F -rationality of two-dimensional graded rings with a rational singularity
aa r X i v : . [ m a t h . A C ] D ec F -RATIONALITY OF TWO-DIMENSIONAL GRADED RINGSWITH A RATIONAL SINGULARITY KOHSUKE SHIBATA
Abstract.
It is known that a two-dimensional F -rational ring has a rationalsingularity. However a two-dimensional ring with a rational singularity is not F -rational in general. In this paper, we investigate F -rationality of a two-dimensional graded ring with a rational singularity in terms of the multiplicity.Moreover, we determine when a two-dimensional graded ring with a rationalsingularity and a small multiplicity is F -rational. Introduction
It is well known by now that there is an interesting connection between F -singularities and singularities in birational geometry. In [8], Hara and Watanabeshowed that a strongly F -regular ring has log terminal singularities and an F -pure ring has log canonical singularities. In [15], Smith showed that an F -rationalring has pseudo-rational singularities. Therefore a two-dimensional excellent F -rational ring has a rational singularity. However two-dimensional excellent ringwith a rational singularity is not F -rational in general. Thus a natural questionis when rings with rational singularities are F -rational.In [7], Hara and Watanabe investigated F -rationality of a two-dimensionalgraded ring with a rational singularity in terms of Pinkham-Demazure construc-tion and gave the necessary and sufficient condition for F -rationality of a two-dimensional graded ring with a rational singularity.In April 2020, Kei-ichi Watanabe asked the author the following question. Question 1.1.
Let D = P ri =1 c i d i P i be an ample Q -divisor on P k , where c i ∈ Z , d i ∈ N and P i are distinct points of P k . Let R = L n ≥ H ( P k , O P k ([ nD ])) t n .Assume that R has a rational singularity and d i > p for all i . Then is R F -rational?In this paper, we give an affirmative answer to this question.
Theorem 1.2.
Let D = P ri =1 c i d i P i be an ample Q -divisor on P k , where c i ∈ Z , d i ∈ N and P i are distinct points of P k . Let R = L n ≥ H ( P k , O P k ([ nD ])) t n .Assume that R has a rational singularity and p does not divide any d i . Then R is F -rational. In particular, Question 1.1 is affirmative. In [6], Hara proved that a two-dimensional log terminal singularity is strongly F -regular if the characteristic is larger than 5. This implies that a two-dimensional Mathematics Subject Classification.
Key words and phrases.
F-rational rings, rational singularities, graded rings. rational double point is F -rational if the characteristic is larger than 5. In this pa-per, we investigate F -rationality of a two-dimensional graded ring with a rationalsingularity in terms of the multiplicity. We prove the following theorem. Theorem 1.3.
Let m ∈ N . There exists a positive integer p ( m ) such that R is F -rational for any two-dimensional graded ring R with a rational singularity, e ( R ) = m and R = k , an algebraically closed field of characteristic p ≥ p ( m ) . Moreover, we can determine p (3) and p (4) in the above theorem. Theorem 1.4.
Let R be a two-dimensional graded ring with a rational singularity. (1) If e ( R ) = 3 and p ≥ , then R is F -rational. (2) If e ( R ) = 4 and p ≥ , then R is F -rational.Furthermore, these inequalities are best possible. The paper is organized as follows. In Section 2, we review definitions and somefacts on F -rational rings, rational singularities and Pinkham-Demazure construc-tion. In Section 3, we investigate F -rationality of a two-dimensional graded ringwith a rational singularity in terms of Pinkham-Demazure construction and givean affirmative answer to Question 1.1. In Section 4, we prove Theorem 1.3. InSection 5, we classify two-dimensional graded rings with a rational singularity andmultiplicity 3 and 4 in terms of Pinkham-Demazure construction. In Section 6,we determine p (3) and p (4) in Theorem 1.3. Acknowledgement.
The author would like to thank Kei-ichi Watanabe for thediscussion and many suggestions. The author are grateful to Alessandro De Ste-fani and Ilya Smirnov for insightful conversations and comments on a rough draftof this paper. The author is partially supported by JSPS KAKENHI Grant Num-ber JP20J00132.
Conventions.
Throughout this paper, p is a prime number and k is an alge-braically closed field of characteristic p . We assume that a ring is essentially offinite type over k . By a graded ring, we mean a ring R = ⊕ n ≥ R n , which is finitelygenerated over the subring R = k .2. Preliminaries
In this section we introduce definitions and some facts on F -rational rings,rational singularities and Pinkham-Demazure construction.2.1. F -rational rings and rational singularities. In this subsection we intro-duce the definitions of F -rational rings and rational singularities. Definition 2.1.
Let R be a ring and I an ideal of R . The tight closure I ∗ of I is defined by x ∈ I ∗ if and only if there exists c ∈ R ◦ such that cx p e ∈ I [ p e ] for e ≫
0, where R ◦ is the set of elements of R which are not in any minimal primeideal and I [ p e ] is the ideal generated by the p e -th powers of the elements of I . Wesay that I is tightly closed if I ∗ = I . -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 3 Definition 2.2.
A local ring ( R, m ) is F -rational if every parameter ideal is tightlyclosed. An arbitrary ring R is F -rational if R m is F -rational for every maximalideal m . Definition 2.3.
A local ring ( R, m ) is F -injective if R -module homomorphism H i m ( F ) : H i m ( R ) → H i m ( R )is injective for all i . An arbitrary ring R is F -injective if R m is F -injective forevery maximal ideal m . Definition 2.4.
Let R be a two-dimensional normal ring, and let f : Y → X :=Spec( R ) be a resolution of singularities. The ring R is said to be (or have) arational singularity if R f ∗ O Y = 0. Remark . It is known that there exists a resolution of singularity even in pos-itive characteristic for any two-dimensional excellent normal ring (see e.g. [13]).2.2.
Hirzebruch-Jung Continued fraction.
In this subsection, we introducethe definition and basic properties of the Hirzebruch-Jung continued fraction.
Definition 2.6.
Let a , a , . . . , a n be real numbers. We denote by [[ a , . . . , a n ]]the Hirzebruch-Jung continued fraction:[[ a , . . . , a n ]] := a − a − a − · · · − a n . Lemma 2.7.
Let m, n be positive integers with m < n , and let a , . . . , a n be realnumbers. Then we have [[ a , . . . , a n ]] = [[ a , . . . , a m , [[ a m +1 . . . , a n ]]]] Proof.
This follows directly from the definition. (cid:3)
Lemma 2.8.
Let a , . . . , a n be positive integers with min { a , . . . , a n } ≥ . Then [[ a , . . . , a n ]] > . Proof.
We prove this by induction on n . If n = 1, then [[ a ]] = a > . If n > a , . . . , a n ]] = [[ a , [[ a , . . . , a n ]]]] > a − ≥ (cid:3) Lemma 2.9.
Let a , . . . , a l , b , . . . , b m , c , . . . , c n be positive integers with b < c and min { a , . . . , a l , b , . . . , b m , c , . . . , c n } ≥ . Then (1) [[ a , . . . , a l , b , . . . , b m ]] < [[ a , . . . , a l ]] . (2) [[ a , . . . , a l , b , . . . , b m ]] < [[ a , . . . , a l , c , . . . , c n ]] . KOHSUKE SHIBATA
Proof. (1) By Lemma 2.7 and Lemma 2.8, we have[[ a , . . . , a l , b , . . . , b m ]] < [[ a , . . . , a l , N ]] = [[ a , . . . , a l − N ]] < [[ a , . . . , a l ]]for a positive integer N > [[ b , . . . , b m ]] > b , . . . , b m ]] < [[ c , . . . , c n ]] . By Lemma 2.7, Lemma 2.8 and Lemma 2.9.(1), we have[[ b , . . . , b m ]] ≤ b ≤ c − < [[ c , [[ c , . . . , c n ]]]] = [[ c , . . . , c n ]] . (cid:3) We denote by (2) l the sequence obtained by repeating l times the number 2. Example 2.10.
Let l be a positive integer. Then we have[[(2) l ]] = l + 1 l . Indeed, if [[(2) n ]] = n +1 n holds for n ∈ N , we have[[(2) n +1 ]] = [[2 , (2) n ]] = [[2 , n + 1 n ]] = 2 − nn + 1 = n + 2 n + 1 . Example 2.11. < [[3 , (2) l ]] = 3 − ll +1 for any l ∈ Z ≥ .2.3. Pinkham-Demazure construction.
In this subsection we introduce theconstruction of a two-dimensional normal graded ring using a Q -divisor on asmooth curve. By a Q -divisor on a variety X , we mean a Q -linear combination ofcodimension-one irreducible subvarieties of X . If D = P a i D i , where a i ∈ Q and D i are distinct irreducible subvarieties, we set [ D ] = P [ a i ] D i , where [ a ] denotesthe greatest integer less than or equal to a .In [14], Pinkham proved the following result. In [1], Demazure generalized thisresult in higher dimensional case. Theorem 2.12 ([1],[14]) . Let R be a two-dimensional normal graded ring over R = k . Then there exists an ample Q -divisor D on C = Proj( R ) such that R ∼ = R ( C, D ) := M n ≥ H ( C, O C ([ nD ])) t n . We call this representation Pinkham-Demazure construction.
