NORMAL AFFINE SURFACES WITH ℂ^{∗}-ACTIONS

HUBERT FLENNER AND MIKHAIL ZAIDENBERG

Abstract. A classification of affine surfaces admitting a ℂ^{∗}-action was given in the work of Białynicki-Birula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of normal quasihomogeneous affine surfaces in terms of their graded rings as well as by defining equations. This is based on a generalization of the Dolgachev-Pinkham-Demazure construction.

Contents

Introduction 1 1. Generalities on graded rings 2 2. The elliptic case 4 3. The parabolic case 6 4. The hyperbolic case 10 References 20

Introduction

A classification of (normal) affine surfaces admitting a ℂ^{∗}-action was given e.g., in [Bi, BiSo, OrWa, Pi, BasHa, Ry] and [FiKa_{1}]-[FiKa_{3}]. Here we obtain a simple alternative description of normal affine surfaces V with a ℂ^{∗}-action in terms of their graded coordinate rings as well as by defining equations. Our approach is based on a generalization of the Dolgachev-Pinkham-Demazure construction [Do, Pi, De]. Recall (see [FiKa_{1}]-[FiKa_{3}]) that a ℂ^{∗}-action on a normal affine surface V is called elliptic if it has a unique fixed point which belongs to the closure of every 1-dimensional orbit, parabolic if the set of its fixed points is 1-dimensional, and hyperbolic if V has only a finite number of fixed points, and these fixed points are of hyperbolic type, that is each one of them belongs to the closure of exactly two 1-dimensional orbits.

In the elliptic case, the complement V^{∗} of the unique fixed point in V is fibered by the 1-dimensional orbits over a projective curve C. In the other two cases V is fibered over an affine curve C, and this fibration is invariant under the ℂ^{∗}-action.

This research started during a visit of the first author at the Institut Fourier of the University of Grenoble, and continued during a stay of both of us at the Max Planck Institute of Mathematics at Bonn and of the second author at the Ruhr University at Bochum. The authors thank these institutions for their support.

1991 Mathematics Subject Classification: 13F20, 13H10, 14J17, 14L30.

Key words: ℂ^{∗}-action, graded algebra, affine surface, cyclic quotient singularity.

Vice versa, given a smooth curve C and a ℚ-divisor D on C, the Dolgachev-Pinkham-Demazure construction provides a normal affine surface V = V_{C,D} with a ℂ^{∗}-action such that C is just the algebraic quotient of V^{∗} or of V, respectively. This surface V is of elliptic type if C is projective and of parabolic type if C is affine.

We remind this construction in sections 1 and 2 below. In section 3 we use it to present any normal affine surface V with a parabolic ℂ^{∗}-action as a normalization of the surface x^{d} − P (z)y = 0 in ℂ^{3} for a certain d ∈ ℕ and a certain polynomial P ∈ ℂ[t] (see Theorem 3.11).

In section 4 we deal with the hyperbolic case. We generalize the Dolgachev-Pinkham-Demazure construction in order to make it work for any hyperbolic ℂ^{∗}-surface. Instead of one ℚ-divisor D on a smooth affine curve C as before, it involves now two ℚ-divisors D_{+} and D_{−} on C. By our result isomorphism classes of normal affine hyperbolic ℂ^{∗}- surfaces are in 1-1-correspondence to equivalence classes of triples (C, D_{+}, D_{−}), where C is a smooth affine curve and D_{+}, D_{−} is a pair of ℚ-divisors on C with D_{+} +D_{−} ≤ 0; two such triples (C, D_{+}, D_{−}) and (C, D_{+}, D_{−}) are considered to be equivalent if and only if C ≅ C′ and D_{±} = D′_{±} ± D_{0} with a principal divisor D_{0}; cf. Theorem 4.5. We also determine the type of singularities and of the special orbits in terms of the divisors D_{±}, see Theorems 4.16 and 4.19.

Using our description it is possible to represent any normal hyperbolic ℂ^{∗}-surface fibered over C = 𝔸_{ℂ}^{1} as the normalization of a surface in ℂ^{4} given by

x^{d} − P (t)y = 0, x^{e}z − Q(t) = 0 and y^{e}z^{d} − R(t) = 0 ,

for certain polynomials P, Q, R ∈ ℂ[t] satisfying the relation P^{e}R = Q^{d}. These polynomials can be easily computed in terms of the data (D_{+}, D_{−}) (see Remark 4.10). For instance, if the divisor D_{−} is integral then this system reduces to one equation x^{e}z − Q(t) = 0 in ℂ^{3}, and vice versa. When gcd(e, d) = 1 then it again reduces to one equation y^{e}z^{d} − R(t) = 0 in ℂ^{3}.

In Proposition 4.13 we show how the pair (D_{+}, D_{−}) is transformed when passing to an equivariant cyclic cover of V. We deduce, in particular, a characterization of normal hyperbolic ℂ^{∗}-surfaces over C = 𝔸_{ℂ}^{1} with the fractional part of D_{−} supported at one point, as normalized cyclic quotients of the surfaces x^{e}z − Q(t) = 0 in ℂ^{3}.

In the forthcoming paper [FlZa], which is actually Part II of the present one, we will apply these results to give a simple description of all normal affine ℂ^{∗}-surfaces equipped in addition by a ℂ^{+}-action. In fact, this class consists of all normal affine surfaces which admit an algebraic group action with an open orbit.

We note that the results of this paper hold m.m. for graded 2-dimensional normal algebras of finite type over a Dedekind domain.

1. Generalities on graded rings

A ℤ-graded ring A = ⨁_{i∈ℤ} A_{i} contains A_{≥0} = ⨁_{i≥0} A_{i} and A_{≤0} = ⨁_{i≤0} A_{i} as subrings. The following lemma is “well known”; in lack of a reference we provide a short argument. Lemma 1.1. If A = ⨁_{i∈Z} A_{i} is a finitely generated A_{0}-algebra, then so are A_{≥0} and A_{≤0}. Moreover, A is normal if and only if so are both A_{≥0} and A_{≤0}.

Proof. Reversing the grading interchanges the subrings A_{≥0} and A_{≤0}. Thus it is sufficient to prove the first part for A_{≥0}. If a_{ij} ∈ A_{i} with −n ≤ i ≤ n, j = 1, …, n_{i}, is a system of homogeneous generators of A, then A_{≥0} is generated (as a module over A_{0}) by the multiplicatively closed system of monomials

a^{k} := ∏_{i,j} a_{ij}^{k_{ij}} ,

where k := (k_{ij}) ∈ ℤ^{N} satisfies the inequalities

k_{ij} ≥ 0, −n ≤ i ≤ n, j = 1, …, n_{i},  ∑_{i,j} ik_{ij} ≥ 0 . (1)

By Gordan’s Lemma (see [Od]) the rational polyhedral lattice cone K ⊆ ℤ^{N} defined by (1) is a finitely generated semigroup. Hence the algebra A_{≥0} is generated by a finite system of monomials a^{k} ∈ A_{≥0}.

