Divisor class groups of graded hypersurfaces
aa r X i v : . [ m a t h . A C ] J a n Contemporary Mathematics
Divisor class groups of graded hypersurfaces
Anurag K. Singh and Sandra Spiroff
Abstract.
We demonstrate how some classical computations of divisor classgroups can be obtained using the theory of rational coefficient Weil divisorsand related results of Watanabe.
1. Introduction
The purpose of this note is to provide a simple technique to compute divisorclass groups of affine normal hypersurfaces of the form k [ z, x , . . . , x d ] / ( z n − g ) , where g is a weighted homogeneous polynomial in x , . . . , x d of degree relativelyprime to n . We use the theory of rational coefficient Weil divisors due to Demazure[ ] and related results of Watanabe [ ]. This provides an alternative approachto various classical examples found in Samuel’s influential lecture notes [ ], aswell as to computations due to Lang [ ] and Scheja and Storch [ ]. While thecomputations we present here are subsumed by those of [ ], our techniques aredifferent. A key point in our approach is that the projective variety defined bya hypersurface as above is weighted projective space over k , and this makes forstraightforward, elementary calculations.Watanabe [ , page 206] pointed out that Q -divisor techniques can be used torecover the classification of graded factorial domains of dimension two, originallydue to Mori [ ]. Robbiano has applied similar methods to a study of factorial andalmost factorial schemes in weighted projective space [ ]. Q -divisors We review some material from [ ] and [ ]. Let k be a field, and let X be anormal irreducible projective variety over k , with rational function field k ( X ).A rational coefficient Weil divisor or a Q -divisor on X is a Q -linear combinationof irreducible subvarieties of X of codimension one. Let D = P n i V i be a Q -divisor,where V i are distinct. Then ⌊ D ⌋ is defined as ⌊ D ⌋ = X ⌊ n i ⌋ V i , Mathematics Subject Classification.
Primary 13C20, Secondary 14C20.The first author was supported by the NSF under grants DMS 0300600 and DMS 0600819. c (cid:13) where ⌊ n ⌋ denotes the greatest integer less than or equal to n . We set O X ( D ) = O X ( ⌊ D ⌋ ) . If each coefficient n i occurring in D is nonnegative, we say that D > Q -divisor D is ample if nD is an ample Cartier divisor for some n ∈ N . Inthis case, the generalized section ring corresponding to D is the ring R ( X, D ) = ⊕ j > H ( X, O X ( jD )) . If R = R ( X, D ), then the n -th Veronese subring of R = R ( X, D ) is the ring R ( n ) = ⊕ j > H ( X, O X ( jnD )) = R ( X, nD ) . The following theorem, due to Demazure, implies that a normal N -graded ring R is determined by a Q -divisor on Proj R . Theorem 2.1. [ , 3.5] . Let R be an N -graded normal domain, finitely generatedover a field R . Let T be a homogeneous element of degree in the fraction field of R . Then there exists a unique ample Q -divisor D on X = Proj R such that R = ⊕ j > H ( X, O X ( jD )) T j . We next recall a result of Watanabe, which expresses the divisor class group of R in terms of the divisor class group of X and a Q -divisor corresponding to R . Theorem 2.2. [ , Theorem 1.6] Let X be a normal irreducible projective varietyover a field. Assume dim X > and let D = P ri =1 ( p i /q i ) V i be a Q -divisor on X where V i are distinct irreducible subvarieties, p i , q i ∈ Z are relatively prime, and q i > . Set R = ⊕ j > H ( X, O X ( jD )) T j . Then there is an exact sequence −−−−→ Z θ −−−−→ Cl( X ) −−−−→ Cl( R ) −−−−→ coker α −−−−→ , where θ (1) = lcm( q i ) · D , and α : Z −→ ⊕ ri =1 Z /q i Z is the map ( p i mod q i ) i . In the exact sequence above, coker α is always a finite group. Moreover, if X is projective space, a Grassmannian variety, or a smooth complete intersection in P n of dimension at least three, then Cl( X ) = Z . It follows that, in these cases, thedivisor class group of R ( X, D ) is finite for any ample Q -divisor D on X , and hencethat R ( X, D ) is almost factorial in the sense of Storch [ ].Lipman proved that the divisor class group of a two-dimensional normal localring R with rational singularities is finite, [ , Theorem 17.4]. While this is a hardresult, the analogous statement for graded rings is a straightforward application ofTheorem 2.2. Indeed, let R be an N -graded normal ring of dimension two, finitelygenerated over an algebraically closed field R , such that R has rational singular-ities. Then R has a negative a -invariant by [ , Theorem 3.3], so H ( X, O X ) = 0where X = Proj R . But then X is a curve of genus 0 so it must be P , and it followsthat the divisor class group of R is finite. Remark 2.3.
