Mori dreamness of blowups of weighted projective planes

We consider the blowup X(a, b, c) of a weighted projective space P(a,b,c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}(a,b,c)$$\end{document} at a general nonsingular point. We give a sufficient condition for a curve to be a negative curve on X(a, b, c) in terms of χ(OX(C))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ({\mathcal {O}}_X(C))$$\end{document}. This can be applied to find the effective cone of X(a, b, c) and can serve as a starting point to prove the Mori dreamness of blowups of many weighted projective planes. We confirm the Mori dreamness of some X(a, b, c) as examples of our method.

1. Introduction. The geometry of blowups of weighted projective spaces at a general nonsingular point has been studied in many articles. Most recently, in [4], the authors find negative curves on such blowups by finding certain Newton polygons, while in [11], the authors compute lower bounds for the effective threshold of an ample divisor P(a, b, c), which is equivalent to finding pseudo effective curves on the blowups. In [6], Hausen, Keicher, and Laface give a criterion for a blowup of a weighted projective plane to be a Mori dream space, that is, whether there exists an orthogonal pair on the blowup.
There is also extensive research on justifying whether certain families X(a, b, c) are Mori dream spaces. In [2,4,6,9], the authors give examples of X(a, b, c) that are Mori dream spaces. In [3,5], there are examples of X(a, b, c) that are not Mori dream spaces.
In this article, instead of studying the associated Rees algebras as in [2,6], or studying certain Newton polygons as in [4], we reduce the question of finding negative curves on X(a, b, c) to applying the orbifold Riemann-Roch formula on X. We give a sufficient condition in Theorem 4.1 for a curve C to be a negative curve. This will enable us to find the effective cone for many X(a, b, c)s. As an application, we confirm in Theorem 6.2 the Mori dreamness of certain weighted projective spaces (one of them is from the table in [6,Theorem 1.3]).

The weighted projective plane and its blow up.
Let a, b, c be three positive integers, and assume that a, b, c are coprime. The weighted projective plane P(a, b, c) is given by the quotient of C 3 \{0} under the C * action We see that on P(a, b, c) there are three possible singular points P 1 = (1, 0, 0), P 2 = (0, 1, 0), P 3 = (0, 0, 1), and they are cyclic quotient singularities of type respectively. Here by saying that a point is a cyclic quotient singularity of type 1 a (b, c), we mean that there is a neighborhood around the point which is locally analytically isomorphic to the quotient C 2 / /μ a , where μ a is the group of the a-th roots of unity and the action of μ a on C 2 is given by ε : (x, y) → (ε b x, ε c y) for ε ∈ μ a (see [12]). By considering the affine patches of P(a, b, c), we can find the singularity type of P 1 , P 2 , and P 3 as claimed.
Let X(a, b, c) be the blow up of P(a, b, c) at a general nonsingular point P . We know that X(a, b, c) is isomorphic to P(a, b, c) outside the point P , so X(a, b, c) has the same singularities as P(a, b, c).
Let f : X(a, b, c) → P(a, b, c) be the morphism of the blow up. Let H be the pullback of O (1), and E be the exceptional curve. Then H and E generate the Picard group of X(a, b, c). The relation between the canonical divisor K X on X(a, b, c) and the canonical divisor K P on P(a, b, c) is given by where ∼ Q represents Q−linear equivalence. In addition, we know that