Next we show that the subalgebra A_{≥0} (and then also A_{≤0}) is normal if so is A. Indeed, the integral closure (A_{≥0})_{norm} ⊆ A = A_{norm} is graded. Take a homogeneous element x ∈ (A_{≥0})_{norm} of degree d := deg x, and let

x^{n} + ∑_{i=1}^{n} b_{i}x^{n−i} = 0, where b_{i} ∈ A_{≥0} , (2)

be an equation of integral dependence. We may assume that b_{i} are also homogeneous, of degree deg b_{i} = di ≥ 0. Since deg b_{i} ≥ 0 we have d ≥ 0, and so x ∈ A_{≥0}.

Conversely, suppose that both A_{≥0} and A_{≤0} are normal. The ring A ⊗_{A_{0}} Frac(A_{0}) is normal and so is equal to Frac(A_{0})[u, u^{−1}] for a homogeneous element u of minimal degree > 0 in A ⊗_{A_{0}} Frac(A_{0}). Hence A_{norm} is contained in this subring of Frac A. If f ∈ A ⊗_{A_{0}} Frac(A_{0}) belongs to the normalization A_{norm} of A then so does its top homogeneous component. Thus it is enough to deal with homogeneous elements. Let a be such an element satisfying an equation of integral dependence (2) over A. We may suppose as above that b_{i} ∈ A_{di} (i = 1, …, n). Since di has the same sign as d := deg a, we have a ∈ (A_{≥0})_{norm} = A_{≥0} if d ≥ 0 and a ∈ (A_{≤0})_{norm} = A_{≤0} if d ≤ 0, respectively. Anyhow, a ∈ A, whence A is normal, as stated. □

Notation 1.2. Let V = Spec A be a normal affine surface over ℂ with an effective ℂ^{∗}-action. The coordinate ring A = ⨁_{i∈ℤ} A_{i} is then naturally graded so that A_{i} is the set of elements of A on which t ∈ ℂ^{∗} acts via t.f = t^{i}f. Thus, A_{0} = A^{ℂ^{∗}} is the subalgebra of invariants, and A_{i} (i ≠ 0) consists of the quasi-invariants of weight i. Up to reversing the grading we may assume that A_{+} := ⨁_{i>0} A_{i} ≠ 0. The subsets A_{+} and A_{−} := ⨁_{i<0} A_{i} of A are ideals in A_{≥0} and A_{≤0}, respectively.

The following lemma is well known (see e.g., [De], [FiKa_{1}, Lemma 1.5]).

Lemma 1.3. (a) If A_{0} ≠ ℂ then the set M := {i ∈ ℤ | A_{i} ≠ 0} coincides either with ℕ or with ℤ, and A_{i} is a locally free A_{0}-module of rank 1 for all i ∈ M. Moreover, if u ∈ Frac(A_{0}) · A_{1} is a non-zero element then

A ⊆ Frac(A_{0})[u, u^{−1}] ,  and even  A ⊆ Frac(A_{0})[u] if M = ℕ .

(b) In particular, if A_{0} ≅ ℂ[t] then A_{i} is a free A_{0}-module of rank 1 for all i ∈ M.

Proof. (a) The K_{0} := Frac(A_{0})-algebra A ⊗_{A_{0}} K_{0} is a 1-dimensional normal graded domain over the field K_{0}. Hence it is isomorphic to the free polynomial ring K_{0}[u] or the ring of Laurent polynomials K_{0}[u, u^{−1}], where u ∈ K_{0}A_{d} and d > 0. As the ℂ^{∗}-action is effective d = 1, and (a) follows.

(b) follows from [Bou, Ch. VII, §4, Corollary 2]. □

Lemma 1.3(a) does not hold in general without the assumption that A_{0} ≠ ℂ as is seen by the Pham-Brieskorn surfaces V_{p,q,r} := {x^{p} + y^{q} + z^{r} = 0} ⊆ ℂ^{3}.

1.4. Usually (cf. [FiKa_{1}]) one distinguishes between the following three cases. (i) The elliptic case: A_{−} = 0, A_{0} = ℂ. (ii) The parabolic case: A_{−} = 0, A_{0} ≠ ℂ. (iii) The hyperbolic case: A_{−} ≠ 0.

Below we provide more information in each of these cases.

2. The elliptic case

In the elliptic case the ℂ^{∗}-action on V is good. In particular, its fixed point set F := V^{ℂ^{∗}} (which is the zero set of the augmentation ideal A_{+} of A) consists of a unique point called the vertex of V, and the surface V is smooth outside the vertex. One considers the smooth projective curve C := Proj A ≅ V^{∗}/ℂ^{∗}, where V^{∗} := V \ F, together with the orbit morphism π : V^{∗} → C (the fibers of π are the orbits of the ℂ^{∗}-action on V^{∗}).

A useful class of examples of normal affine surfaces with a good ℂ^{∗}-action is provided by the affine cones over projective curves. For an ample divisor D on a smooth projective curve C the ring

A_{C,D} := ⨁_{k≥0} H^{0}(C, O_{C}(kD))) · u^{k} ⊆ Frac(C)[u] ,

where u is an indeterminate, is the coordinate ring of a normal affine surface V := Spec A_{C,L} with a good ℂ^{∗}-action. This surface V is a cone over C obtained by blowing down the zero section of the line bundle associated to O_{C}(−D).

Let furthermore a finite group G act on V freely off the vertex, and assume that this action commutes with the given good ℂ^{∗}-action on V. Then the quotient V/G is again a normal affine surface with a good ℂ^{∗}-action. Conversely, the following result is true.

Theorem 2.1. ([Do, Pi, De, Ru]) Every normal affine surface with a good ℂ^{∗}-action appears as the quotient of an affine cone over a smooth projective curve by a finite group acting freely off the vertex of the cone.

Generalizing the construction above, for a smooth projective curve C and a ℚ-divisor D on C one considers the graded ring

A_{C,D} := ⨁_{k≥0} H^{0}(C, O(⌊kD⌋)) · u^{k} ,

where ⌊E⌋ denotes the integral part of a ℚ-divisor E. We have the following result.

Theorem 2.2. ([Pi], [De, Theorem 3.5]) Given a normal affine surface V = Spec A with a good ℂ^{∗}-action there exists a ℚ-divisor D on the curve C = Proj A such that A ≅ A_{C,D}.

The affine toric surfaces provide an interesting family of elliptic ℂ^{∗}-surfaces.

Example 2.3. ([Od, Co]) We remind that a normal affine toric surface V = V_{σ} is associated to a strictly convex rational polyhedral cone σ ⊆ ℝ^{2}. If dim σ = 0 or = 1 then V_{σ} ≅ ℂ^{∗} × ℂ^{∗} or V_{σ} ≅ 𝔸_{ℂ}^{1} × ℂ^{∗}, respectively, and so A^{×} ≠ ℂ^{∗}. Consequently, these two cannot be elliptic ℂ^{∗}-surfaces. Otherwise, if dim σ = 2 then choosing an appropriate base e_{1}, e_{2} of the lattice one may suppose that σ is the cone C(e_{2}, de_{1}−ee_{2}), where d ≥ 1, 0 ≤ e < d and gcd(e, d) = 1. We denote V_{d,e} := V_{σ}; then V_{d,e} = Spec A_{d,e}, where

A_{d,e} := ⨁_{b≥0, ad−be≥0} ℂ · x^{a}y^{b} ⊆ ℂ[x, x^{−1}, y, y^{−1}]

is the semigroup algebra of the dual cone σ^{∨} = C (e_{1}, ee_{1} + de_{2}).

