \in \Omega ,\) and \({\mathcal {E}}_p \cdot {\mathcal {E}}_q=0\) for \(p,q \in \Omega \), such that \(p \ne q.\)

3 The finiteness of \((-1)\)-curves and \((-2)\)-curves of \(X_n^{r_1,r_2}\)

In this section, we prove that there are only a finite number of \((-1)\)-curves and \((-2)\)-curves of \(X_n^{r_1,r_2}\) under the assumption that \((n+2)^2+nr_2+4r_2-nr_1-r_1r_2\) is positive. Here, a \((-1)\)-curve, respectively, a \((-2)\)-curve, 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:

Theorem 3.1

If \((n+2)^2+nr_2+4r_2-nr_1-r_1r_2\) is positive, then \(X_n^{r_1,r_2}\) has only a finite number of \((-1)\)-curves and \((-2)\)-curves.

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 \(X_n^{r_1,r_2}\), such that the class of D does not belong to \(\{ {\mathcal {C}}_n - \sum _{j=1}^{r_2} {\mathcal {E}}_{Q_j}, {\mathcal {C}}_n + (n+2){\mathcal {F}} - \sum _{i=1}^{r_1} {\mathcal {E}}_{P_i} \}\). One may write the class of D in NS\((X_n^{r_1,r_2})\) as \( {\mathcal {D}} = a {\mathcal {C}}_n +b {\mathcal {F}} - \sum _{j=1}^{r_1} \gamma _j {\mathcal {E}}_{P_j} - \sum _{\ell =1}^{r_2} \mu _{\ell } {\mathcal {E}}_{Q_{\ell }}, \) for some integers a, b,  \( \gamma _1, \gamma _2, \ldots , \gamma _{r_1}, \mu _1, \mu _2, \ldots , \mu _{r_2}.\) We have the equalities \({\mathcal {D}}^2=-2\) and \({\mathcal {D}}\cdot (-{\mathcal {K}}_{X_n^{r_1,r_2}})=0,\) and \(-{\mathcal {K}}_{X_n^{r_1,r_2}}= \big ({\mathcal {C}}_n - \sum _{\ell =1}^{r_2} {\mathcal {E}}_{Q_{\ell }}\big ) + {\mathcal {G}} \) is the class of an anticanonical divisor in \(\textrm{NS}(X_n^{r_1,r_2}),\) where \({\mathcal {G}}\) is the class of the strict transform of G in \(\textrm{NS}(X_n^{r_1,r_2});\) this implies the following equalities:

$$\begin{aligned} 2ab-a^2n -\sum _{j=1}^{r_1}\gamma _j^2 - \sum _{\ell =1}^{r_2}\mu _{\ell }^2 =-2, \end{aligned}$$
(1)
$$\begin{aligned} b-an - \sum _{\ell =1}^{r_2}\mu _{\ell }=0,\quad \text {and}\quad 2a+b-\sum _{j=1}^{r_1}\gamma _j=0, \end{aligned}$$

since \({\mathcal {D}}^2=-2\), \({\mathcal {D}} \cdot \big ({\mathcal {C}}_n - \sum _{\ell =1}^{r_2} {\mathcal {E}}_{Q_{\ell }}\big ) = 0\) and \({\mathcal {D}} \cdot {\mathcal {G}} = 0.\)

Using the above equalities, and the fact that \(\gamma _j \ge 0\) and \(\mu _{\ell } \ge 0\) for every \(j \in \{1,2, \ldots ,r_1 \}\) and \(\ell \in \{1,2,\ldots ,r_2\},\) it is sufficient to prove that a and b are bounded. Now, let \({{\bar{\gamma }}}_j=\gamma _j - \frac{\displaystyle {2a+b}}{\displaystyle {r_{1}}}\) for all \(j \in \{1,2,\ldots ,r_1\},\) and let \({\bar{\mu }}_{\ell }=\mu _{\ell }-\frac{\displaystyle {b-an}}{\displaystyle {r_{2}}}\) for each \(\ell \in \{1,2,\ldots ,r_2\}.\) Therefore, \(\sum _{j=1}^{r_1}{\bar{\gamma }}_j = 0,\) \(\sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell } =0,\) and

$$\begin{aligned} \sum _{j=1}^{r_1}\gamma _j^2 + \sum _{\ell =1}^{r_2} \mu _{\ell }^2= \sum _{j=1}^{r_1}{\bar{\gamma }}_j^2 + \sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell }^2 +\dfrac{(2a+b)^2}{r_1} +\dfrac{(b-an)^2}{r_2}. \end{aligned}$$
(2)