Remark . (1) A divisor D on a smooth curve is ample if and only ifdeg D > D , D be ample Q -divisors on a smooth curve C . If D − D is aprincipal divisor on C , then R ( C, D ) ∼ = R ( C, D ). Indeed, let f be therational function on C with div( f ) = D − D , and let g be a rationalfunction on C with div( g ) + nD ≥
0. Then div( f n g ) + nD = div( g ) + nD ≥ . Therefore we have an isomorphism R ( C, D ) ∼ = R ( C, D ) definedby gt n f n gt n . -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 5 (3) If C = P k , we can put D = sP − P ri =1 a i P i in Theorem 2.12, where s ∈ N and a i ∈ Q > with 0 < a i <
1, and P i are distinct points of P k .Indeed, since P is linearly equivalent to Q for any points P, Q of P k by [9,II.Proposition 6.4], this remark holds by the above remark.A resolution of a singularity is said to be good if the exceptional divisor hasnormal crossing and each irreducible components of the exceptional divisor issmooth. A resolution of a surface singularity is called a minimal good resolution ifthe resolution is the smallest resolution of good resolutions. An exceptional divisor E of the minimal good resolution of a two-dimensional singularity is said to be acentral curve if E has positive genus or E meets at least three other exceptionaldivisors of the minimal good resolution. The dual graph of the minimal goodresolution is said to be star-shaped if the dual graph has at most one centralcurve.In [14], Pinkham determined the exceptional set of the minimal good resolutionof Spec( R ( P k , D )). Theorem 2.14 ([14]) . Let D = sP − P ri =1 c i d i P i be an ample Q -divisor on P k ,where s, c i , d i ∈ N with < c i < d i , and P i are distinct points of P k . Let b i , . . . , b im i be positive integers with d i c i = [[ b i , . . . , b im i ]] . Then the exceptionalset of the minimal good resolution of Spec( R ( P k , D )) consists of (1) unique central curve E ∼ = P k with E = − s and (2) r branches of P k ’s E i − E i −· · ·− E im i corresponding to P i with E ij = − b ij and E E i = 1 .Thus the dual graph is star-shaped as follows: WVUTPQRS − b E WVUTPQRS − b E · · · _^]\XYZ[ − b m E m WVUTPQRS − b E WVUTPQRS − b E · · · _^]\XYZ[ − b m E m ONMLHIJK − sE ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ rrrrrrrr ❀❀❀❀❀❀❀❀❀❀❀❀❀ ... ... ... WVUTPQRS − b r E r WVUTPQRS − b r E r · · · _^]\XYZ[ − b rm r E rm r Definition 2.15.
Let ( R, m ) be a d -dimensional normal graded ring. The a -invariant a ( R ) of R is defined by a ( R ) := max (cid:8) n ∈ Z | [ H d m ( R )] n = 0 (cid:9) , where [ H d m ( R )] n denotes the n -th graded piece of the highest local cohomologymodule of H d m ( R ).Theorem 2.16 is a very useful characterization of a rational singularity. KOHSUKE SHIBATA
Theorem 2.16 ([14, Corollary 5.8],[5, Korollary 3.10],[16, Theorem 2.2] ) . Let C be a smooth curve, D an ample Q -divisor on C and R = R ( C, D ) . Then thefollowing conditions are equivalent. (1) R has a rational singularity. (2) C = P k and deg[ nD ] ≥ − for any positive integer n . (3) a ( R ) < . Lemma 2.17.
Let D = sP − P ri =1 c i d i P i be an ample Q -divisor on P k , where s, c i , d i ∈ N with < c i < d i , and P i are distinct points of P k . If s + 2 ≤ r , then R ( P k , D ) does not have a rational singularity.Proof. Since deg[ D ] = s − r ≤ − R ( P k , D ) does not have a rational singularityby Theorem 2.16. (cid:3) In [10], Hochster and Huneke gave a necessary and sufficient condition for agraded ring to be F -rational. Theorem 2.18 ([10, Theorem 7.12]) . Let R be a two-dimensional normal gradedring. Then R is F -rational if and only if R is F -injective and a ( R ) < . Fundamental cycle.
In this subsection, we introduce the definition anduseful properties of the fundamental cycle.Let (
X, x ) be a two-dimensional normal singularity, and let f : Y → X be aresolution of singularity. We denote by Exc( f ) the exceptional set of f . We callthe minimum element of the set (cid:26) Z ∈ Div( Y ) \ { } (cid:12)(cid:12)(cid:12)(cid:12) Supp( Z ) ⊂ Exc( f ) and ZE ≤ E of f (cid:27) . the fundamental cycle of f . Proposition 2.19.
Let R be a two-dimensional local ring with a rational singu-larity, f : X → Spec( R ) the minimal good resolution and E , . . . , E r the primeexceptional divisors of f . Let Z = P ri =1 n i E i be the fundamental cycle of f . Then e ( R ) = − Z = r X i =1 n i ( − E i −
2) + 2 . Proof.
We can compute Z by a computation sequence of cycles0 < Z < . . . < Z s = Z defined by Z = F (we can take any prime exceptional divisor of f ) and Z i = Z i − + F i , where F i is any prime exceptional divisor f with Z i − F i > p ( Z i ) = p ( Z i − ) + p ( F i ) + Z i − F i − ≥ p ( Z i − )and Z i = Z i − + 2 Z i − F i + F i since p ( F i ) ≥ p ( D ) := D + K X D + 1 is the virtualgenus of D for an exceptional divisor D on X . Since p ( Z ) = 0 by [11, Proposition -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 7 p ( F i ) = 0 and Z i − F i = 1 for all i . Hence we have e ( R ) = − Z = r X i =1 n i ( − E i −
2) + 2 . by [11, Proposition 7.3.5]. (cid:3) Remark . Note that the dual graph of the minimal good resolution of a two-dimensional rational singularity does not have a ( − p ( F i ) = 0 and Z i − F i = 1 in the above proof, which implies thatall irreducible components of the exceptional set have to be smooth rational curves,pairwise intersecting transversally in at most one point (see [11, Proposition 7.2.8.(ii)]).Once we have the coefficient of the central curve of the fundamental cycle ofthe minimal good resolution of Spec( R ( P k , D )), the fundamental cycle can becomputed by the following formula.For a divisor D = P ri =1 a i E i , where E i is a prime divisor, we denote by Coeff E i D the coefficient a i . Lemma 2.21.
Let D = sP − P ri =1 c i d i P i be an ample Q -divisor on P k , where s, c i , d i ∈ N with < c i < d i , and P i are distinct points of P k . Let f : X → Spec( R ( P k , D )) be the minimal good resolution. Let F be a non-zero effectivedivisor on X with Supp( F ) ⊂ Exc( f ) and n the coefficient of the central curve E on F . Let E i − E i − · · · − E im i be the branch of P k ’s corresponding to P i suchthat d i c i = [[ b i , b i , . . . , b im i ]] ,E ij = − b ij and E E i = 1 . Define e i , . . . , e im i ∈ Q by e ij = [[ b ij , b i,j +1 , . . . , b im i ]] . We assume that the coefficient n ij of E ij on F is given inductively, n i = ⌈ n e i ⌉ = ⌈ n c i d i ⌉ , . . . , n i,j +1 = ⌈ n ij e i,j +1 ⌉ , . . . , n im i = ⌈ n i,m i − e im i ⌉ . Then F is the smallest element of the set G ∈ Div( X ) \ { } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Supp( G ) ⊂ Exc( f ) , Coeff E G = n and GE ≤ for any prime exceptionaldivisor E of f with E = E . Moreover if n is equal to the coefficient of the central curve of the fundamentalcycle of f , then F is the the fundamental cycle.Proof. The statement follows from [12, Lemma 1.1]. (cid:3)
Corollary 2.22.
Let D = sP − P ri =1 c i d i P i be an ample Q -divisor on P k , where s, c i , d i ∈ N with < c i < d i , and P i are distinct points of P k . Let Z be the KOHSUKE SHIBATA fundamental cycle of the minimal good resolution of
Spec( R ( P k , D )) , and let E be the central curve of the minimal good resolution. Then Coeff E Z = min { n ∈ N | deg[ nD ] ≥ } . In particular, if s + 1 ≤ r , then Coeff E ( Z ) ≥ . Proof.