The 2-torus 𝕋 = (ℂ^{∗})^{2} acts on V_{d,e} with an open orbit V_{d,e}^{∗} := V_{d,e}\{¯0}. Thus one can introduce on V_{d,e} a number of elliptic, parabolic as well as hyperbolic ℂ^{∗}-actions by choosing appropriate 1-parameter algebraic subgroups of the torus 𝕋.

In [Ri, BeRi_{1}, BeRi_{2}, Co] one can find a description of minimal sets of generators of the algebras A_{d,e} as above, as well as defining equations for the affine varieties V_{d,e} = Spec A_{d,e} ↪ ℂ^{N}.

We would like to emphasize the well known relation between affine toric surfaces and cyclic quotient singularities [Od, Proposition 1.24].

Lemma 2.4. If B is the normalization of A := A_{d,e} in the field L := Frac(A)[u] with u := \sqrt[d]{x}, then B is the polynomial ring B = ℂ[u, v] with v := u^{e}y. The Galois group ⟨ζ⟩ ≅ ℤ_{d} of L : Frac(A) acts on B via

ζ.u = ζu, ζ.v = ζ^{e}v ,

and A = B^{ℤ_{d}}. Consequently, there is an isomorphism

V_{d,e} ≅ 𝔸_{ℂ}^{2}/G_{d,e} = 𝔸_{ℂ}^{2}/ℤ_{d} .

Proof. For the convenience of the reader we give a short argument. By definition, A is generated over ℂ by the monomials

x^{a}y^{b} with b ≥ 0, ad − be ≥ 0 .

As x^{a}y^{b} = u^{ad−be}v^{b}, this shows that A embeds naturally into ℂ[u, v] and that even A = ℂ[x, x^{−1}, y] ∩ ℂ[u, v]. In particular A is a normal domain. Because of u^{d} = x ∈ A and v^{d} = x^{e}y^{d} ∈ A the ring B is integral over A, whence it is the normalization of A.

The second part follows from the first one, since L is a cyclic extension of Frac(A) with Galois group ℤ_{d} acting via ζ.u = ζu and ζ.x = x for all x ∈ A. □

Remark 2.5. Assuming that e > 0 and letting ξ := ζ^{e} one obtains

(ζu, ζ^{e}v) = (ξ^{e′}u, ξv) ,

where 0 ≤ e′ < d and ee′ ≡ 1 mod d (note that for d = 1 this means e′ = 0). Hence, with τ(u, v) := (v, u) the conjugate ℤ_{d}-action G′_{d,e′} := τ^{−1}G_{d,e′}τ on 𝔸_{ℂ}^{2}

ξ.(u, v) = (ξ^{e′}u, ξv)

has the same orbits as G_{d,e} thus providing an isomorphism of toric surfaces

V_{d,e} ≅ 𝔸_{ℂ}^{2}/G_{d,e} ≅ 𝔸_{ℂ}^{2}/G′_{d,e′} ≅ 𝔸_{ℂ}^{2}/G_{d,e′} ≅ V_{d,e′} .

Moreover, V_{d,e} ≅ V_{d′,e′} if and only if d = d′ and either e = e′ or ee′ ≡ 1 mod d.

3. The parabolic case

In the parabolic case one considers a normal affine surface V with a ℂ^{∗}-action such that the coordinate ring A = ⨁_{i≥0} A_{i} is positively graded and A_{0} is a 1-dimensional domain. Thus A_{0} corresponds to a smooth affine curve C = Spec A_{0}, which can be identified with the algebraic quotient V//ℂ^{∗} (indeed, A_{0} = A^{ℂ^{∗}} is the ring of invariants of the ℂ^{∗}-action on A). The embedding A_{0} ↪ A corresponds to the quotient morphism π : V → C, and the projection A → A_{0} gives an embedding ι : C ↪ V which provides a retraction of π and whose image is the fixed point set. Every fiber of π : V → C is the closure of a non-trivial orbit; it contains a unique fixed point (a source of this orbit) [FiKa_{1}, Lemma 1.7].

A simple example of a parabolic ℂ^{∗}-surface is the cylinder C × 𝔸_{ℂ}^{1} over a smooth affine curve C, where ℂ^{∗} acts on the second factor. More examples can be produced by applying equivariant affine modifications to C × 𝔸_{ℂ}^{1} (see [KaZa_{1}, Theorem 1.1]). Actually, one obtains in this way all normal affine surfaces with a parabolic ℂ^{∗}-action.

3.1. The Dolgachev-Pinkham-Demazure construction (see Theorem 2.2) is available also in the parabolic case. Let C = Spec A_{0} be an affine curve over ℂ with function field K_{0} := Frac(A_{0}), and let D be a ℚ-Cartier divisor on C. Similarly as in the elliptic case we can introduce the algebra

A_{0}[D] := A_{C,D} = ⨁_{n≥0} H^{0}(C, O_{C}(⌊nD⌋)) · u^{n} ⊆ K_{0}[u] .

More explicitly, if f ∈ K_{0} then

fu^{n} ∈ A := A_{0}[D] ⇔ divf + nD ≥ 0. (3)

We note that the algebra A is normal (see Corollary 3.8(b) below). Notice also that u ∈ A_{1} if and only if D ≥ 0.

The following theorem is well known (cf. [De, Theorem 3.5]); for the convenience of the reader we include a short proof.

Theorem 3.2. Let C = Spec A_{0} be a normal affine algebraic curve with function field K_{0} := Frac(A_{0}). If A = ⨁_{i≥0} A_{i} is a normal finitely generated A_{0}-algebra of dimension 2 then the following hold.

(a) A is isomorphic to A_{0}[D] for some ℚ-divisor D on C. More precisely, if u ∈ K_{0} · A_{1} is a non-zero element and if the divisor D is defined by the equality

π^{∗}D = div u − ι(C) ,

then A and A_{0}[D] are equal when considered as subrings of K_{0}[u].

(b) For two ℚ-divisors D and D′ on C, the rings A = A_{0}[D] and A′ = A_{0}[D′] are isomorphic as graded A_{0}-algebras if and only if D and D′ are linearly equivalent.

Proof. (a) Since u ∈ K_{0} · A_{1} is homogeneous, the divisor div u on the normal surface V = Spec A is invariant under the induced ℂ^{∗}-action on V, and so we have

div u = ∑_{i=1}^{m} p_{i}F_{i} + ι(C)

with p_{i} ∈ ℤ, where F_{i} = π^{−1}(x_{i}) are the fibers of π over distinct points x_{i} ∈ C, i = 1, …, m. Letting π^{∗}x_{i} = q_{i}F_{i} with q_{i} ∈ ℕ (i = 1, …, m), the ℚ-divisor D := ∑_{i=1}^{m} p_{i}/q_{i}x_{i} on V satisfies

div u = π^{∗}(D) + ι(C) .