From Eqs. (1) and (2), we have

$$\begin{aligned} \sum _{j=1}^{r_1}{\bar{\gamma }}_j^2 + \sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell }^2 = 2ab-a^2n+2-\dfrac{(2a+b)^2}{r_1} - \dfrac{(b-an)^2}{r_2}. \end{aligned}$$

Therefore, the following inequality is satisfied:

$$\begin{aligned} 2abr_1r_2-a^2nr_1r_2+2r_1r_2-(b+2a)^2r_2-(b-an)^2r_1 \ge 0. \end{aligned}$$

Then, we obtain after completing the square in a

$$\begin{aligned} \left( a-\dfrac{(r_1r_2-2r_2+nr_1)b}{n^2r_1+nr_1r_2+4r_2} \right) ^2 -\dfrac{(r_1r_2-2r_2+nr_1)^2b^2}{(n^2r_1+nr_1r_2+4r_2)^2} - \dfrac{2r_1r_2-b^2r_2-b^2r_1}{n^2r_1+nr_1r_2+4r_2} \le 0. \end{aligned}$$

This implies that

$$\begin{aligned} \dfrac{(r_1r_2-2r_2+nr_1)^2b^2}{(n^2r_1+nr_1r_2+4r_2)^2} + \dfrac{2r_1r_2-b^2r_2-b^2r_1}{n^2r_1+nr_1r_2+4r_2} \ge 0. \end{aligned}$$

Therefore

$$\begin{aligned} b^2-\dfrac{2(n^2r_1+nr_1r_2+4r_2)}{(n+2)^2+nr_2+4r_2-nr_1-r_1r_2} \le 0. \end{aligned}$$

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

$$\begin{aligned} \left( a-\dfrac{(r_1r_2-2r_2+nr_1)b}{n^2r_1+nr_1r_2+4r_2} \right) ^2 -\dfrac{(r_1r_2-2r_2+nr_1)^2b^2}{(n^2r_1+nr_1r_2+4r_2)^2} - \dfrac{2r_1r_2-b^2r_2-b^2r_1}{n^2r_1+nr_1r_2+4r_2} \le 0. \end{aligned}$$

Therefore, \(X_n^{r_1,r_2}\) contains a finite number of \((-2)\)-curves.

For the finiteness of the set of \((-1)\)-curves, let \({\mathcal {N}}\) be a class of a \((-1)\)-curve on \(X_n^{r_1,r_2},\) so \({\mathcal {N}}=a {\mathcal {C}}_n +b {\mathcal {F}} - \sum _{j=1}^{r_1} \gamma _j {\mathcal {E}}_{P_j} - \sum _{\ell =1}^{r_2} \mu _{\ell } {\mathcal {E}}_{Q_{\ell }}, \) for some integers a, b,  \( \gamma _1, \gamma _2, \ldots , \gamma _{r_1}, \mu _1, \mu _2, \ldots , \mu _{r_2},\) such that \({\mathcal {N}}\) does not belong to \(\{ {\mathcal {C}}_n - \sum _{j=1}^{r_2} {\mathcal {E}}_{Q_j}, {\mathcal {C}}_n + (n+2){\mathcal {F}} - \sum _{i=1}^{r_1} {\mathcal {E}}_{P_i}, {\mathcal {F}}-{\mathcal {E}}_{P_1}, \ldots , {\mathcal {F}}-{\mathcal {E}}_{P_{r_1}}, {\mathcal {F}}-{\mathcal {E}}_{Q_1}, \ldots , {\mathcal {F}}-{\mathcal {E}}_{Q_{r_2}}, {\mathcal {E}}_{P_1}, \ldots , {\mathcal {E}}_{P_{r_1}}, {\mathcal {E}}_{Q_1}, \ldots , {\mathcal {E}}_{Q_{r_2}} \}\). Then, \({\mathcal {N}}\cdot (-{\mathcal {K}}_{X_n^{r_1,r_2}})=1\) and \({\mathcal {N}}^2=-1,\) so we have the following two cases to study:

  1. Case (1)

    \({\mathcal {N}}\cdot ({\mathcal {C}}_n-\sum _{\ell =1}^{r_2} {\mathcal {E}}_{Q_{\ell }})=1\) and \({\mathcal {N}}\cdot {\mathcal {G}}=0,\) and

  2. Case (2)

    \({\mathcal {N}}\cdot ({\mathcal {C}}_n-\sum _{\ell =1}^{r_2} {\mathcal {E}}_{Q_{\ell }})=0,\) and \({\mathcal {N}}\cdot {\mathcal {G}}=1.\)