Let f : X → Spec( R ( P k , D )) be the minimal good resolution. For l ∈ N ,let F l be the smallest element of the set G ∈ Div( X ) \ { } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Supp( G ) ⊂ Exc( f ) , Coeff E G = l and GE ≤ E of f with E = E . Then we have F l E = − ls + r X i =1 ⌈ lc i d i ⌉ = − deg[ lD ]for l ∈ N by Lemma 2.21. Let n be the coefficient of E in Z . Then F n = Z byLemma 2.21. Therefore n = min { n ∈ N | deg[ nD ] ≥ } . If s + 1 ≤ r , then deg[ D ] ≤ −
1. Therefore we have Coeff E ( Z ) ≥ (cid:3) F -rationality of R ( P k , D )In this section, we investigate F -rationality of a two-dimensional graded ringwith a rational singularity in terms of Pinkham-Demazure construction and givean affirmative answer to Question 1.1.The following criterion for F -rationality is given in [7]. Theorem 3.1 ([7, Theorem 2.9]) . Let D be an ample Q -divisor on P k and R = R ( P k , D ) . Assume that R has a rational singularity. Let B n = − p [ − nD ]+[ − pnD ] for a positive integer n , and let ( B n ) red be the reduced divisor with the same supportas B n . Then R is F -rational if and only if for every positive integer n , we have deg[ − pnD ] + deg( B n ) red ≤ . Remark . Let D = P ri =1 a i P i with a i ∈ Q . Thendeg( B n ) red ≤ ♯ { i | na i Z } . In general, deg( B n ) red = ♯ { i | na i Z } . For example, if p = 2 and D = P , thendeg( B ) red = 0 and ♯ { i | a i Z } = 1. Proposition 3.3.
Let D = sP − P ri =1 c i d i P i be an ample Q -divisor on P k , where s, c i , d i ∈ N with < c i < d i , and P i are distinct points of P k . (1) If s ≥ r , then R ( P k , D ) is F -rational. (2) If R ( P k , D ) has a rational singularity and is not F -rational, then s +1 = r . (3) If R ( P k , D ) has a rational singularity, deg D ≥ and p ≥ r − , then R ( P k , D ) is F -rational. -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 9 Proof.
Let B n = − p [ − nD ] + [ − pnD ] for a positive integer n .(1) Since deg[ nD ] ≥ n , R ( P k , D ) has a rational singu-larity by Theorem 2.16. Note that deg[ − pnD ] ≤ − s and deg( B n ) red ≤ r for anypositive integer n . Therefore R ( P k , D ) is F -rational by Theorem 3.1.(2) This statement follows from Lemma 2.17 and Proposition 3.3.(1).(3) Since deg[ − pnD ] ≤ deg( − pnD ) ≤ − pn and deg( B n ) red ≤ r for any positiveinteger n , R ( P k , D ) is F -rational by Theorem 3.1. (cid:3) Example 3.4.
Let D = 2 P − ( P + P + P ), where P i are distinct points of P k . Then R ( P k , D ) is F -rational for all p . Indeed, since deg[ nD ] ≥ − n , R ( P k , D ) has a rational singularity by Theorem 2.16. Therefore R ( P k , D ) is F -rational by Proposition 3.3.(3).Watanabe asked the following question. Question 3.5.
Let D = P ri =1 c i d i P i be an ample Q -divisor on P k , where c i ∈ Z , d i ∈ N and P i are distinct points of P k . Let R = R ( P k , D ). Assume that R has arational singularity and d i > p for all i . Then is R F -rational?
Theorem 3.6.
Let D = P ri =1 c i d i P i be an ample Q -divisor on P k , where c i ∈ Z , d i ∈ N and P i are distinct points of P k . Let R = R ( P k , D ) . Assume that R hasa rational singularity and p does not divide any d i . Then R is F -rational. Inparticular, Question 3.5 is affirmative.Proof. We assume that R is not F -rational. Then there exists a positive integer m such that deg[ − pmD ] + deg( B m ) red ≥ , where B m = − p [ − mD ] + [ − pmD ] by Theorem 3.1. Since R has a rational singu-larity, we have for every positive integer n ,deg[ nD ] ≥ − l = ♯ { i ∈ N | mc i d i Z } . Then we have ⌈ pmc j d j ⌉ − [ pmc j d j ] = 1for j ∈ { i ∈ N | mc i d i N } and deg( B m ) red ≤ l (see Remark 3.2). We havedeg[ − pmD ] = r X i =1 [ − pmc i d i ] = − r X i =1 ⌈ pmc i d i ⌉ = − l − r X i =1 [ pmc i d i ]= − l − deg[ pmD ] ≤ − l + 1 . Hence we have 2 ≤ deg[ − pmD ] + deg( B m ) red ≤ − l + 1 + l = 1 , which is a contradiction. Therefore R is F -rational. (cid:3) F -rationality of two-dimensional graded rings with a rationalsingularity In this section we show that for a positive integer m , any two-dimensionalgraded ring with multiplicity m and a rational singularity is F -rational if thecharacteristic of the base field is sufficiently large depending on m .An irreducible curve E on a smooth surface is called a ( − i )-curve if E ∼ = P k with E = − i . Definition 4.1.
Let R be a two-dimensional normal ring and i a positive integer.We define C i ( R ) to be the number of ( − i )-curves in the exceptional set of theminimal good resolution of Spec( R ). Definition 4.2.
Let a be a rational number with a >
1. Let a = [[ b , . . . , b m ]]be the Hirzebruch-Jung continued fraction of a and n = ♯ { i | b i = 2 } . Let { j , j , . . . , j m − n } be the set of numbers such that b j l = 2 and j l < j l +1 . Wedefine T ( a ) = ( b j , b j , . . . , b j m − n ) ∈ N m − n if m = n , and T ( a ) = ∅ if m = n . Example 4.3. T ([[2 , , , , , , , , Lemma 4.4.
For any positive integer l , let D l = sP − P ri =1 c ( l ) i d ( l ) i P i be an ample Q -divisor on P k , where s, c ( l ) i , d ( l ) i ∈ N with < c ( l ) i < d ( l ) i , and P i are distinct pointsof P k . We assume that c ( l ) i d ( l ) i ≤ c ( l +1) i d ( l +1) i for any i, l . Let a i = lim l →∞ c ( l ) i d ( l ) i . Assume that a i ∈ Q . Let D = sP − P ri =1 a i P i . If R ( P k , D l ) has a rational singularity for any l , then R ( P k , D ) has a rational singularity.Proof. If the lemma fails, then by Theorem 2.16, there exists a positive integer n such that deg[ nD ] = sn − r X i =1 ⌈ na i ⌉ ≤ − . Since R ( P k , D l ) has a rational singularity for any l , we havedeg[ mD l ] = sm − r X i =1 (cid:6) mc ( l ) i d ( l ) i (cid:7) ≥ − m . Since c ( l ) i d ( l ) i ≤ c ( l +1) i d ( l +1) i for any i, l , we have ⌈ na i ⌉ = ⌈ nc ( l ) i d ( l ) i ⌉ for any i and any sufficiently large number l . Therefore for any sufficiently large l , we have − ≤ deg[ nD l ] = deg[ nD ] ≤ − , which is contradiction. (cid:3) Theorem 4.5.
Let m ∈ N . There exists a positive integer p ( m ) such that R is F -rational for any two-dimensional graded ring R with a rational singularity, e ( R ) = m and R = k , an algebraically closed field of characteristic p ≥ p ( m ) . -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 11 Proof.
If the theorem fails, then by Theorem 2.12 and Theorem 2.16, there exista positive integer m and a sequence { D l } l ∈ N of ample Q -divisors on P k l suchthat R ( P k l , D l ) is a two-dimensional non- F -rational graded ring with a rationalsingularity, e ( R ( P k l , D l )) = m and lim l →∞ p l = ∞ , where p l is the characteristic of the field k l . Let E l be the central curve ofthe exceptional set of the minimal good resolution of Spec( R ( P k l , D l )). Thenwe may assume that ( E l ) is constant for any l by Proposition 2.19. Hence byProposition 3.3.(2), we may put D l = sP ( l )0 − P s +1 i =1 c ( l ) i d ( l ) i P ( l ) i , where s, c ( l ) i , d ( l ) i ∈ N with 0 < c ( l ) i < d ( l ) i , and P ( l ) i are distinct points of P k l .By Proposition 2.19, we may assume that C j ( R ( P k l , D l )) is constant for any l when we fix j with j ≥ C j ′ ( R ( P k l , D l )) = 0 for any l, j ′ with j ′ ≥ m + 1.Therefore we may assume that T ( d ( l ) i c ( l ) i ) is constant for any i, l by Theorem 2.14.Hence we may assume that c ( l ) i d ( l ) i ≤ c ( l +1) i d ( l +1) i for any i, l by Lemma 2.9. Thus theHirzebruch-Jung continued fraction of d ( l ) i c ( l ) i can be expressed as follows: d ( l ) i c ( l ) i = [[ b i , b i , . . . , b im i , (2) e ( l ) i , f ( l ) i , . . . , f ( l ) in ( l ) i ]]with lim l →∞ e ( l ) i = ∞ . Note that b ij is independent of l . We have[[ b i , b i , . . . , b im i , (2) e ( l ) i + n ( l ) i ]] ≤ d ( l ) i c ( l ) i ≤ [[ b i , b i , . . . , b im i , (2) e ( l ) i ]]by Lemma 2.9. Therefore we havelim l →∞ d ( l ) i c ( l ) i = [[ b i , b i , . . . , b i,m i − , b im i − . Let c i , d i be positive integers with 0 < c i ≤ d i and d i c i = [[ b i , . . . , b i,m i − , b im i − . Let D = sP − P s +1 i =1 c i d i P i , where P i are distinct points of P k . Then R ( P k , D ) hasa rational singularity by Lemma 4.4. Since deg D > − nD l ] ≤ deg[ − nD ] ≤ − s for any positive integer l and any sufficiently large number n ∈ N . Hence, sincelim l →∞ p l = ∞ , we have for any positive integer n and any sufficiently largenumber l deg[ − p l nD l ] + deg( B ln ) red ≤ , where B ln = − p l [ − nD l ] + [ − p l nD l ]. By Theorem 3.1, R ( P k l , D l ) is F -rational forany sufficiently large number l , which is contradiction. (cid:3) Example 4.6.