Since V is normal, for a rational function φ ∈ K_{0} on C the following equivalences hold:

φu^{n} ∈ A_{n} ⇔ div (φu^{n}) ≥ 0 ⇔ π^{∗}div φ + ndiv u ≥ 0 ⇔
π^{∗}div φ + nπ^{∗}(D) + nι(C) ≥ 0 ⇔ div φ + nD ≥ 0 ⇔ φ ∈ H^{0}(C, O_{C}(⌊nD⌋)) .

Hence A_{n} = H^{0}(C, O_{C}(⌊nD⌋)) · u^{n} for all n ≥ 0, as desired.

(b) Any isomorphism of graded A_{0}-algebras

φ : A_{0}[D] = ⨁_{n≥0} H^{0}(C, O_{C}(⌊nD⌋)) · u^{n} ⟶ A_{0}[D′] = ⨁_{n≥0} H^{0}(C, O_{C}(⌊nD′⌋)) · u^{′n} ,

extends to an isomorphism of graded K_{0}-algebras

φ_{K_{0}} : K_{0}[u] → K_{0}[u′]

and so has the form u^{n} ↦ f^{n}u^{′n}, n ≥ 0, for some non-zero f ∈ K_{0}. Conversely, such a morphism φ_{K_{0}} maps A_{0}[D] isomorphically onto A_{0}[D′] if and only if

H^{0}(C, O_{C}(⌊nD′⌋)) = f^{n} · H^{0}(C, O_{C}(⌊nD⌋)) ∀n.

As

f^{n} · H^{0}(C, O_{C}(⌊nD⌋)) = H^{0}(C, O_{C}(⌊nD − ndivf⌋)) ,

the existence of an isomorphism φ as above is equivalent to the existence of an element f ∈ K_{0} with D′ = D − divf. □

3.3. We denote {D} = D − ⌊D⌋ the fractional part of a ℚ-divisor D. Since principal divisors are ℤ-divisors, we have {D} = {D′} as soon as D ∼ D′.

If C = Spec ℂ[t] = 𝔸_{ℂ}^{1} then the converse is also true. Indeed, any ℤ-divisor on 𝔸_{ℂ}^{1} is principal, and so the linear equivalence class of a ℚ-divisor D on 𝔸_{ℂ}^{1} is uniquely determined by the fractional part {D} of D. Thus we obtain the following corollary.

Corollary 3.4. For every normal parabolic ℂ^{∗}-surface V = Spec A with A = ⨁_{n≥0} A_{n} and A_{0} = ℂ[t], there is a unique isomorphism A ≅ A_{0}[D] of graded A_{0}-algebras, where D = 0 or D = ∑_{i=1}^{n} \frac{p_{i}}{q_{i}} x_{i} with 0 < p_{i} < q_{i}, gcd(p_{i}, q_{i}) = 1 ∀i = 1, …, n and x_{i} ∈ 𝔸_{ℂ}^{1}, x_{i} ≠ x_{j} for i ≠ j.

The next lemma is also well known; in lack of a reference we provide a short argument.

Lemma 3.5. Let D be a ℚ-divisor on a normal affine variety S and consider the graded ring A := ⨁_{i≥0} A_{i}, where A_{i} := H^{0}(S, O_{S}(⌊iD⌋)) · u^{i}. For d ∈ ℕ the following conditions are equivalent.

(i) dD is integral.

(ii) A_{d+m} = A_{d}A_{m} for all m ≥ 0.

(iii) The d-th Veronese subring A^{(d)} := ⨁_{m≥0} A_{md} is isomorphic to the symmetric algebra S_{A_{0}}(A_{d}) i.e., A_{kd} = S_{A_{0}}^{k}A_{d}.

Proof. Condition (ii) is equivalent to

O_{S}(⌊(m + d)D⌋) ≅ O_{S}(⌊mD⌋) ⊗ O_{S}(⌊dD⌋) ∀m ≥ 0 ,

and the latter condition is equivalent to

(ii′)  ⌊(m + d)D⌋ = ⌊mD⌋ + ⌊dD⌋ ∀m ≥ 0 .

Similarly, (iii) is equivalent to

(iii′)  ⌊kdD⌋ = k⌊dD⌋ ∀k ≥ 0 .

The equivalence of (i), (ii′) and (iii′) now follows from the elementary fact that for a rational number r = \frac{p}{q} the following conditions are equivalent:

(1) dr ∈ ℤ (2) ⌊mr + d⌋ = ⌊mr⌋ + ⌊d⌋ ∀m ≥ 0 (3) ⌊kdr⌋ = k⌊dr⌋ ∀m ≥ 0.

□

Notation 3.6. We denote d(A) the smallest positive integer d satisfying the equivalent conditions of Lemma 3.5.

Remark 3.7. In the situation of Theorem 3.2, one can recover D from the graded ring A = A_{0}[D] more algebraically as follows. Consider d ∈ ℕ with A_{d}A_{i} = A_{d+i} for all i ≥ 0 (or, equivalently, A_{id} = S^{i}(A_{d}), see Lemma 3.5) and let v be a generator of A_{d} as A_{0}-module; this exists after a suitable localization of A_{0}. If u^{d} = fv with f ∈ Frac A_{0}, then D = div(f)/d. In fact, the ideal vA is equal to A_{≥d} and so its set of zeros has no irreducible components in the fibres of π. Thus div v = d · ι(C) on V. Since

π^{∗}(D) = div u − ι(C) and d · div u = div v + div f

as divisors on V, we obtain D = div(f)/d.

A parabolic ℂ^{∗}-surface V = Spec A_{0}[D] has at most cyclic quotient singularities, as follows from Miyanishi’s Theorem (see [Miy, Lemma 1.4.4(1)]). In the next result (see [De, Section 5]) we describe their structure in terms of the divisor D.

Proposition 3.8. (a) If A_{0} = ℂ[t] and if D is supported on the origin in Spec A_{0} = 𝔸_{ℂ}^{1} so that D = −\frac{e}{d}[0] with gcd(e, d) = 1, then A := A_{0}[−\frac{e}{d}[0]] is naturally isomorphic to the semigroup algebra

A = A_{d,e} = ⨁_{b≥0, ad−be≥0} ℂ · t^{a}v^{b}

graded via deg t = 0, deg v = 1 (cf. Example 2.3). Consequently, V := Spec A is isomorphic to the toric surface V_{d,e′} = Spec A_{d,e′} ≅ 𝔸_{ℂ}^{2}/G_{d,e′}, where e′ ≡ e mod d and 0 ≤ e′ < d.

(b) If C = Spec A_{0} is any normal affine curve over ℂ and D is a ℚ-divisor on C, then the surface V = Spec A_{0}[D] is normal with at most cyclic quotient singularities. More precisely, if D(a) = −e/d with gcd(e, d) = 1 then V has a quotient singularity of type (d, e′) at ι(a), where e′ is as in (a).

Proof. The first part of (a) follows immediately from (3) in 3.1, whereas the second one is a consequence of Lemma 2.4.