Assume that we are in \(Case \; (1),\) then

$$\begin{aligned} \sum _{\ell =1}^{r_2}\mu _{\ell }= b-an-1, \, \sum _{j=1}^{r_1}\gamma _j=b+2a, \quad \text {and}\quad \sum _{j=1}^{r_1}\gamma _j^2 + \sum _{\ell =1}^{r_2}\mu _{\ell }^2 = 2ab-a^2n+1, \end{aligned}$$

since \({\mathcal {N}} \cdot \big ({\mathcal {C}}_n - \sum _{\ell =1}^{r_2} {\mathcal {E}}_{Q_{\ell }}\big )= 1,\) \({\mathcal {N}} \cdot {\mathcal {G}} = 0 \) and \({\mathcal {N}}^2=-1.\)

Now, let \({{\bar{\gamma }}}_j=\gamma _j - \frac{\displaystyle {2a+b}}{\displaystyle {r_{1}}}\) for all \(j \in \{1,2,\ldots ,r_1\},\) and let \({\bar{\mu }}_{\ell }=\mu _{\ell }-\frac{\displaystyle {b-an-1}}{\displaystyle {r_{2}}}\) for each \(\ell \in \{1,2,\ldots ,r_2\}.\) Therefore, \(\sum _{j=1}^{r_1}{\bar{\gamma }}_j = 0,\) \(\sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell } =0,\) and

$$\begin{aligned} 0 \le \sum _{j=1}^{r_1}{\bar{\gamma }}_j^2 + \sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell }^2 = 2ab-a^2n+1-\dfrac{(2a+b)^2}{r_1} - \dfrac{(b-an-1)^2}{r_2}. \end{aligned}$$

This implies that

$$\begin{aligned} 2ab-a^2n+1-\dfrac{(2a+b)^2}{r_1} - \dfrac{(b-an-1)^2}{r_2} \ge 0. \end{aligned}$$

Then, we obtain that

$$\begin{aligned} \left( a-\dfrac{r_1r_2b-2r_2b+nr_1b-nr_1}{nr_1r_2+4r_2+n^2r_1} \right) ^{2} \end{aligned}$$

is less than or equal to

$$\begin{aligned} \dfrac{r_1r_2-r_2b^2-r_1b^2+2r_1b-r_1}{nr_1r_2+4r_2+n^2r_1} +\dfrac{(r_1r_2b-2r_2b+nr_1b-nr_1)^2}{(nr_1r_2+4r_2+n^2r_1)^2}. \end{aligned}$$

Therefore, we have the following inequality:

$$\begin{aligned} \dfrac{r_1r_2-r_2b^2-r_1b^2+2r_1b-r_1}{nr_1r_2+4r_2+n^2r_1} +\dfrac{(r_1r_2b-2r_2b+nr_1b-nr_1)^2}{(nr_1r_2+4r_2+n^2r_1)^2} \ge 0, \end{aligned}$$

and then

$$\begin{aligned} b^2-\dfrac{2(n+4)b}{(n+2)^2+nr_2+4r_2-nr_1-r_1r_2} - \dfrac{nr_1r_2+4r_2+n^2r_1-nr_1-4}{(n+2)^2+nr_2+4r_2-nr_1-r_1r_2} \le 0. \end{aligned}$$

Thus, completing the square in b and using our hypothesis, it follows that b is bounded. Consequently, a is bounded too. Therefore, in \(Case \; (1)\), \(X_n^{r_1,r_2}\) contains a finite number of \((-1)\)-curves.

Now, assume that we are in \(Case \; (2)\), that is, \({\mathcal {N}}\cdot ({\mathcal {C}}_n-\sum _{\ell =1}^{r_2} {\mathcal {E}}_{Q_{\ell }})=0,\) and \({\mathcal {N}}\cdot {\mathcal {G}}=1,\) then

$$\begin{aligned} \sum _{\ell =1}^{r_2}\mu _{\ell }= b-an, \, \sum _{j=1}^{r_1}\gamma _j=b+2a-1,\quad \text {and}\quad \sum _{j=1}^{r_1}\gamma _j^2 + \sum _{\ell =1}^{r_2}\mu _{\ell }^2 = 2ab-a^2n+1. \end{aligned}$$