We note some aspects of Watanabe’s proof of Theorem 2.2. LetDiv( X ) be the group of Weil divisors on X , and letDiv( X, Q ) = Div( X ) ⊗ Z Q IVISOR CLASS GROUPS OF GRADED HYPERSURFACES 3 be the group of Q -divisors. For D as in Theorem 2.2, set Div( X, D ) to be thesubgroup of Div( X, Q ) generated by Div( X ) and the divisors1 q V , . . . , q r V r . Each element E ∈ Div(
X, D ) gives a divisorial ideal ⊕ j > H ( X, O X ( E + jD )) T j of R , and hence an element of Cl( R ). The map Div( X, D ) −→ Cl( R ) induces asurjective homomorphismDiv( X, D ) / Div( X ) −→ Cl( R ) / image(Cl( X )) .
3. Computing divisor class groups
The divisor class groups of affine surfaces of characteristic p defined by equa-tions of the form z p n = g ( x, y ) have been studied in considerable detail; suchsurfaces are sometimes called Zariski surfaces . In [ ] Lang computed the divisorclass group of hypersurfaces of the form z p n = g ( x , . . . , x d ) where g is a homoge-neous polynomial of degree relatively prime to p . The proposition below recovers[ , Proposition 3.11].Let A = k [ x , . . . , x d ] be a polynomial ring over a field. We say g ∈ A is a weighted homogeneous polynomial if there exists an N -grading on A , with A = k ,for which g is a homogeneous element. Proposition 3.1.
Let R = k [ z, x , . . . , x d ] / ( z n − g ) be a normal hypersurface overa field k , where g ∈ k [ x , . . . , x d ] is a weighted homogeneous polynomial with degreerelatively prime to n . Let g = h · · · h r , where h i ∈ k [ x , . . . , x d ] are irreduciblepolynomials. Then Cl( R ) = ( Z /n Z ) r − , and the images of ( z, h ) , . . . , ( z, h r − ) form a minimal generating set for Cl( R ) . Note that if n >
2, then the hypothesis that R is normal forces h , . . . , h r tobe pairwise coprime irreducible polynomials. Proof of Proposition 3.1.
The polynomial ring k [ x , . . . , x d ] has a gradingunder which deg x i = c i for c i ∈ N , and the degree of g is an integer m relativelyprime to n . We assume, without any loss of generality, that gcd( c , . . . , c d ) = 1.Consider the N -grading on R where deg x i = nc i and deg z = m . Note that underthis grading deg g = P deg h i = mn . The n -th Veronese subring of R is R ( n ) = k [ z n , x , . . . , x d ] / ( z n − g ) = k [ x , . . . , x d ] , which is a polynomial ring in x , . . . , x d . Let X = Proj R ( n ) = Proj R .There exist integers s i , a , and b such that P di =1 s i c i = 1 and am + bn = 1.Consider the Q -divisor on X given by D = b div( x ) + an div( g ) = b d X i =1 s i V ( x i ) + an r X i =1 V ( h i ) , where x = x s · · · x s d d . We claim that(3.1.1) R = ⊕ j > H ( X, O X ( jD )) T j , ANURAG K. SINGH AND SANDRA SPIROFF where T = z a x b is a homogeneous degree 1 element of the fraction field of R . Firstnote that ⌊ am/n ⌋ = ⌊ (1 − bn ) /n ⌋ = − b , so ⌊ mD ⌋ = bm div( x ) + j amn k div( g ) = bm div( x ) − b div( g ) . Consequently deg ⌊ mD ⌋ = 0, and H ( X, O X ( mD )) T m is the k -vector space spannedby the element x − bm g b T m = x − bm ( z n ) b ( z a x b ) m = z bn + am = z . Let c = c t for an integer 1 t d . Then ncD = bnc div( x ) + ac div( g ) has degree nc , and H ( X, O X ( ncD )) T nc contains the element x t x − bnc g − ac T nc = x t x − bnc ( z n ) − ac ( z a x b ) nc = x t . To prove the claim (3.1.1), it remains to verify that z, x , . . . , x d are k -algebragenerators for the ring ⊕ j > H ( X, O X ( jD )) T j . An arbitrary positive integer j can be written as um + vn for 0 u n −