A criterion for X(a, b, c)
to be a Mori dream space. According to [7], X is a Mori dream space (MDS) if and only if the semiample cone and the nef cone are equal and they are polyhedral in Cl Q (X  Proof. Assume C is not a reduced irreducible curve, i.e., there exist C 1 > 0 and C 2 > 0 such that If n i are positive integers, then we have C 2 i ≥ 0 and μ i > 0 since otherwise it will contradict the minimality of n. We have also C 1 ·C 2 ≥ 0 because otherwise C 1 and C 2 have a common component C 0 ∼ n 0 H − μ 0 E with C 2 0 < 0. Then n 0 ≥ 0 is impossible under the assumption of minimality of n; if n 0 = 0, then C 0 ∼ kE for some positive integer k, but this would contradict with assumption 1.
Then one of the n i has to be zero. Assume that C 1 ∼ n 1 H − μ 1 E and C 2 ∼ μ 2 E, where μ i are positive integers. In this case μ 1 > μ, contradicting assumption 1. Therefore C is a reduced irreducible curve.
=⇒ If X is a MDS, the effective cone of X is polyhedral, which is generated by E and an irreducible curve So C is a component of F . This gives l > n. This gives condition 2 in Lemma 3.1. Since C is the boundary of the effective cone, C − kE, for any positive integer k, will lie outside the effective cone. This gives condition 1 in Lemma 3.1.
If X is a MDS, then its nef cone equals its semiample cone. Since the nef cone is the dual of the effective cone, there exists a divisor D > 0 such that D · C = 0 and D is a generator of the nef cone. D is in addition semiample and therefore C is not a fixed component of |D|.
⇐= If there exists a curve C ∼ nH − μE such that h 0 (C) = 1 and C 2 < 0 satisfying the conditions in Lemma 3.1, then C is irreducible by Lemma 3.1. According to [8,Lemma 1.22], C is in the extremal ray of the effective cone of X. That means C and E generate the effective cone. Since D · C = 0, D and H generate the nef cone of X. We just need to check that D is also semiample. Under the condition C 2 < 0, we have that D cannot lie in the extremal ray generated by C. Suppose D ∼ n 2 H − μ 2 E, then μ n > μ2 n2 , i.e., μ · n 2 > n · μ 2 . We can then find some multiple kD such that kD n2 > kμ2−μ kn2−n and (kμ 2 − μ) > 0 . In fact, any k satisfying kn 2 > n and kμ 2 > μ will do. Since C is not a base component of D, kD is a divisor with at most finite point base locus. This gives that D is semiample ([14, Theorem 6.2]). Therefore the nef cone is also the semiample cone, and X is a MDS.
We conclude from the proposition above that to check whether X(a, b, c), where a, b, c are coprime and √ abc / ∈ Z, is a MDS, we just need to check the following: 1. Find C ∼ nH − μE such that h 0 (C) = 1 and C 2 < 0, and n is minimal for this property as stated in Lemma 3.1. 2. Find D > 0 such that D · C = 0 and h 0 (D) = h 0 (D − C).

Effective cone.
In general, it is difficult to find C ∼ nH − μE such that h 0 (O X (D)) = 1 and C 2 < 0 as in Lemma 3.1. The following theorem gives a sufficient condition.  X(a, b, c) with n, μ positive integers. If C is the minimal divisor that satisfies χ(O X (C)) = 1 and C 2 < 0 (the minimality of C means that there is no l < n such that χ(O X (lH − kE)) = 1 and (lH − kE) 2 < 0 for any positive integer k), then C is a divisor that satisfies the assumptions in Lemma 3.1.
Proof. By the Serre duality, we have But for positive integers n, μ, we have H 0 (O X (−(a+b+c+n)H+(1+μ)E)) = 0. Therefore χ(O X (C)) = 1, together with C 2 < 0, implies h 0 (O X (C)) = 1 and h 1 (O X (C)) = 0. Consider the short exact sequence Next we need to show that there are no divisors C ∼ n H − μ E such that h 0 (C ) = 1, C 2 < 0, and n < n. Assume there is such a C and assume in addition n is minimal for such divisors and μ is the highest possible multiple of E such a C with given n can have. We know C is irreducible by Lemma 3.1, and then C generates the extremal ray of the effective cone X. Since C 2 < 0, C 2 < 0, we have C · C < 0. This indicates that C is a component of C. Assume C ∼ C + C . If C · C < 0, C is also a component of C . We can therefore assume C ∼ kC + C , where C · C ≥ 0. We want to show that C = 0 and k = 1.
Suppose C > 0. Since C · E ≥ 0 (since H 0 (O X (C − kE)) = 0 for k > 0) and C · C ≥ 0, we have that C is a nef divisor and therefore C · C ≥ 0. Then h 0 (O C (C − C )) = 0. Together with the exact sequence this will imply h 0 (O X (kC )) = 0, which is absurd. Therefore C = 0, and C ∼ kC .
The following exact sequence This implies χ(O X (C )) = 1. This gives C ∼ C , that is, C has to be the minimal divisor in the sense of Lemma 3.1.
Once we have the above theorem, we can reduce the question of finding a negative curve on X(a, b, c) to applying the orbifold Riemann-Roch formula on X (a, b, c). One remark here is that we are aware that the conditions in the above theorem are not necessary conditions, i.e., we do not always have χ(O X (C)) = 1 for a negative curve satisfying conditions Lemma 3.1. For example, X(5, 33, 49) has a negative curve C ∼ 1617H −18E and χ(O X (C)) = 0; X (8,15, 43) has a negative curve C ∼ 645H − 9E and χ(O X (C)) = 0 (see [9]).