Tensoring the isomorphism in (a) with − ⊗_{ℂ[t]} ℂ[[t]] we obtain that (b) holds if A_{0} ≅ ℂ[[t]]. The general case follows from this by taking completions at the maximal ideals of A_{0}. □

The algebra A_{0}[D] is finitely generated over A_{0}, so there exist f_{1}, …, f_{n} ∈ K_{0} and m_{1}, …, m_{n} ∈ ℕ such that

A = A_{0}[f_{1}u^{m_{1}}, …, f_{n}u^{m_{n}}] ⊆ K_{0}[u] .

In the next result we show how to compute D from such a representation.

Proposition 3.9. Let C = Spec A_{0} be a smooth affine curve and K_{0} := Frac A_{0}. If a 2-dimensional subring B of the polynomial ring K_{0}[u] is represented as

B = A_{0}[f_{1}u^{m_{1}}, …, f_{n}u^{m_{n}}] ⊆ K_{0}[u] , m_{i} > 0 ∀i

with f_{1}, …, f_{n} ∈ K_{0}, then its normalization A = B_{norm} coincides as an A_{0}-subalgebra of K_{0}[u] with A_{0}[D], where

D := − min_{1≤i≤n} \frac{div f_{i}}{m_{i}} .

Proof. By definition of D we have div f_{i} + m_{i}D ≥ 0 so by (3) f_{i}u^{m_{i}} ∈ A_{0}[D] and B is a subring of A_{0}[D]. As A_{0}[D] is normal (see Proposition 3.8(b)), A is also contained in A_{0}[D]. Let us show that these subrings coincide.

According to Theorem 3.2, we can represent A as A = A_{0}[D′] with π^{∗}(D′) = div u − ι(C). In particular f_{i}u^{m_{i}} ∈ A = A_{0}[D′], so again by (3) divf_{i} + m_{i}D′ ≥ 0 or, equivalently, D′ ≥ −\frac{1}{m_{i}} div f_{i}. Thus D′ ≥ D and A_{0}[D] ⊆ A_{0}[D′] = A. As we have already shown the converse inclusion we obtain that A = A_{0}[D], as desired. □

The following examples of parabolic ℂ^{∗}-surfaces ruled over 𝔸_{ℂ}^{1} are basic (see Theorem 3.11 below).

Example 3.10. For a unitary polynomial P ∈ ℂ[t] and for an integer d ≥ 1 we let

B_{d,P}^{+} := ℂ[t, u, v]/(u^{d} − P (t)v) ≅ ℂ [t, u, \frac{u^{d}}{P (t)}]

graded via

deg t = 0, deg u = 1, deg v = d .

The normalization

A_{d,P}^{+} := (B_{d,P}^{+} )_{norm}

is a positively graded finitely generated ℂ-algebra of dimension 2 with A_{0} = ℂ[t]. By Proposition 3.9 and Corollary 3.4 we have

A_{d,P}^{+} ≅ A_{0}[D] ≅ A_{0}[\{D\}], where D = D(d, P) := \frac{div(P)}{d} .

For P (t) = ∏_{i=1}^{n}(t − x_{i})^{r_{i}} (where x_{i} ≠ x_{j} if i ≠ j) we obtain

D = ∑_{i=1}^{n} \frac{r_{i}}{d} x_{i} , and \{D\} = ∑_{i=1}^{n} \{\frac{r_{i}}{d} \} x_{i} ,

whereas D = 0 if P = 1. Replacing D by {D} we may suppose that

(∗) gcd(d, r_{1}, …, r_{n}) = 1, 0 < r_{i} < d ∀i = 1, …, n, if d ≥ 2, and P = 1 if d = 1.

If two pairs (d, P) and (d̃, P̃) satisfy this condition (∗) and A_{d,P}^{+} ≅ A_{d̃,P̃}^{+} as graded

A_{0}-algebras then by Corollary 3.4 we have \frac{div(P)}{d} = \frac{div(P̃)}{d̃}, and so d = d̃ and P = P̃.

Thus we obtain the following classification result.

Theorem 3.11. For every normal affine surface V = Spec A, where A = ⨁_{i≥0} A_{i} with A_{0} = ℂ[t], there is a unique pair (d, P) satisfying condition (∗) and an equivariant isomorphism of A_{0}-schemes

φ : V ⟶ V_{d,P}^{+} := Spec A_{d,P}^{+} .

Remark 3.12. 1. In the situation of Theorem 3.11 above, the Veronese subring A^{(d)} is equal to A_{0}[v] = ℂ[t, v]. The cyclic group ℤ_{d} acts on A via the ℂ^{∗}-action and A^{(d)} coincides with the ring of invariants A^{ℤ_{d}}, whereas A is the normalization of A^{(d)} in the fraction field Frac(A). Thus the morphism V → 𝔸_{ℂ}^{2} = Spec ℂ[t, v] induced by the inclusion ℂ[t, v] ⊆ A represents V as a cyclic covering of the plane branched along the curve u = 0, and V is the normalization of a surface {u^{d} − P (t)v = 0} in ℂ^{3}.

2. More generally, let C be any affine smooth curve and let A = ⨁_{i≥0} A_{i} be a normal 2-dimensional A_{0}-algebra of finite type. If A_{1} = u · A_{0} and A_{d} = v · A_{0}, d := d(A_{≥0}), for suitable elements u ∈ A_{1} and v ∈ A_{d} then A is the normalization of an algebra A_{0}[u, v]/(u^{d} − P_{+}v) graded via deg u = 1, deg v = d, for a certain d ∈ ℕ and a certain element P_{+} ∈ A_{0}.

4. The hyperbolic case

Let A = ⨁_{i∈ℤ} A_{i} be the coordinate ring of a normal affine surface V = Spec A with ℂ^{∗}-action such that A_{+}, A_{−} are both non-zero. Here again there is a quotient morphism π : V → C = Spec A_{0} induced by the inclusion A_{0} ↪ A. Every fiber of π is either a non-trivial orbit or a union of two 1-dimensional orbits and a hyperbolic fixed point, which is a source for one of them and a sink for the other one [FiKa_{1}, Lemma 1.7]. Thus the fixed point set F is finite and contains Sing V.

By Lemma 1.1 the proper subalgebras A_{≥0} and A_{≤0} of A are normal and finitely generated, and so V_{+} := Spec A_{≥0} and V_{−} := Spec A_{≤0} are normal affine surfaces with a parabolic ℂ^{∗}-action birationally dominated by V. The natural embeddings A_{0} ↪ A_{≥0} ↪ A and A_{0} ↪ A_{≤0} ↪ A yield the commutative diagram

[figure: V_{+}; σ_{+}; V; σ_{−}; V_{−}; π_{+}; π_{−}; C; π] (4)

where σ_{±} are equivariant birational morphisms. Hence σ_{±} are equivariant affine modifications [KaZa_{1}, Theorem 1.1]. More precisely the following result holds.

Proposition 4.1. V can be obtained from V_{±} by blowing up a ℂ^{∗}-invariant subscheme and deleting the proper transform of a ℂ^{∗}-invariant divisor D^{±} on V_{±}, which contains the fixed point curve ι_{±}(C) ⊆ V_{±}.