1. We then have ⌊ jD ⌋ = b ( um + vn ) div( x ) + (cid:22) a ( um + vn ) n (cid:23) div( g )= b ( um + vn ) div( x ) + ( va − ub ) div( g ) , which has degree vn . Consequently H ( X, O X ( jD )) T j vanishes if v is negative,and for nonnegative v , it is spanned by elements µ x − b ( um + vn ) g − va + ub T um + vn = µz u , for monomials µ in x i of degree v . This completes the proof of (3.1.1).Since nD has integer coefficients, the exact sequence of Theorem 2.2 for thedivisor nD and corresponding ring R ( n ) reduces to0 −−−−→ Z θ −−−−→ Cl( X ) −−−−→ Cl( R ( n ) ) −−−−→ , where θ (1) = nD . Since R ( n ) is a polynomial ring, and hence factorial, it followsthat nD generates Cl( X ). Next, consider the exact sequence applied to the divisor D and corresponding ring R , i.e., the sequence0 −−−−→ Z θ −−−−→ Cl( X ) −−−−→ Cl( R ) −−−−→ coker α −−−−→ . The lcm of the denominators occurring in D is n , so we once again have θ (1) = nD .Consequently θ is an isomorphism and Cl( R ) = coker α , where α : Z −→ r M Z /n Z with α (1) = ( a, . . . , a ) . Since a and n are relatively prime, it follows thatCl( R ) = ( Z /n Z ) r − . We next determine explicit generators for Cl( R ) by Remark 2.3. The Q -divisors E t = − n V ( h t ) for 1 t r give a generating set for Div( X, D ) / Div( X ) which surjects onto Cl( R ). Hence thedivisorial ideals p t = ⊕ j > H ( X, O X ( E t + jD )) T j where 1 t d , IVISOR CLASS GROUPS OF GRADED HYPERSURFACES 5 generate Cl( R ). The computation of p t is straightforward, and we give a briefsketch. First note that ⌊ E t + mD ⌋ = bm div( x ) + (cid:22) am − n (cid:23) V ( h t ) + X i = t j amn k V ( h i )= bm div( x ) − b div( g ) , so H ( X, O X ( E t + mD )) T m is the k -vector space spanned by x − bm g b T m = z. Since the degree of each x i is a multiple of n , we have deg h t = nγ for some integer γ . We next compute the component of p t in degree nγ . Note that ⌊ E t + nγD ⌋ = − V ( h t ) + bnγ div( x ) + aγ div( g ) , so H ( X, O X ( E t + nγD )) T nγ is the k -vector space spanned by h t x − bnγ g − aγ T nγ = h t . It is now a routine verification that z, h t are generators for the ideal p t , which, wenote, is a height one prime of R . Consequently Cl( R ) is generated by p , . . . , p r .Using ∼ to denote linear equivalence, we have nE t + nγD ∼ r X i =1 E i + mD ∼ , implying that n [ p t ] = 0 and P i [ p i ] = 0 in Cl( R ). These correspond to the calcula-tions with divisorial ideals, p ( n ) t = h t R and r \ i =1 p i = zR , and imply, in particular, that [ p ] , . . . , [ p r − ] is a generating set for Cl( R ). (cid:3) Example 3.2.