Let \({{\bar{\gamma }}}_j=\gamma _j - \frac{\displaystyle {2a+b-1}}{\displaystyle {r_{1}}}\) for all \(j \in \{1,2,\ldots ,r_1\},\) and let \({\bar{\mu }}_{\ell }=\mu _{\ell }-\frac{\displaystyle {b-an}}{\displaystyle {r_{2}}}\) for each \(\ell \in \{1,2,\ldots ,r_2\}.\) Therefore, \(\sum _{j=1}^{r_1}{\bar{\gamma }}_j = 0,\) \(\sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell } =0,\) and

$$\begin{aligned} 0\le \sum _{j=1}^{r_1}{\bar{\gamma }}_j^2 + \sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell }^2 = 2ab-a^2n+1-\dfrac{(2a+b-1)^2}{r_1} - \dfrac{(b-an)^2}{r_2}. \end{aligned}$$

This implies that

$$\begin{aligned} 2ab-a^2n+1-\dfrac{(2a+b-1)^2}{r_1} - \dfrac{(b-an)^2}{r_2} \ge 0. \end{aligned}$$

Then, after completing the square in a, we get that

$$\begin{aligned} \left( a-\dfrac{r_1r_2b-2r_2b+nr_1b+2r_2}{nr_1r_2+4r_2+n^2r_1} \right) ^{2} \end{aligned}$$

is less than or equal to

$$\begin{aligned} \dfrac{r_1r_2-r_2b^2+2r_2b-r_1b^2-r_2}{nr_1r_2+4r_2+n^2r_1}+ \dfrac{(r_1r_2b+nr_1b+2r_2-2r_2b)^2}{(nr_1r_2+4r_2+n^2r_1)^2}. \end{aligned}$$

Therefore, we have the following inequality:

$$\begin{aligned} b^2-\dfrac{2(nr_2+n^2+2r_2+2n)b}{(n+2)^2+nr_2+4r_2-nr_1-r_1r_2} + \dfrac{nr_1r_2+4r_2+n^2r_1-nr_2-n^2}{(n+2)^2+nr_2+4r_2-nr_1-r_1r_2} \le 0. \end{aligned}$$

Thus, a and b are bounded. Therefore, in case \(Case \; (2)\), \(X_n^{r_1,r_2}\) contains a finite number of \((-1)\)-curves. \(\square \)

The following lemma is the special case of Theorem 3.1, when \(r_1\) and \(r_2\) are zero.

Lemma 3.2

With the above notation, the surface \(\Sigma _n\) has finitely many \((-1)\)-curves and \((-2)\)-curves.

Proof

Since \(K_{\Sigma _n}^2 = 4,\) \(\Sigma _n\) is an anticanonical rational surface (see [3, Lemma 2.1, p. 3]). Then, the result holds from [3] and [31]. \(\square \)

The following result gives the list of \((-1)\)-curves and \((-2)\)-curves on the surface \(X_n^{n+4,r_2}\).

Corollary 3.3

With notation as above. The \((-1)\)-curves and \((-2)\)-curves on \(X_n^{n+4,r_2}\) are those given in Tables 1, 2, and 3.

Proof

It follows from the bounds given in the proof of the last theorem. \(\square \)

Remark 3.4

It is worth noting that all the \((-1)\)-curves (which are not exceptional) and \((-2)\)-curves on \(X_n^{n+4,r_2}\) come from smooth curves in \(\Sigma _n\) for every non-negative integers n and \(r_2 \).

Consequently, we show that the surface \(X_0^{4,10}\) has no \((-2)\)-curves, as in the case of blowing up the projective plane \({\mathbb {P}}_k^{2}\) at points in general position.

Example 3.5

With the notation of Theorem 3.1, the surface \(X_0^{4,10}\) has not \((-2)\)-curves. However, it has 556 \((-1)\)-curves.

Now, we handle the finite generation of the effective monoid of \(X_n^{r_1,r_2}\).

Corollary 3.6

With notation as above, if \((n+2)^2+nr_2+4r_2-nr_1-r_1r_2\) is a positive integer, then the effective monoid of the surface \(X_n^{r_1,r_2}\) is finitely generated.

Proof

It follows from Theorem 3.1 and [31]. \(\square \)

Table 1 List of \((-1)\) and \((-2)\)-curves of \(X_0^{4,r_2}\)
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Table 2 List of \((-1)\) and \((-2)\)-curves of \(X_1^{5,r_2}\)
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Table 3 List of \((-1)\) and \((-2)\)-curves of \(X_n^{n+4,r_2}\) with \(n \ge 2\) and \(r_2 \ge 0\)
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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 \({\mathbb {P}}^2_k\), 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 \({\mathbb {P}}^2_k\) 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 \({\mathbb {P}}^2_k\); see [8, Table 1, p. 34] and also Table 2. It is worth nothing that this surface is the blow-up of \(\Sigma _1\) 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 \({\mathbb {P}}^2_k\) at the node of an irreducible cubic \(r_2\) times do not affect the finite generation of the effective monoid.