Let D = 2 P − p +12 p P − p − p P − P , where P i are distinct pointsof P k . Then R = R ( P k , D ) has a rational singularity with e ( R ) = (cid:6) p +12 (cid:7) but isnot F -rational. Indeed, we have for m ∈ N ,deg[2 mD ] = (cid:2) mp (cid:3) − (cid:6) mp (cid:7) ≥ − m − D ] = (cid:2) m − p (cid:3) − (cid:6) p + 2 m − p (cid:7) ≥ − . Therefore R has a rational singularity by Theorem 2.16. Sincedeg[ − pD ] + deg( B ) red = 2 , where B = − p [ − D ] + [ − pD ], R is not F -rational by Theorem 3.1.If p = 2, then the dual graph of the minimal good resolution of Spec( R ) is thefollowing: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Here the number next to a vertex means the coefficient of the relevant exceptionaldivisor in the fundamental cycle. Therefore we have e ( R ) = 2 by Proposition 2.19.If p ≥
3, then the dual graph of the minimal good resolution of Spec( R ) is thefollowing: ONMLHIJK − p +12 ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − p − ONMLHIJK − p − ONMLHIJK − p ONMLHIJK − p +12 _^]\XYZ[ − p +12 Note that min { n ∈ N | deg[ nD ] ≥ } = p . Therefore we can compute thefundamental cycle by Lemma 2.21 and Corollary 2.22. We have e ( R ) = p +12 byProposition 2.19. Remark . Example 4.6 implies that p ( m ) > m −
1, where p ( m ) is the positiveinteger in Theorem 4.5. Remark . Theorem 4.5 does not hold for higher dimensional graded rings. Infact, consider the ring R = k [ x, y, z, w ] / ( x + y + z + w p ). Then R has rationalsingularities but R is not F -rational if 3 does not divide p − -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 13 Classification of R ( P k , D ) which is a rational triple point andrational fourth point In this section we classify normal graded rings R ( P k , D ) with a rational singu-larity and e ( R ( P k , D )) = 3 and 4.5.1. Preliminaries of classification of R ( P k , D ) . In this subsection, we give re-sults for the classification of R ( P k , D ) with a rational singularity and e ( R ( P k , D )) =3 and 4. Lemma 5.1.
Let D = P ri =1 a i P i and D = P ri =1 b i P i be ample Q -divisors on P k , where P i are distinct points of P k . Assume a i ≥ b i for any i . If R ( P k , D ) hasa rational singularity, then R ( P k , D ) has a rational singularity.Proof. Since deg[ nD ] ≥ deg[ nD ] for any n ∈ N , R ( P k , D ) has a rational singu-larity by Theorem 2.16. (cid:3) Lemma 5.2.
Let D = sP − P ri =1 c i d i P i be an ample Q -divisor on P k , where s, c i , d i ∈ N with < c i < d i , and P i are distinct points of P k . Let R = R ( P k , D ) and f : X → Spec( R ) the minimal good resolution. Assume that R has a rationalsingularity. (1) If e ( R ) = 3 , then the dual graph of f has the following property;There is unique ( − -curve and others are ( − -curves. In this case, D isone of the following: for some n i , a, b ∈ Z ≥ , P − X i =1 n i n i + 1 P i or P − X i =1 n i n i + 1 P i − a , , (2) b ]] P . (2) If e ( R ) = 4 , then the dual graph of f has one of the following properties; (a) There is unique ( − -curve and others are ( − -curves. In this case, D is one of the following: for some n i , a, b ∈ Z ≥ , P − X i =1 n i n i + 1 P i or P − X i =1 n i n i + 1 P i − a , , (2) b ]] P . (b) There is unique ( − -curve and others are ( − -curves. In this case, D is one of the following: for some n i , a, b ∈ Z ≥ , P − X i =1 n i n i + 1 P i or P − X i =1 n i n i + 1 P i − a , , (2) b ]] P . (c) There are two ( − -curves and others are ( − -curves. In this case, D is one of the following: for some n i , n, a, b, c, d ∈ Z ≥ , P − X i =1 n i n i + 1 P i − a , , (2) b ]] P , P − X i =1 n i n i + 1 P i − a , , (2) b , , (2) c ]] P or P − nn + 1 P − a , , (2) b ]] P − c , , (2) d ]] P . Proof.
We prove only (2), as (1) is proved similarly. By Proposition 2.19, the dualgraph of f has one of the following properties;(a) There is unique ( − − − − − − s +1 ≥ r . Let Z be the fundamental cycle of f , and let E be the central curve of f . If s + 1 = r , then Coeff E ( Z ) ≥ E is a ( − r = 5, then e ( R ) ≥ − E is a ( − r = 4,then e ( R ) ≥ m ]] = m +1 m for m ∈ N by Example 2.10. By Theorem 2.14, wecan determine the coefficients of D . (cid:3) Lemma 5.3.
Let n, a, b ∈ Z ≥ with n ≥ . Then we have [[(2) a , n, (2) b ]] = (cid:0) ( a + 1) n − (2 a + 1) (cid:1) b + ( a + 1) n − a (cid:0) an − (2 a − (cid:1) b + an − ( a − . Proof.
Note that [[(2) m ]] = m +1 m for m ∈ N by Example 2.10. We prove this byinduction on a . If a = 0, then[[(2) a , n, (2) b ]] = [[ n, b + 1 b ]] = n − bb + 1 = ( n − b + nb + 1 . If a >
0, then[[(2) a +1 , n, (2) b ]] = [[2 , (2) a , n, (2) b ]]= [[2 , (cid:0) ( a + 1) n − (2 a + 1) (cid:1) b + ( a + 1) n − a (cid:0) an − (2 a − (cid:1) b + an − ( a −
1) ]]= 2 − (cid:0) an − (2 a − (cid:1) b + an − ( a − (cid:0) ( a + 1) n − (2 a + 1) (cid:1) b + ( a + 1) n − a = (cid:0) ( a + 2) n − (2 a + 3) (cid:1) b + ( a + 2) n − ( a + 1) (cid:0) ( a + 1) n − (2 a + 1) (cid:1) b + ( a + 1) n − a . (cid:3) -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 15 Lemma 5.4.
Let a, b, c ∈ Z ≥ . Then we have [[(2) a , , (2) b , , (2) c ]] = (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8 (cid:0) ( a + 1) b + 3 a + 2 (cid:1) c + (2 a + 2) b + 5 a + 3 . Proof.
We prove this by induction on a . If a = 0, then by Lemma 5.3[[(2) a , , (2) b , , (2) c ]] = [[3 , ( b + 2) c + 2 b + 3( b + 1) c + 2 b + 1 ]] = (2 b + 5) c + 4 b + 8( b + 2) c + 2 b + 3 . If a >
0, then[[(2) a +1 , , (2) b , , (2) c ]] = [[2 , (2) a , , (2) b , , (2) c ]]= [[2 , (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8 (cid:0) ( a + 1) b + 3 a + 2 (cid:1) c + (2 a + 2) b + 5 a + 3 ]]= 2 − (cid:0) ( a + 1) b + 3 a + 2 (cid:1) c + (2 a + 2) b + 5 a + 3 (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8= (cid:0) ( a + 3) b + 3 a + 8 (cid:1) c + (2 a + 6) b + 5 a + 13 (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8 . (cid:3) We will use the following result to check whether R ( P k , D ) has a rational sin-gularity for D in the list of Lemma 5.2 by Lemma 5.1 in next subsections. Lemma 5.5.
Let D = 2 P − a P − a P − a P be a Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Then R ( P k , D ) has a rational singularity, if ( a , a , a ) is equal to ( , , nn +1 ) for some n ∈ Z ≥ or ( , , ) .Proof. If ( a , a , a ) = ( , , nn +1 ) for some n ∈ Z ≥ , then for any l ∈ N ,deg[ lD ] = 2 l − ⌈ l ⌉ − ⌈ l ⌉ − ⌈ lnn + 1 ⌉ ≥ [ l − ⌈ l ⌉ ≥ − , which implies that R ( P k , D ) has a rational singularity by Theorem 2.16.If ( a , a , a ) = ( , , ), then deg[ lD ] ≥ − l ∈ N with 1 ≤ l ≤ D ] = 1. Therefore deg[ lD ] ≥ − l ∈ N . Hence R ( P k , D ) has arational singularity by Theorem 2.16. (cid:3) In next subsections, we determine D in the list of Lemma 5.2 such that R ( P k , D )has a rational singularity with e ( R ( P k , D )) = 3 and 4 using the following steps:(1) We will check whether R ( P k , D ) has a rational singularity by Theorem2.16 or Lemma 5.1.(2) We will determine the fundamental cycle of the minimal good resolutionof Spec( R ( P k , D )) by Theorem 2.14, Lemma 2.21 and Corollary 2.22.(3) We will determine e ( R ( P k , D )) by Proposition 2.19.(4) We will compute the Hirzebruch-Jung continued fractions [[(2) a , , (2) b ]],[[(2) a , , (2) b ]], [[(2) a , , (2) b , , (2) c ]] by Lemma 5.3 and Lemma 5.4. The case there is unique ( − -curve. In this subsection we classify the R ( P k , D ) with a rational singularity such that there is unique ( − R ( P k , D )) and others are ( − − Proposition 5.6.