Proof. Let us show this for V_{+}, the proof for V_{−} being similar. Choose a system of homogeneous generators a_{1}, …, a_{n} of the finitely generated A_{0}-subalgebra A_{≤0} and let f_{0} ∈ A_{+} be a non-zero element of degree m = − min_{i} deg a_{i}. Letting f_{i} := a_{i}f_{0} for i = 1, …, n we obtain

A = A_{≥0} [\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ] = A_{≥0}[I/f_{0}] := \{\frac{x_{k}}{f_{0}^{k}} | x_{k} ∈ I^{k}, k ≥ 0\} ,

where I is the graded ideal of A_{≥0} generated by f_{0}, …, f_{n}. Thus V = Spec A is obtained by blowing up V_{+} = Spec A_{≥0} with center I and deleting the proper transform of the ℂ^{∗}-invariant divisor div f_{0} on V_{+}. As this divisor contains ι_{+}(C), the result follows. □

Remark 4.2. The exceptional divisor of σ_{±} contains extra components of the reducible fibers of π. If the center V (I_{±}) of σ_{±} is nonempty then σ_{±} contracts exactly one irreducible curve at each point of the center V (I_{±}). It might happen that the center is empty, and V = V_{±} \ ι_{±}(C). But in general (in contrary to the situation considered in [De, 2.2]) a blowup is present, as the following simple example shows.

Example 4.3. If A = ℂ[x, y] is graded via deg x = 1, deg y = −1 then A_{0} = ℂ[xy] and A_{≥0} = ℂ[x, xy]. Thus V ≅ 𝔸_{ℂ}^{2} with the hyperbolic ℂ^{∗}-action t.(x, y) = (tx, t^{−1}y) and V_{+} ≅ 𝔸_{ℂ}^{2} with the parabolic ℂ^{∗}-action t.(u, v) = (tu, v). Furthermore, C ≅ 𝔸_{ℂ}^{1}, π(x, y) = xy, π_{+}(u, v) = v and σ(x, y) = (x, xy) is the standard affine modification of 𝔸_{ℂ}^{2} at the origin along a coordinate axes.

4.4. The Dolgachev-Pinkham-Demazure construction is still available in the hyperbolic case. In [De, Theorem 3.5] it is done under the additional assumption that A_{−n}⊗A_{n} → A_{0} is an isomorphism for all n. Here we generalize the construction in order to make it work for any hyperbolic ℂ^{∗}-surface.

Let D_{+}, D_{−} be ℚ-divisors on the smooth affine curve C := Spec A_{0}. For n ≥ 0 we consider the A_{0}-submodules

A_{−n} := H^{0}(C, O_{C}(⌊nD_{−}⌋)) · u^{−n} and A_{n} := H^{0}(C, O_{C}(⌊nD_{+}⌋)) · u^{n}

of Frac(A_{0})[u, u^{−1}], where u is an indeterminate of degree 1. If D_{+} + D_{−} ≤ 0 then for n ≥ m ≥ 0 we have

⌊nD_{+}⌋ + ⌊mD_{−}⌋ ≤ ⌊(n − m)D_{+}⌋,

whence A_{n} · A_{−m} ⊆ A_{n−m}. Similarly, for 0 ≤ n ≤ m we have A_{n} · A_{−m} ⊆ A_{n−m}. Thus

A := A_{0}[D_{+}, D_{−}] := ⨁_{n∈ℤ}A_{n}

is a finitely generated A_{0}-subalgebra of Frac(A_{0})[u, u^{−1}] with A_{≥0} = A_{0}[D_{+}] and A_{≤0} = A_{0}[D_{−}]. The grading on A defines a natural hyperbolic ℂ^{∗}-action on the surface V := Spec A. The latter surface is normal as so are the algebras A_{0}[D_{+}] and A_{0}[D_{−}] (see Lemma 1.1 and Corollary 3.8(b)). Conversely, we have the following theorem.

Theorem 4.5. If C = Spec A_{0} is a smooth affine curve and A = ⨁_{i∈ℤ} A_{i} is a normal graded finitely generated domain of dimension 2 with A_{±} ≠ 0, then the following hold.

(a) A is isomorphic to A_{0}[D_{+}, D_{−}], where D_{+}, D_{−} are ℚ-divisors on C satisfying D_{+} + D_{−} ≤ 0. More precisely, if u ∈ Frac(A_{0}) · A_{1} and if the divisors D_{+}, D_{−} on C are defined by

(5) π_{+}^{∗}(D_{+}) = div(u) − ι_{+}(C) and π_{−}^{∗}(D_{−}) = div(u^{−1}) − ι_{−}(C),

where π_{±} are as in diagram (4) above and ι_{±} : C ↪ V_{±} are the natural embeddings, then D_{+} + D_{−} ≤ 0 and A ≅ A[D_{+}, D_{−}].

(b) A_{0}[D_{+}, D_{−}] ≅ A_{0}[D′_{+}, D′_{−}] as graded A_{0}-algebras if and only if, for a rational function φ ∈ Frac(A_{0}), one has

D′_{+} = D_{+} + div φ  and  D′_{−} = D_{−} − div φ .

Proof. (a) By Theorem 3.2 and its proof we have equalities

A_{≥0} = A_{0}[D_{+}] and A_{≤0} = A_{0}[D_{−}]

as subalgebras of Frac(A_{0})[u, u^{−1}], whence A = A_{0}[D_{+}, D_{−}]. It remains to show that D_{+} + D_{−} ≤ 0. Applying in (5) the functors σ_{+}^{∗} and σ_{−}^{∗} respectively, we obtain

π^{∗}(D_{+}) = div(u) − σ_{+}^{∗}ι_{+}^{∗}(C) and π^{∗}(D_{−}) = div(u^{−1}) − σ_{−}^{∗}ι_{−}^{∗}(C).

Taking the sum of these equalities yields π^{∗}(D_{+} + D_{−}) = −(σ_{+}^{∗}ι_{+}^{∗}(C) + σ_{−}^{∗}ι_{−}^{∗}(C)), whence D_{+} + D_{−} ≤ 0, as required. Finally (b) follows from Theorem 3.2(b) and its proof. □

Consequently, if A_{0} = ℂ[t] then A admits a unique presentation A = A_{0}[D_{+}, D_{−}] with D_{+} = {D_{+}} and D_{+} + D_{−} ≤ 0.

It follows from Theorem 4.5 that outside |D_{+}| ∪ |D_{−}|, the map π : V → C is a locally trivial principal ℂ^{∗}-bundle. More generally, the Dolgachev-Pinkham-Demazure construction shows the following result (cf. [BasHa], [FiKa_{1}, Proposition 1.11]).

Corollary 4.6. In all three cases, outside of a finite subset of the curve C the projection π : V^{∗} → C and π : V → C, respectively, defines a locally trivial fiber bundle. This is a principal ℂ^{∗}-bundle in the elliptic and hyperbolic cases, and a line bundle in the parabolic case.

Note that if u ∈ A_{1} ∪ A_{−1} is a non-zero element then its restriction to a general fiber of π gives a fiber coordinate and so a trivialization over a Zariski open subset of C.

Remarks 4.7. 1. If v ∈ A is an invertible element of degree d > 0 then we can write

v = fu^{d} ∈ A_{d} and v^{−1} = f^{−1}u^{−d} ∈ A_{−d} ,

where f ∈ Frac(A_{0}) satisfies

div(f) + dD_{+} ≥ 0 and − div(f) + dD_{−} ≥ 0 .