We use Proposition 3.1 to compute the divisor class group of diag-onal hypersurfaces R = k [ z, x , . . . , x d ] / ( z n − x m − · · · − x m d d )where n is relatively prime to m i for 1 i d , and k is a field of characteristiczero, or of characteristic not dividing each m i .By the Jacobian criterion, R has an isolated singularity at the homogeneousmaximal ideal m . Hence if d >
4, then R , as well as its m -adic completion b R , arefactorial by Grothendieck’s parafactoriality theorem [ ]; see [ ] for a simple proofof Grothendieck’s theorem. Case d = 3 . The polynomial g = x m + x m + x m is irreducible since k [ x , x , x ] / ( g ) is a normal domain by the Jacobian criterion. We set deg x i to be m m m /m i . Then g is a weighted homogeneous polynomial of degree m m m ,which is relatively prime to n , so Proposition 3.1 implies that R is factorial. Since R satisfies the Serre conditions ( R ) and ( S ), the completion b R is factorial as well by[ , Korollar 1.5]. The divisor class groups of rational three-dimensional Brieskornsingularities are computed in [ , Chapter IV]; see also [ ]. Case d = 2 . Let g = x m + x m . If c = gcd( m , m ), let m = ac and m = bc ,and set deg x = b and deg x = a . Let f be an irreducible factor of g . Then f ishomogeneous, and hence has the form P a ij x i x j where a ij ∈ k and bi + cj = deg f ANURAG K. SINGH AND SANDRA SPIROFF for each term occurring in the summation. Since x and x do not divide g , we seethat f must contain nonzero terms of the form a j x j and a i x i . Hence deg f is amultiple of ab , and it follows that f is a polynomial in x a and x b . Consequentlythe number of factors of g in k [ x , x ] is the number of factors of s c + t c in k [ s, t ]or, equivalently, the number of factors of 1 + t c in k [ t ].In particular, if m and m are relatively prime, then g is irreducible andProposition 3.1 implies that R is factorial. As is well-known, b R need not be factorial;see for example, [ , Theorem III.5.2].If k is algebraically closed, then g is a product of c irreducible factors, and soProposition 3.1 implies that Cl( R ) = ( Z /n Z ) c − . Remark 3.3.
The condition that the degree of g is relatively prime to n is certainlycrucial in Proposition 3.1. In the absence of this, Cl( R ) need not be finite, forexample C [ z, x , x , x ] / ( z − x − x − x ) has divisor class group Z . However,one can drop the relatively prime condition when considering hypersurfaces of theform z n − x g ( x , . . . , x d ), see also [ , Proposition 3.12]: Corollary 3.4.
Let R = k [ z, x , . . . , x d ] / ( z n − x g ) be a normal hypersurface overa field k , where g is a weighted homogeneous polynomial in x , . . . , x d . Let g = h · · · h r , where h i ∈ k [ x , . . . , x d ] are irreducible. Then Cl( R ) = ( Z /n Z ) r , and the images of ( z, h ) , . . . , ( z, h r ) form a minimal generating set for Cl( R ) . Proof.
We may choose the degree of x such that deg( x g ) is relatively primeto n . The result then follows from Proposition 3.1. (cid:3) We conclude with the following example.
Example 3.5.
Let k be a field. Corollary 3.4 implies that the divisor class groupof the ring R = k [ xy, x n , y n ] is Z /n Z , since R is isomorphic to the hypersurface k [ z, x , x ] / ( z n − x x ) . In [ , Chapter III], the divisor class group of R is computed by Galois descent if n is relatively prime to the characteristic of k , and by using derivations if n equalsthe characteristic of k . References [1] J. Bingener and U. Storch,
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Lectures on unique factorization domains , Tata Inst. Fund. Res. Stud. Math. ,Tata Inst. Fund. Res., Bombay, 1964.[11] G. Scheja and U. Storch, Uwe Zur Konstruktion faktorieller graduierter Integrit¨atsbereiche ,Arch. Math. (Basel) (1984), 45–52.[12] U. Storch, Fastfaktorielle Ringe , Schr. Math. Inst. Univ. M¨unster , 1967.[13] U. Storch, Die Picard-Zahlen der Singularit¨aten t r + t r + t r + t r = 0, J. Reine Angew.Math. (1984), 188–202.[14] K.-i. Watanabe, Some remarks concerning Demazure’s construction of normal graded rings ,Nagoya Math. J. (1981), 203–211. Department of Mathematics, University of Utah, 155 South 1400 East, Salt LakeCity, UT 84112, USA
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