Riemann-Roch on blowups of weighted projective planes.
In this section, we want to prepare ourselves to use the Riemann-Roch formula to find curves C that satisfy the conditions in Theorem 4.1. We first introduce some notations.
For a cyclic quotient singularity of type 1 r (a 1 , a 2 ), we denote There are, as mentioned earlier, three possible singular points on X, which are given by f −1 (P 1 ), f −1 (P 2 ), and f −1 (P 3 ), and they are cyclic quotient singularities of type 1 X(a, b, c), we can apply the orbifold Riemann-Roch formula ([13, Section 8]) and get (a, b)).
Taking into consideration the intersection matrix of H and E, the RR formula for D can be written as (a, b)).
The contributions from the singularities are given by Dedekind sums. To have some control over these sums, we first recall a formula for δ n ( 1 r (a)) and σ n ( 1 r (a)) for r, a coprime integers ([15, Lemma 3.2.1]). Here δ n ( 1 r (a)) and σ n ( 1 r (a)) are given by . (5.4) Lemma 5.1. Given two coprime integers a and r, where α is the inverse of a mod r, i.e., a · α ≡ 1 mod r, for all n ∈ Z. In particular, this gives (a 1 , a 2 )) as in 5.1, assume r, a 1 , a 2 are coprime, we have Proof. As r, a 1 are coprime, there exists k > 0 ∈ Z such that a 1 · k ≡ 1 mod r. Then Since r, ka 2 are coprime, there exists q > 0 ∈ Z such that q · ka 2 ≡ 1 mod r. By Lemma 5.1, we can further write the above expression as When q = 1, the first inequality above becomes an equality, and when kn = r−2 2 , the second equality becomes an equality. The final inequality is strict as long as r > 1. We also have the lower bound as follows: One sufficient condition for the first inequality to be an equality is that q = r − 1. The last inequality is equal when kn = r 2 .

Mori dreamness of some blowups
Proof. We use the Riemann-Roch formula 5.3 and the bounds in Lemma 5.2 for the δ n 's.
This proposition says that for a given μ, we only need to check n in a certain range to search for C ∼ nH − μE such that χ(O X (C)) = 1 and C 2 < 0. We then increase μ gradually to find such a C that satisfies the conditions in Theorem 4.1. We use the Magma program [10] to help us with the search. See the algorithms here [16].
In the remark after [6, Theorem 1.3], the authors suspect that the blowups of the weighted projective spaces P(7, 10, 19), P(7, 19, 22), P(7, 23, 27), and P(7, 26, 29) are Mori dream spaces. As an application of our theorem, we can show that P (7, 10, 19) is a Mori dream space and this rests heavily on the Kawamata-Viehweg vanishing theorem. We can not do the same for P(7, 19, 22), P(7, 23, 27), and P(7, 26, 29). But we could also confirm the Mori dreamness of the blow up of P(7, 19, 60) and P(7, 23, 59), which as far as we are aware of are new. Proof. We use the algorithm [16] to search for the negative curves. We can easily check that the given divisors generate the corresponding effective cones according to Theorem 4.1 and Lemma 3.1. We then find the corresponding dual cone in each case, which gives the nef cone respectively. What is remaining is to check whether each nef cone is also the semiample cone. We see this case by case. 1. In the case of X (7,10,19), we see that H and D ∼ 840H − 23E generate the nef cone. It remains to check whether the divisor D ∼ 840H − 23 is also semiample. To justify that, we need to prove that the negative curve C ∼ 437H − 12E is not a fixed component of D. We see that