Thus 0 ≥ D_{+} + D_{−} ≥ 0, whence D_{−} = −D_{+}. Conversely, if D_{+} = −D_{−} and if vA_{0} = A_{d} with d := d(A_{≥0}) = d(A_{≤0}), then v = fu^{d} with divf + dD_{+} = 0 by Remark 3.7. Hence also divf^{−1} − dD_{−} = 0, so f^{−1}u^{−d} ∈ A and fu^{d} is a unit in A.

2. Suppose now that A_{0} ≅ ℂ[t] and D_{+} is the ℚ-divisor −\frac{e}{d}[0], as in Lemma 2.4, where 0 ∈ 𝔸_{ℂ}^{1} is the the origin. Consider the element

v := t^{e}u^{d} ∈ A := A_{0}[D_{+}].

As A has an isolated singularity, the localized ring A_{v} is smooth and graded of hyperbolic type. Since v^{−1} = t^{−e}u^{−d} ∈ A_{v}, the associated divisor D_{−} is just D_{−} = \frac{e}{d}[0]. Thus

A_{v} = A_{0}[D_{+}, −D_{+}].

Of course, the same result holds if A_{0} = ℂ[t] is replaced by the formal power series ring A_{0} = ℂ[[t]].

3. Similarly, if C = Spec A_{0} is any 1-dimensional smooth curve and if D_{+} is a ℚdivisor on C, then A_{0}[D_{+}, −D_{+}] is a smooth algebra. This is easily reduced to the case A_{0} = ℂ[[t]] in (2) using completions at the maximal ideals of A_{0}.

The following analogue of Proposition 3.8 holds with a similar proof.

Lemma 4.8. Let C = Spec A_{0} be a smooth affine curve with function field K_{0} = Frac(A_{0}). If a graded 2-dimensional domain B ⊆ K_{0}[u, u^{−1}] is represented as

B = A_{0}[h_{1}u^{−n_{1}}, …, h_{k}u^{−n_{k}}, f_{1}u^{m_{1}}, …, f_{n}u^{m_{n}}] (where n_{i}, m_{j} > 0 ∀i, j)

with h_{1}, …, h_{k}, f_{1}, …, f_{n} ∈ K_{0} and B_{0} = A_{0}, then its normalization A = B_{norm} coincides (as a graded A_{0}-subalgebra of K_{0}[u]) with A_{0}[D_{+}, D_{−}], where

D_{−} = − min_{1≤i≤k} \frac{div h_{i}}{n_{i}}  and  D_{+} = − min_{1≤j≤n} \frac{div f_{j}}{m_{j}} .

We notice that the assumption A_{0} = B_{0} amounts to the inequalities

\frac{div h_{i}}{n_{i}} + \frac{div f_{j}}{m_{j}} ≥ 0 ∀i, j ,

which in turn are equivalent to D_{+} + D_{−} ≤ 0.

The following lemma provides a useful additional information in the particular case where V is ruled over the affine line 𝔸_{ℂ}^{1}.

Lemma 4.9. Let A = ⨁_{i∈ℤ} A_{i} = A_{0}[D_{+}, D_{−}] ⊆ Frac(A_{0})[u, u^{−1}] with A_{0} ≅ ℂ[t], and let d_{±} = d_{±}(A) be the minimal positive integer such that the divisor d_{±}D_{±} is integral. If A_{±1} = u_{±} · A_{0}, A_{±d_{±}} = v_{±} · A_{0} and

u_{+}u_{−} = Q(t), u_{±}^{d_{±}} = P_{±}(t)v_{±}

for some unitary polynomials Q, P_{±} ∈ ℂ[t] then

(6)  D_{+} = \frac{div P_{+}}{d_{+}} + D_{0}  and  D_{−} = \frac{div P_{−}}{d_{−}} − D_{0} − div Q ,

where D_{0} is the integral divisor D_{0} = div(u/u_{+}) on 𝔸_{ℂ}^{1}. Consequently,

(7) \frac{div P_{+}}{d_{+}} + \frac{div P_{−}}{d_{−}} ≤ div Q .

Furthermore, P_{+} and P_{−} are uniquely determined by D_{+} and D_{−} through

(8)  \{D_{+}\} = \frac{div P_{+}}{d_{+}}  and  \{D_{−}\} = \frac{div P_{−}}{d_{−}} .

Proof. We have u^{d_{+}} = P_{+} · (u/u_{+})^{d_{+}}v_{+} and u^{−d_{−}} = P_{−} · (u/u_{+})^{−d_{−}}Q^{−d_{−}}v_{−} and so by Remark 3.7

D_{+} = \frac{div(P_{+} · (u/u_{+})^{d_{+}})}{d_{+}} = \frac{div P_{+}}{d_{+}} + D_{0}

D_{−} = \frac{div(P_{−} · (u/u_{+})^{−d_{−}}Q^{−d_{−}}}{d_{−}} = \frac{div P_{−}}{d_{−}} − D_{0} − div Q .

Now (7) follows from the inequality D_{+} + D_{−} ≤ 0. To show (8) we let P_{±} = S_{±}^{d_{±}}T_{±}, where S_{±}, T_{±} ∈ ℂ[t] are polynomials with

div S_{±} = ⌊\frac{div P_{±}}{d_{±}} ⌋ and div T_{±} = \{\frac{div P_{±}}{d_{±}} \}

respectively. The relation (u_{±}/S_{±})^{d_{±}} = T_{±}v_{±} then shows that u_{±}/S_{±} is integral over A and so by the normality of A is contained in A_{±1}. As u_{±} is a generator of A_{±1} this forces S_{±} = 1, proving (8). □

Remark 4.10. The relation u_{+}^{d_{+}} = v_{+}P_{+} shows that u_{+} is integral over A_{0}[u_{−}, v_{±}] ⊆ A. Moreover, A and its subring A_{0}[u_{−}, v_{±}] have the same field of fractions. Therefore A is the normalization of A_{0}[u_{−}, v_{±}]. To find the relations between the generators of A_{0}[u_{−}, v_{±}], write d_{±} = d′_{±}k, where k := gcd(d_{+}, d_{−}). From v_{±} = u_{±}^{d_{±}}/P_{±} we get

v_{+}^{d′_{−}}v_{−}^{d′_{+}} = \frac{u_{+}^{d_{+}d′_{−}}u_{−}^{d′_{+}d_{−}}}{P_{+}^{d′_{−}}P_{−}^{d′_{+}}} = \frac{Q^{kd′_{+}d′_{−}}}{P_{+}^{d′_{−}}P_{−}^{d′_{+}}}

(in particular, Q^{kd′_{+}d′_{−}} is divisible by P_{+}^{d′_{−}}P_{−}^{d′_{+}}; cf. (7)). Similarly,

v_{+}u_{−}^{d_{+}} = \frac{u_{+}^{d_{+}}u_{−}^{d_{+}}}{P_{+}} = \frac{Q^{d_{+}}}{P_{+}} ∈ A_{0} .