Let D = 3 P − P i =1 a i P i be an ample Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Assume that a ≤ a ≤ a ≤ a , a = aa +1 , a = bb +1 , a = cc +1 and a = dd +1 for a, b, c, d ∈ Z ≥ and R ( P k , D ) hasa rational singularity. Then ( a , a , a , a ) = (0 , bb +1 , cc +1 , dd +1 ) for ≤ b ≤ c ≤ d or ( , , cc +1 , dd +1 ) for ≤ c ≤ d . Moreover if ( a , a , a , a ) = (0 , bb +1 , cc +1 , dd +1 ) for ≤ b ≤ c ≤ d , then e ( R ( P k , D )) = 3 and if ( a , a , a , a ) = ( , , cc +1 , dd +1 ) for ≤ c ≤ d , then e ( R ( P k , D )) = 4 .Proof. Since R ( P k , D ) has a rational singularity, we have deg[2 D ] ≥ −
1. Therefore a = 0 or a = b = 1.Assume ( a , a , a , a ) = (0 , bb +1 , cc +1 , dd +1 ) for 0 ≤ b ≤ c ≤ d . R ( P k , D ) hasa rational singularity since deg[ lD ] ≥ l ∈ N . The dual graph of theminimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ❉❉❉❉❉❉ ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ③③③③③③ Therefore e ( R ( P k , D )) = 3.Assume ( a , a , a , a ) = ( , , cc +1 , dd +1 ) for 1 ≤ c ≤ d . Thendeg[ lD ] = 3 l + [ − l − l − lcc + 1 ] + [ − ldd + 1 ] ≥ [ l − l ≥ − l ∈ N . Therefore R ( P k , D ) has a rational singularity. The dual graph ofthe minimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ❉❉❉❉❉❉ ONMLHIJK − ONMLHIJK − ❉❉❉❉❉❉ ③③③③③③ ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ③③③③③③ ONMLHIJK − Therefore e ( R ( P k , D )) = 4. (cid:3) Next, we consider the case the central curve is a ( − Proposition 5.7.
Let D = 2 P − P i =1 a i P i be an ample Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Assume that a ≤ a , a = mm +1 , a = nn +1 , a = [[(2) a , , (2) b ]] for m, n, a, b ∈ Z ≥ and R ( P k , D ) has a rationalsingularity. -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 17 (i) If e ( R ( P k , D )) = 3 , then ( a , a , a ) is one of the following: (1) (0 , nn + 1 , ( a + 1) b + 2 a + 1( a + 2) b + 2 a + 3 ) for n ≥ , a ≥ , b ≥ , (2) ( 12 , nn + 1 , b + 12 b + 3 ) for n ≥ , b ≥ , , ( a + 1) b + 2 a + 1( a + 2) b + 2 a + 3 ) for a ≥ , b ≥ , (4) ( 12 , , b + 33 b + 5 ) for b ≥ , (5) ( 12 , , b + 54 b + 7 ) for b ≥ , (6) ( 12 , ,
79 ) , (7) ( 12 , ,
35 ) , (8) ( 12 , ,
35 ) , (9) ( 23 , nn + 1 ,
13 ) for n ≥ . (ii) If e ( R ( P k , D )) = 4 , then ( a , a , a ) is one of the following: (1) ( 12 , , b + 75 b + 9 ) for b ≥ , (2) ( 12 , , b + 33 b + 5 ) for b ≥ , (3) ( 12 , , b + 33 b + 5 ) for b ≥ , (4) ( 12 , ,
35 ) , (5) ( 12 , ,
35 ) , (6) ( 23 , , b + 12 b + 3 ) for b ≥ , (7) ( 23 , , b + 12 b + 3 ) for b ≥ , (8) ( 23 , ,
25 ) , (9) ( 34 , ,
13 ) , (10) ( 34 , ,
13 ) , (11) ( 34 , ,
13 ) . Proof.
Case 1.
We assume that m = 0. Then R ( P k , D ) has a rational singularitysince deg[ lD ] ≥ l ∈ N . The dual graph of the minimal good resolutionof Spec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 3. Case 2.
We assume that m = 1 and a = 0. Note that D ≥ P − P − nn +1 P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity by Lemma 5.1 andLemma 5.5. The dual graph of the minimal good resolution of Spec( R ( P k , D )) isthe following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 3. Case 3.
We assume that m = n = 1 and a ≥
1. Note that D ≥ P − P − P − a + b +1 a + b +2 P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity byLemma 5.1 and Lemma 5.5. The dual graph of the minimal good resolution ofSpec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 3. Case 4.
We assume that m = 1, n = 2 and 1 ≤ a ≤
3. Note that D ≥ P − P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularityby Lemma 5.1 and Lemma 5.5. The dual graphs of the minimal good resolutionof Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 3 when 1 ≤ a ≤ a = 3 and b = 0 and e ( R ( P k , D )) =4 when a = 3 and b ≥ Case 5.
We assume that m = 1, n = 2 and a ≥
4. Then R ( P k , D ) does not havea rational singularity since deg[5 D ] ≤ − Case 6.
We assume that m = 1, 3 ≤ n ≤ a = 1. Note that D ≥ P − P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularityby Lemma 5.1 and Lemma 5.5. The dual graphs of the minimal good resolutionof Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 19 ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 3 when n = 3 , b = 0 and e ( R ( P k , D )) = 4 when n = 3 , b ≥ Case 7.
We assume that m = 1, 5 ≤ n ≤ a = 1 and b = 0. Let D ′ =2 P − P − P − P . Then deg[ lD ′ ] ≥ − l ∈ N since deg[ lD ′ ] ≥ − ≤ l ≤
69 and deg[70 D ′ ] ≥
1. Note that D ≥ D ′ by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity by Lemma 5.1. The dual graphs of the minimalgood resolution of Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 8.
We assume that m = 1, n ≥ a = 1 and b ≥
1. Then R ( P k , D ) doesnot have a rational singularity since deg[5 D ] ≤ − Case 9.
We assume that m = 1, n ≥ a ≥
1. Then R ( P k , D ) does not havea rational singularity since deg[7 D ] ≤ − Case 10.
We assume that m = 2, a = 0 and b = 0. Then R ( P k , D ) has a rationalsingularity since deg[ lD ] ≥ l + [ − l ] − l + [ − l ] = [ l ] + [ − l ] ≥ − l ∈ N .The dual graph of the minimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 3. Case 11.
We assume that m = 2, 2 ≤ n ≤ a = 0 and b ≥
1. Note that D ≥ P − P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rationalsingularity by Lemma 5.1 and Lemma 5.5. The dual graphs of the minimal good resolution of Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4 when n = 2 , n = 4 and b = 1 and e ( R ( P k , D )) = 5when n = 4 and b ≥ Case 12.
We assume that m = 2, n ≥ a = 0 and b ≥
1. Let E be the centralcurve of the minimal good resolution of Spec( R ( P k , D )) and E be the ( − Z be the fundamental cycle of the minimal goodresolution of Spec( R ( P k , D )). Then Coeff E ( Z ) ≥ E ( Z ) ≥
3. Hence e ( R ( P k , D )) ≥ Case 13.
We assume that m ≥ a ≥
1. Then R ( P k , D ) does not have arational singularity since deg[2 D ] ≤ − Case 14.
We assume that m = 3, 3 ≤ n ≤ a = b = 0. Let D ′ =2 P − P − P − P . Then deg[ lD ′ ] ≥ − l ∈ N since deg[ lD ′ ] ≥ − ≤ l ≤
83 and deg[84 D ′ ] ≥
1. Note that D ≥ D ′ by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity by Lemma 5.1. The dual graphs of the minimalgood resolution of Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 21 ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4 when 3 ≤ n ≤ e ( R ( P k , D )) = 5 when n = 6. Case 15.
We assume that m ≥ n ≥ a = 0. Then R ( P k , D ) does nothave a rational singularity since deg[7 D ] ≤ − Case 16.
We assume that m ≥ b ≥
1. Then R ( P k , D ) does not have arational singularity since deg[3 D ] ≤ − (cid:3) The case there is unique ( − -curve. In this subsection we classify the R ( P k , D ) with a rational singularity such that there is unique ( − R ( P k , D )) and others are ( − − Proposition 5.8.