Fig. 3
figure 3

The configuration of the ordinary points of a nodal cubic of \({\mathbb {P}}_k^2\)

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4 The finiteness of the Cox ring of \(X_n^{r_1,r_2}\)

In this section, we prove that the surface \(X_n^{r_1,r_2}\) satisfies the anticanonical orthogonal property, and we use Theorem 3.1 to prove the finite generation of the Cox ring of \(X_n^{r_1,r_2}\).

Lemma 4.1

With notation as above, let \({\mathcal {D}}\) be the class of a nef divisor in \(\textrm{NS}(X_n^{r_1,r_2})\), such that \({\mathcal {D}}\cdot {\mathcal {K}}_{X_n^{r_1,r_2}}=0.\) If \((n+2)^2+nr_2+4r_2-nr_1-r_1r_2\) is a positive integer, then \({\mathcal {D}}=0.\)

Proof

Let \({\mathcal {D}}\) be a class of a nef divisor in \(NS(X_n^{r_1,r_2}),\) so \({\mathcal {D}}=a {\mathcal {C}}_n +b {\mathcal {F}} - \sum _{j=1}^{r_1} \gamma _j {\mathcal {E}}_{P_j} - \sum _{\ell =1}^{r_2} \mu _{\ell } {\mathcal {E}}_{Q_{\ell }}, \) for some integers a, b,  \( \gamma _1, \gamma _2, \ldots , \gamma _{r_1}, \mu _1, \mu _2, \ldots , \mu _{r_2}.\) Therefore, \({\mathcal {D}}^2 \ge 0, \, {\mathcal {D}}\cdot ({\mathcal {C}}_n-\sum _{\ell =1}^{r_2} {\mathcal {E}}_{Q_{\ell }})=0,\) and \( {\mathcal {D}}\cdot {\mathcal {G}}= 0.\) From these, we have the following:

$$\begin{aligned} 2ab-a^2n -\sum _{j=1}^{r_1}\gamma _j^2 - \sum _{\ell =1}^{r_2}\mu _{\ell }^2 \ge 0, \end{aligned}$$
(3)
$$\begin{aligned} 2a+b-\sum _{j=1}^{r_1}\gamma _j=0, \text { and} \end{aligned}$$
(4)
$$\begin{aligned} b-an - \sum _{\ell =1}^{r_2}\mu _{\ell }=0. \end{aligned}$$

Now, let \({{\bar{\gamma }}}_j=\gamma _j - \dfrac{2a+b}{r_1}\) for all \(j \in \{1,2,\ldots ,r_1\},\) and let \({\bar{\mu }}_{\ell }=\mu _{\ell }-\dfrac{b-an}{r_2}\) for each \(\ell \in \{1,2,\ldots ,r_2\}.\) Therefore

$$\begin{aligned} 0 \le \sum _{j=1}^{r_1}{\bar{\gamma }}_j^2 + \sum _{\ell =1}^{r_2}{\bar{\mu }}_{\ell }^2 \le 2ab-a^2n-\dfrac{(2a+b)^2}{r_1} - \dfrac{(b-an)^2}{r_2}. \end{aligned}$$

This implies that

$$\begin{aligned} 2ab-a^2n-\dfrac{(2a+b)^2}{r_1} - \dfrac{(b-an)^2}{r_2} \ge 0. \end{aligned}$$

Consequently, after completing the square in a, we obtain that

$$\begin{aligned} \left( a- \dfrac{(r_1r_2-2r_2+nr_1)b}{nr_1r_2+4r_2+n^2r_1} \right) ^2 \le \dfrac{(r_1r_2-2r_2+nr_1)^2b^2}{(nr_1r_2+4r_2+n^2r_1)^2} - \dfrac{(r_1+r_2)b^2}{nr_1r_2+4r_2+n^2r_1}. \end{aligned}$$

Therefore

$$\begin{aligned} -((n+2)^2+nr_2+4r_2-nr_1-r_1r_2)b^2 \ge 0. \end{aligned}$$

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, \({\mathcal {D}}=0.\) Therefore, we are done.

\(\square \)

In the following theorem, the numerical condition \((n+2)^2+nr_2+4r_2-nr_1-r_1r_2 > 0\) gives us a family of smooth projective rational surfaces whose Cox rings are finitely generated.

Theorem 4.2

With the above notation, if \((n+2)^2+nr_2+4r_2-nr_1-r_1r_2\) is a positive integer, then the Cox ring of the surface \(X_n^{r_1,r_2}\) is finitely generated.

Proof

It follows from Theorem 3.1, Lemma 4.1, and Theorem 1 of [7]. \(\square \)