Let D = 4 P − P i =1 a i P i be an ample Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Assume that a = aa +1 , a = bb +1 , a = cc +1 and a = dd +1 for a, b, c, d ∈ Z ≥ . Then R ( P k , D ) has a rational singularitywith e ( R ( P k , D )) = 4 .Proof. R ( P k , D ) has a rational singularity since deg[ lD ] ≥ l ∈ N . Thedual graph of the minimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ❉❉❉❉❉❉ ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ❉❉❉❉❉❉ ③③③③③③ ONMLHIJK − · · · ONMLHIJK − ③③③③③③ ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 4. (cid:3) Next, we consider the case the central curve is a ( − Proposition 5.9.
Let D = 2 P − P i =1 a i P i be an ample Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Assume that a ≤ a , a = mm +1 , a = nn +1 , a = [[(2) a , , (2) b ]] for m, n, a, b ∈ Z ≥ and R ( P k , D ) has a rational singularity with e ( R ( P k , D )) = 4 . Then ( a , a , a ) is one of the following: (1) (0 , nn + 1 , (2 a + 1) b + 3 a + 1(2 a + 3) b + 3 a + 4 ) for n ≥ , a ≥ , b ≥ , (2) ( 12 , nn + 1 , b + 13 b + 4 ) for n ≥ , b ≥ , , (2 a + 1) b + 3 a + 1(2 a + 3) b + 3 a + 4 ) for a ≥ , b ≥ , (4) ( 12 , , b + 45 b + 7 ) for b ≥ , (5) ( 12 , , b + 77 b + 10 ) for b ≥ , (6) ( 12 , , b + 109 b + 13 ) for b ≥ , (7) ( 12 , , b + 45 b + 7 ) for b ≥ , (8) ( 12 , , b + 45 b + 7 ) for b ≥ , (9) ( 12 , ,
47 ) , (10) ( 12 , ,
47 ) , (11) ( 23 , nn + 1 , b + 13 b + 4 ) for n ≥ , b ≥ , (12) ( 34 , nn + 1 ,
14 ) for n ≥ . Proof.
Case 1.
We assume that m = 0. Then R ( P k , D ) has a rational singularitysince deg[ lD ] ≥ l ∈ N . The dual graph of the minimal good resolutionof Spec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 2.
We assume that m = 1 and a = 0. Note that D ≥ P − P − nn +1 P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity by Lemma 5.1 andLemma 5.5. The dual graph of the minimal good resolution of Spec( R ( P k , D )) isthe following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 3.
We assume that m = n = 1 and a ≥
1. Note that D ≥ P − P − P − a + b +1 a + b +2 P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity byLemma 5.1 and Lemma 5.5. The dual graph of the minimal good resolution ofSpec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 23 Case 4.
We assume that m = 1, n = 2 and 1 ≤ a ≤
3. Note that D ≥ P − P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularityby Lemma 5.1 and Lemma 5.5. The dual graphs of the minimal good resolutionof Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 5.
We assume that m = 1, n = 2 and a ≥
4. Then R ( P k , D ) does not havea rational singularity since deg[5 D ] ≤ − Case 6.
We assume that m = 1, 3 ≤ n ≤ a = 1. Note that = [[2 , D ≥ P − P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rationalsingularity by Lemma 5.1 and Proposition 5.7(ii)(5). The dual graphs of theminimal good resolution of Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4 when n = 3 , n = 5 , b = 0 and e ( R ( P k , D )) =6 when n = 5 , b ≥ Case 7.
We assume that m = 1, n ≥ a ≥
2. Then R ( P k , D ) does not havea rational singularity since deg[3 D ] ≤ − E be the central curve of the minimal good resolution of Spec( R ( P k , D ))and E be the ( − Z be the fundamental cycle ofthe minimal good resolution of Spec( R ( P k , D )). Case 8.
We assume that m = 1, 7 ≤ n ≤ a = 1 and b = 0. ThenCoeff E ( Z ) ≥ E ( Z ) ≥ e ( R ( P k , D )) ≥ Case 9.
We assume that m = 1, n ≥ a ≥ b ≥
1. Then R ( P k , D ) doesnot have a rational singularity since deg[7 D ] ≤ − Case 10.
We assume that m = 1, n ≥ a = 1. Then R ( P k , D ) does nothave a rational singularity since deg[9 D ] ≤ − Case 11.
We assume that m = 2 and a = 0. Note that D ≥ P − P − nn +1 P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity by Lemma5.1 and Proposition 5.7(i)(9). The dual graph of the minimal good resolution ofSpec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 12.
We assume that m ≥ a ≥
1. Then R ( P k , D ) does not have arational singularity since deg[2 D ] ≤ − Case 13.
We assume that m = 3 and a = b = 0. Then R ( P k , D ) has a rationalsingularity since deg[ lD ] ≥ l + [ − l ] − l + [ − l ] = [ l ] + [ − l ] ≥ − l ∈ N .The dual graph of the minimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 25 Case 14.
We assume that m = 3, a = 0 and b ≥
1. Then Coeff E ( Z ) ≥ E ( Z ) ≥ e ( R ( P k , D )) ≥ Case 15.
We assume that m ≥ a = 0. Then Coeff E ( Z ) ≥ E ( Z ) ≥ e ( R ( P k , D )) ≥ (cid:3) The case there are two ( − -curves. In this subsection we classify the R ( P k , D ) with a rational singularity such that there are two ( − R ( P k , D )) and others are ( − − Proposition 5.10.
Let D = 3 P − P i =1 a i P i be an ample Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Assume that a = mm +1 , a = nn +1 , a = [[(2) a , , (2) b ]] for m, n, a, b ∈ Z ≥ . Then R ( P k , D ) has a rational singularitywith e ( R ( P k , D )) = 4 .Proof. R ( P k , D ) has a rational singularity since deg[ lD ] ≥ l ∈ N . Thedual graph of the minimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ❉❉❉❉❉❉ ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ③③③③③③ Therefore e ( R ( P k , D )) = 4. (cid:3) Next, we consider the case the central curve is a ( − − Proposition 5.11.
Let D = 2 P − P i =1 a i P i be an ample Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Assume that a ≤ a , a = mm +1 , a = nn +1 , a = [[(2) a , , (2) b , , (2) c ]] for m, n, a, b, c ∈ Z ≥ and R ( P k , D ) has arational singularity with e ( R ( P k , D )) = 4 . Then ( a , a , a ) is one of the following: (1) (0 , nn + 1 , (cid:0) ( a + 1) b + 3 a + 2 (cid:1) c + (2 a + 2) b + 5 a + 3 (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8 ) for n, a, b, c ≥ , (2) ( 12 , nn + 1 , ( b + 2) c + 2 b + 3(2 b + 5) c + 4 b + 8 ) for n ≥ , b ≥ , c ≥ , (3) ( 12 , , (cid:0) ( a + 1) b + 3 a + 2 (cid:1) c + (2 a + 2) b + 5 a + 3 (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8 ) for a ≥ , b ≥ , c ≥ , (4) ( 12 , , (2 b + 5) c + 4 b + 8(3 b + 8) c + 6 b + 13 ) for b ≥ , c ≥ , (5) ( 12 , , (3 b + 8) c + 6 b + 13(4 b + 11) c + 8 b + 18 ) for b ≥ , c ≥ . Proof.
Case 1.
We assume that m = 0. Then R ( P k , D ) has a rational singularitysince deg[ lD ] ≥ l ∈ N . The dual graph of the minimal good resolutionof Spec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 2.
We assume that m = 1 and a = 0. Note that D ≥ P − P − nn +1 P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity by Lemma 5.1 andLemma 5.5. The dual graph of the minimal good resolution of Spec( R ( P k , D )) isthe following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 3.
We assume that m = n = 1 and a ≥
1. Note that D ≥ P − P − P − a + b + c +2 a + b + c +3 P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity byLemma 5.1 and Lemma 5.5. The dual graph of the minimal good resolution ofSpec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 4.
We assume that m = 1, n = 2 and 1 ≤ a ≤
3. Note that D ≥ P − P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularityby Lemma 5.1 and Lemma 5.5. The dual graph of the minimal good resolutionof Spec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − Therefore e ( R ( P k , D )) = 4 when a = 1 , e ( R ( P k , D )) = 5 when a = 3. -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 27 Case 5.
We assume that m = 1, n = 2 and a ≥
4. Then R ( P k , D ) does not havea rational singularity since deg[5 D ] ≤ − E be the central curve of the minimal good resolution of Spec( R ( P k , D ))and E , E be the ( − Z be the fundamental cycleof the minimal good resolution of Spec( R ( P k , D )). Case 6.
We assume that m = 1, n ≥ a = 1. Then Coeff E ( Z ) ≥ E ( Z ) + Coeff E ( Z ) ≥ e ( R ( P k , D )) ≥ Case 7.
We assume that m = 1, n ≥ a ≥
2. Then R ( P k , D ) does not havea rational singularity since deg[3 D ] ≤ − Case 8.
We assume that m ≥ a = 0. Then Coeff E ( Z ) ≥ E ( Z ) + Coeff E ( Z ) ≥ e ( R ( P k , D )) ≥ Case 9.
We assume that m ≥ a ≥
1. Then R ( P k , D ) does not have arational singularity since deg[2 D ] ≤ − (cid:3) Finally, we consider the case the central curve is a ( − − Proposition 5.12.
Let D = 2 P − P i =1 a i P i be an ample Q -divisor on P k , where a i ∈ Q ≥ and P i are distinct points of P k . Assume that a ≤ a , a = mm +1 , a = [[(2) a , , (2) b ]] , a = [[(2) c , , (2) d ]] for m, a, b, c, d ∈ Z ≥ and R ( P k , D ) has arational singularity with e ( R ( P k , D )) = 4 . Then ( a , a , a ) is one of the following: (1) (0 , ( a + 1) b + 2 a + 1( a + 2) b + 2 a + 3 , ( c + 1) d + 2 c + 1( c + 2) d + 2 c + 3 ) for a ≥ , b ≥ , c ≥ , d ≥ mm + 1 , b + 12 b + 3 , d + 12 d + 3 ) for m ≥ , b ≥ , d ≥ , b + 12 b + 3 , ( c + 1) d + 2 c + 1( c + 2) d + 2 c + 3 ) for b ≥ , c ≥ , d ≥ , b + 33 b + 5 , d + 33 d + 5 ) for b ≥ , d ≥ , (5) ( 12 , , d + 54 d + 7 ) for d ≥ , (6) ( 12 , , d + 75 d + 9 ) for d ≥ , (7) ( 23 , , ( c + 1) d + 2 c + 1( c + 2) d + 2 c + 3 ) for c ≥ , d ≥ , (8) ( mm + 1 , , d + 33 d + 5 ) for m ≥ , d ≥ . Proof.
Note that a ≤ c by Lemma 2.9. Case 1.
We assume that m = 0. Then R ( P k , D ) has a rational singularity sincedeg[ lD ] ≥ l ∈ N . The dual graph of the minimal good resolution ofSpec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 2.
We assume that m ≥ a = c = 0. Note that D ≥ P − mm +1 P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularityby Lemma 5.1 and Lemma 5.5. The dual graph of the minimal good resolutionof Spec( R ( P k , D )) is the following: ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ❉❉❉❉❉❉ ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ③③③③③③ Therefore e ( R ( P k , D )) = 4. Case 3.
We assume that m = 1, a = 0 and c ≥
1. Note that D ≥ P − P − P − c + d +1 c + d +2 P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularity byLemma 5.1 and Lemma 5.5. The dual graph of the minimal good resolution ofSpec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 4.
We assume that m = 1, a = 1 and 1 ≤ c ≤
3. Note that D ≥ P − P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rational singularityby Lemma 5.1 and Lemma 5.5. The dual graphs of the minimal good resolutionof Spec( R ( P k , D )) are the following: ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 29 ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 4 when c = 1 or c = 2 , b = 0, e ( R ( P k , D )) = 5when c = 2 and b ≥ e ( R ( P k , D )) = 6 when c = 3 and b ≥ E be the central curve of the minimal good resolution of Spec( R ( P k , D ))and E be the ( − P in its dual graph. Let Z be the fundamental cycle of the minimal good resolution of Spec( R ( P k , D )). Case 5.
We assume that m = 1, a = 1 and c ≥
4. Then Coeff E ( Z ) ≥ E ( Z ) ≥
2. Hence e ( R ( P k , D )) ≥ Case 6.
We assume that m ≥ a ≥ c ≥
2. Then R ( P k , D ) does not havea rational singularity since deg[3 D ] ≤ − Case 7.
We assume that m = 2, a = b = 0 and c ≥
1. Note that D ≥ P − P − P − c + d +1 c + d +2 P by Lemma 2.9. Therefore R ( P k , D ) has a rationalsingularity by Lemma 5.1 and Proposition 5.7.(i).(9). The dual graph of theminimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − Therefore e ( R ( P k , D )) = 4. Case 8.
We assume that m ≥ a = 0, b ≥ c ≥
1. Then Coeff E ( Z ) ≥ E ( Z ) ≥
2. Hence e ( R ( P k , D )) ≥ Case 9.
We assume that m ≥ a ≥
1. Then R ( P k , D ) does not have arational singularity since deg[2 D ] ≤ − Case 10.
We assume that m ≥ a = b = 0 and c = 1. Note that D ≥ P − mm +1 P − P − P by Lemma 2.9. Therefore R ( P k , D ) has a rationalsingularity by Lemma 5.1 and Proposition 5.7.(i).(9). The dual graph of the minimal good resolution of Spec( R ( P k , D )) is the following: ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ❉❉❉❉❉❉ ONMLHIJK − ONMLHIJK − ONMLHIJK − ONMLHIJK − · · · ONMLHIJK − ONMLHIJK − ③③③③③③ Therefore e ( R ( P k , D )) = 4. Case 11.
We assume that m ≥ a = b = 0 and c ≥
2. Then Coeff E ( Z ) ≥ E ( Z ) ≥
2. Hence e ( R ( P k , D )) ≥ (cid:3) Classification of R ( P k , D ) which is a rational triple point and rationalfourth point. In this subsection, we summarize our results of this section in thefollowing theorem.
Theorem 5.13.
Let D = sP − P ri =1 a i P i be an ample Q -divisor on P k , where s ∈ N and a i ∈ Q with ≤ a i < and P i are distinct points of P k . Assumethat R ( P k , D ) has a rational singularity. Suppose if T ( a i ) = T ( a j ) for i < j , then a i ≤ a j , and if T ( a i ) = ∅ and T ( a j ) = ∅ , then i < j , where T ( ∗ ) is defined inDefinition 4.2. (1) If e ( R ( P k , D )) = 3 , then ( s, a , . . . , a r ) is one of the following: Here, n , a , b , c are any non-negative integers.
1. (3 , aa +1 , bb +1 , cc +1 ), 2. (2 , , nn +1 , ( a +1) b +2 a +1( a +2) b +2 a +3 ),3. (2 , , n +1 n +2 , b +12 b +3 ), 4. (2 , , , ( a +2) b +2 a +3( a +3) b +2 a +5 ),5. (2 , , , b +33 b +5 ), 6. (2 , , , b +54 b +7 ), 7. (2 , , , ),8. (2 , , , ), 9. (2 , , , ), 10. (2 , , n +2 n +3 , ),(2) If e ( R ( P k , D )) = 4 , then ( s, a , . . . , a r ) is one of the following: Here, m , n , a , b , c , d are any non-negative integers.
1. (3 , , , cc +1 , dd +1 ), 2. (2 , , , b +115 b +14 ), 3. (2 , , , b +53 b +8 ),4. (2 , , , b +53 b +8 ), 5. (2 , , , ), 6. (2 , , , ),7. (2 , , , b +22 b +5 ), 8. (2 , , , b +22 b +5 ), 9. (2 , , , ),10. (2 , , , ), 11. (2 , , , ), 12. (2 , , , ),13. (4 , aa +1 , bb +1 , cc +1 , dd +1 ), 14. (2 , , nn +1 , (2 a +1) b +3 a +1(2 a +3) b +3 a +4 ),15. (2 , , n +1 n +2 , b +13 b +4 ), 16. (2 , , , (2 a +3) b +3 a +4(2 a +5) b +3 a +7 ),17. (2 , , , b +45 b +7 ), 18. (2 , , , b +77 b +10 ), 19. (2 , , , b +109 b +13 ),20. (2 , , , b +45 b +7 ), 21. (2 , , , b +45 b +7 ), 22. (2 , , , ),23. (2 , , , ), 24. (2 , , n +2 n +3 , b +13 b +4 ), 25. (2 , , n +3 n +4 , ),26. (3 , mm +1 , nn +1 , ( a +1) b +2 a +1( a +2) b +2 a +3 ), 27. (2 , , nn +1 , (cid:0) ( a +1) b +3 a +2 (cid:1) c +(2 a +2) b +5 a +3 (cid:0) ( a +2) b +3 a +5 (cid:1) c +(2 a +4) b +5 a +8 ), -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 31
28. (2 , , n +1 n +2 , ( b +2) c +2 b +3(2 b +5) c +4 b +8 ), 29. (2 , , , (cid:0) ( a +2) b +3 a +5 (cid:1) c +(2 a +4) b +5 a +8 (cid:0) ( a +3) b +3 a +8 (cid:1) c +(2 a +6) b +5 a +13 ),30. (2 , , , (2 b +5) c +4 b +8(3 b +8) c +6 b +13 ), 31. (2 , , , (3 b +8) c +6 b +13(4 b +11) c +8 b +18 ),32. (2 , , ( a +1) b +2 a +1( a +2) b +2 a +3 , ( c +1) d +2 c +1( c +2) d +2 c +3 ), 33. (2 , m +1 m +2 , b +12 b +3 , d +12 d +3 ),34. (2 , , b +12 b +3 , ( c +2) d +2 c +3( c +3) d +2 c +5 ), 35. (2 , , b +33 b +5 , d +33 d +5 ),36. (2 , , , d +54 d +7 ), 37. (2 , , , d +75 d +9 ),38. (2 , , , ( c +2) d +2 c +3( c +3) d +2 c +5 ), 39. (2 , m +1 m +2 , , d +33 d +5 ).6. F -rationality of two-dimensional graded rings with rationaltriple point and rational fourth point In this section, we determine p (3) and p (4) in Theorem 1.3 using the classifica-tion in Section 5.We can reduce the calculation to check the F -rationality of R ( P k , D ) using thefollowing lemma when we prove the theorems in this section. Lemma 6.1.
Let D = 2 P − P i =1 b i P i be an ample Q -divisor on P k , where b i ∈ Q > and P i are distinct points of P k . (1) If ( b , b , b ) = ( , , nn +1 ) for n ∈ N , then deg[ − lD ] ≤ − for l ∈ N \ N and deg[ − lD ] ≤ − for l ∈ N . (2) If ( b , b , b ) = ( , , ) , then deg[ − lD ] ≤ − for l ∈ N with l = 2 , , , , , . (3) If ( b , b , b ) = ( , , ) , then deg[ − lD ] ≤ − for l ∈ N with l = 2 , , , , , , , , , , , , , , . (4) If ( b , b , b ) = ( , , nn +1 ) for n ∈ N , then deg[ − lD ] ≤ − for l ∈ N \ N and deg[ − lD ] ≤ − for l ∈ N . (5) If ( b , b , b ) = ( , , nn +1 ) for n ∈ N , then deg[ − lD ] ≤ − for l ∈ N \ N and deg[ − lD ] ≤ − for l ∈ N .Proof. This lemma follows immediately from the calculation. (cid:3)
Theorem 6.2.
Let R be a two-dimensional graded ring with e ( R ) = 3 and arational singularity. If p ≥ , then R is F -rational. Furthermore, this inequalityis best possible.Proof. Example 4.6 shows that there exists a two-dimensional non- F -rationalgraded ring R with a rational singularity, e ( R ) = 3 and p = 5.From now on, we assume that p ≥
7. By Theorem 2.12, Theorem 2.16 andTheorem 5.13, there exists an ample Q -divisor D on P k in the list of Theorem5.13.(1) with R ∼ = R ( P k , D ). Let D = sP − P i =1 a i P i , where s ∈ N , 0 ≤ a i < P i are distinct points of P k . Let n, a, b, c be non-negative integers. If necessary,we switch the order of ( a , a , a ). Case 1.
We assume that ( s, a , a , a ) is one of the followings:(3 , aa + 1 , bb + 1 , cc + 1 ) , (2 , , nn + 1 , ( a + 1) b + 2 a + 1( a + 2) b + 2 a + 3 ) . Then R ( P k , D ) is F -rational by Proposition 3.3.(1). Case 2.
We assume that s = 2 and ( a , a , a ) is one of the followings:( 12 , ,
79 ) , ( 12 , ,
35 ) , ( 12 , ,
35 ) . Then R ( P k , D ) is F -rational by Theorem 3.6. Case 3.
We assume that s = 2 and ( a , a , a ) is one of the followings:( 12 , b + 12 b + 3 , n + 1 n + 2 ) , ( 12 , , ( a + 2) b + 2 a + 3( a + 3) b + 2 a + 5 ) . Then D ≥ P − P − P − ll +1 P for sufficiently large number l . Therefore R ( P k , D ) is F -rational by Theorem 3.1 and Lemma 6.1. Case 4.
We assume that s = 2 and ( a , a , a ) is one of the followings:( 12 , , b + 33 b + 5 ) , ( 12 , , b + 54 b + 7 ) . Then D ≥ P − P − P − P . Therefore R ( P k , D ) is F -rational by Theorem3.1 and Lemma 6.1. Case 5.
We assume that ( s, a , a , a ) = (2 , , , n +2 n +3 ) . Then D ≥ P − P − P − ll +1 P for sufficiently large number l . Therefore R ( P k , D ) is F -rational byTheorem 3.1 and Lemma 6.1.By the above discussion, if p ≥
7, then R is F -rational. (cid:3) Theorem 6.3.
Let R be a two-dimensional graded ring with e ( R ) = 4 and arational singularity. If p ≥ , then R is F -rational. Furthermore, this inequalityis best possible.Proof. Example 4.6 shows that there exists a two-dimensional non- F -rationalgraded ring R with a rational singularity, e ( R ) = 4 and p = 7.From now on, we assume that p ≥
11. By Theorem 2.12, Theorem 2.16 andTheorem 5.13, there exists an ample Q -divisor D on P k in the list of Theorem5.13.(2) with R ∼ = R ( P k , D ). Let D = sP − P ri =1 a i P i , where s ∈ N , 0 ≤ a i < P i are distinct points of P k . Let m, n, a, b, c, d be non-negative integers. Case 1.
We assume that ( s, a , . . . , a r ) = (3 , , , cc +1 , dd +1 ) . we have deg[ − lD ] ≤− l ∈ N and deg[ − lD ] ≤ − l ∈ N \ N . Then R ( P k , D ) is F -rationalby Theorem 3.1. Case 2.
We assume that s = 2 and ( a , a , a ) is one of the followings:( 12 , ,
35 ) , ( 12 , ,
35 ) , ( 23 , ,
25 ) , ( 34 , ,
13 ) , ( 34 , ,
13 ) , ( 34 , ,
13 ) , ( 12 , ,
47 ) , ( 12 , ,
47 ) . Then R ( P k , D ) is F -rational by Theorem 3.6. -RATIONALITY OF 2-DIMENSIONAL GRADED RINGS 33 Case 3.
We assume that ( s, a , . . . , a r ) is one of the followings:(4 , aa + 1 , bb + 1 , cc + 1 , dd + 1 ) , (2 , , nn + 1 , (2 a + 1) b + 3 a + 1(2 a + 3) b + 3 a + 4 ) , (3 , mm + 1 , nn + 1 , ( a + 1) b + 2 a + 1( a + 2) b + 2 a + 3 ) , (2 , , nn + 1 , (cid:0) ( a + 1) b + 3 a + 2 (cid:1) c + (2 a + 2) b + 5 a + 3 (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8 ) , (2 , , ( a + 1) b + 2 a + 1( a + 2) b + 2 a + 3 , ( c + 1) d + 2 c + 1( c + 2) d + 2 c + 3 ) . Then R ( P k , D ) is F -rational by Proposition 3.3.(1).In the rest of this proof, we always assume that s = 2 and r = 3. If necessary,we switch the order of ( a , a , a ). Case 4.
We assume that ( a , a , a ) is one of the followings:( 12 , , b + 115 b + 14 ) , ( 12 , b + 53 b + 8 ,
34 ) , ( 12 , b + 53 b + 8 ,
45 ) , ( b + 22 b + 5 , ,
23 ) , ( b + 22 b + 5 , ,
34 ) , ( 12 , , b + 45 b + 7 ) , ( 12 , , b + 77 b + 10 ) , ( 12 , , b + 109 b + 13 ) , ( 12 , b + 45 b + 7 ,
34 ) , ( 12 , b + 45 b + 7 ,
45 ) , ( 12 , , (2 b + 5) c + 4 b + 8(3 b + 8) c + 6 b + 13 ) , ( 12 , , (3 b + 8) c + 6 b + 13(4 b + 11) c + 8 b + 18 ) , ( 12 , b + 33 b + 5 , d + 33 d + 5 ) , ( 12 , , d + 54 d + 7 ) , ( 12 , , d + 75 d + 9 ) . Then D ≥ P − P − P − P . Therefore R ( P k , D ) is F -rational by Theorem3.1 and Lemma 6.1. Case 5.
We assume that ( a , a , a ) is one of the followings:( 12 , b + 13 b + 4 , n + 1 n + 2 ) , ( 12 , , (2 a + 3) b + 3 a + 4(2 a + 5) b + 3 a + 7 ) , ( 12 , ( b + 2) c + 2 b + 3(2 b + 5) c + 4 b + 8 , n + 1 n + 2 ) , ( 12 , , (cid:0) ( a + 2) b + 3 a + 5 (cid:1) c + (2 a + 4) b + 5 a + 8 (cid:0) ( a + 3) b + 3 a + 8 (cid:1) c + (2 a + 6) b + 5 a + 13 ) , ( b + 12 b + 3 , d + 12 d + 3 , mm + 1 ) , ( 12 , b + 12 b + 3 , ( c + 2) d + 2 c + 3( c + 3) d + 2 c + 5 ) . Then D ≥ P − P − P − ll +1 P for sufficiently large number l . Note that if( a , a , a ) = ( b +12 b +3 , d +12 d +3 , mm +1 ), then we have deg[ − lD ] ≤ − l ∈ N . Therefore R ( P k , D ) is F -rational by Theorem 3.1 and Lemma 6.1. Case 6.
We assume that ( a , a , a ) is one of the followings:( b + 13 b + 4 , , n + 2 n + 3 ) , ( 13 , , ( c + 2) d + 2 c + 3( c + 3) d + 2 c + 5 ) , ( 13 , d + 33 d + 5 , m + 1 m + 2 ) . Then D ≥ P − P − P − ll +1 P for sufficiently large number l . Therefore R ( P k , D ) is F -rational by Theorem 3.1 and Lemma 6.1. Case 7.
We assume that ( a , a , a ) = ( , n +3 n +4 , ). Then R ( P k , D ) is F -rationalby Theorem 3.1 and Lemma 6.1.By the above discussion, if p ≥
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