Thus A as above is the normalization of the algebra

B := ℂ[t, u_{−}, v_{+}, v_{−}]/ (u_{−}^{d_{−}} − v_{−}P_{−}, v_{+}^{d′_{−}}v_{−}^{d′_{+}} − \frac{Q^{kd′_{+}d′_{−}}}{P_{+}^{d′_{−}}P_{−}^{d′_{+}}} , v_{+}u_{−}^{d_{+}} − \frac{Q^{d_{+}}}{P_{+}} ) (9)

graded via deg t = 0, deg u_{−} = −1, deg v_{±} = ±d_{±}. Indeed, Spec B is irreducible because the general fibers of the map t : Spec B → Spec A_{0} are irreducible, and every fiber is 1-dimensional and in the closure of the generic fiber.

Since u_{−} satisfies the relation u_{−}^{d_{−}} = v_{−}P_{−}, it is integral over the ring A_{0}[v_{±}]. Hence A is integral over A_{0}[v_{±}]. The normalization of A_{0}[v_{±}] is just the Veronese subring A^{(k)} of A, since the fraction fields of both rings coincide. In other words, A^{(k)} is the normalization of the hypersurface algebra

B̃ := ℂ[t, v_{+}, v_{−}]/ (v_{+}^{d′_{−}}v_{−}^{d′_{+}} − P (t)) , where P := \frac{Q^{kd′_{+}d′_{−}}}{P_{+}^{d′_{−}}P_{−}^{d′_{+}}} .

The cyclic group ℤ_{d} acts on A via the ℂ^{∗}-action with invariant ring A^{(k)} or, geometrically speaking, V = Spec A is a cyclic branched covering of degree k of the normalization of the hypersurface {v_{+}^{d′_{−}}v_{−}^{d′_{+}} = P (t)} in 𝔸_{C}^{3}.

In particular, in the case gcd(d_{+}, d_{−}) = 1 the algebra A is the normalization of A_{0}[v_{+}, v_{−}]/(v_{+}^{d_{−}}v_{−}^{d_{+}} − P (t)). It is worthwhile to remark how to get a representation A ≅ A_{0}[D_{+}, D_{−}] in terms of P. Choose p, q ∈ ℤ with |{d_{+} \atop d_{−}} {p \atop q}| = 1 so that u′ = v_{+}^{p}v_{−}^{q} has degree 1. By an easy calculation u^{′d_{+}} = v_{+}P^{p} and u^{′−d_{−}} = v_{−}/P^{q}, whence by Remark 3.7 A ≅ A_{0}[D_{+}, D_{−}] with

D_{+} = \frac{p}{d_{+}} div P and D_{−} = −\frac{q}{d_{−}} div P .

The special case d_{+} = 1 leads to the following example.

Example 4.11. (Cf. [Be, Example 4.11]) For a unitary polynomial P ∈ ℂ[t], we let A = A_{d,P} = B_{norm} be the normalization of the ℂ-algebra

B = B_{d,P} := ℂ[t, u, v]/(u^{d}v − P (t))

graded via

deg t = 0, deg u = 1, deg v = −d ,

so that the normal affine surface V := Spec A is equipped with a hyperbolic ℂ^{∗}- action. The regular function t on V is ℂ^{∗}-invariant, so the ℂ^{∗}-action preserves its fibers, and the general fibers of t are irreducible. Hence t gives the quotient morphism π : V → C ≅ ℂ = Spec ℂ[t]. As A ≅ A_{0}[u, P u^{−d}] we can write

A ≅ A_{0}[D_{+}, D_{−}], where D_{+} = 0 and D_{−} = −div(P)/d

(see Lemma 4.8).

Moreover, we can recover P_{±} and Q in the representation of Remark 4.10. Indeed, by the construction given there P_{+} = 1 and by (8) {D_{−}} = div(P_{−})/d_{−} (note that d = d_{−}). This gives

(10)  div P_{−} = d \{−\frac{div P}{d} \}  and div Q = \frac{div P + div P_{−}}{d}

(see (6)). In particular,

A_{≥0} ≅ A_{0}[u] ≅ ℂ[t, u] and A_{≤0} ≅ A_{d,P_{−}}^{+}

as graded A_{0}-algebras, where for the second isomorphism we have to reverse the grading of one of the rings.

This discussion provides the following characterization of the algebras A_{d,P}.

Proposition 4.12. If A = A_{0}[D_{+}, D_{−}], where A_{0} ≅ ℂ[t] and D_{+}, D_{−} are ℚ-divisors on 𝔸_{ℂ}^{1} with D_{+} + D_{−} ≤ 0, then the following conditions are equivalent.

(i) D_{+} is integral i.e., {D_{+}} = 0.

(ii) A_{≥0} ≅ A_{0}[u] as graded A_{0}-algebras, where deg u = 1.

(iii) A ≅ A_{d,P} as graded A_{0}-algebras, where D_{+} + D_{−} = −\frac{div P}{d}.

Next we study the effect of a base change to the Dolgachev-Pinkham-Demazure representation.

Proposition 4.13. Let C = Spec A_{0} be an affine curve with function field K_{0} = Frac(A_{0}) and let

A := A_{0}[D_{+}, D_{−}] ⊆ K_{0}[u, u^{−1}] ,

where D_{±} are ℚ-divisors on C satisfying D_{+} + D_{−} ≤ 0. Let L be the field L := Frac(A)[\sqrt[d]{tu^{b}}], where t ∈ K_{0} and b, d ∈ ℕ. If A′ is the normalization of A in L then the following hold.

1. A′_{0} is the normalization of A_{0} in K_{0}[s] with s := \sqrt[k]{t}, where k := gcd(b, d).

2. A′ ≅ A′_{0}[D′_{+}, D′_{−}] with

D′_{±} := \frac{k}{d} (p^{∗}(D_{±}) ± βdiv s) ,

where p : C′ := Spec A′_{0} → C is the projection and β is defined by βb ≡ k mod d.

Proof. We let b = b′k and d = d′k. The normalization A′ admits a natural \frac{1}{d}-grading, and the element u^{∗} := \sqrt[d]{tu^{b}} is of degree b/d = b′/d′. If we write k = βb + δd, then the element u′ := u^{∗β}u^{δ} ∈ Frac(A′) has minimal possible positive degree 1/d′. Thus

A′ ⊆ Frac(A′_{0})[u′, u^{′−1}].

To compute A′_{0}, we note that u^{∗n}u^{−m} with n, m ∈ ℕ has degree 0 if and only if nb′/d′ = m. In particular, n = n′d′ is an integer multiple of d′. Thus K′_{0} := Frac A′_{0} is generated over K_{0} by u^{∗d′}u^{−b′} = t^{d′/k} (i.e., n′ = 1). As d′ and k are coprime, it follows that s = \sqrt[k]{t} also belongs to K′_{0} and that this field is actually generated by s over K_{0}, proving (1).

After localizing A_{0} we may assume that there is an element v_{+} ∈ A of degree d_{+} = d(A_{≥0}) with A_{d_{+}} = v_{+}A_{0} (see 3.6). We claim that then A′_{sd_{+}} = v_{+}^{s}A′_{0} for all s ≥ 0. If not, then for some s > 0 and some non-unit x ∈ A′_{0} the element v_{+}^{s}/x belongs to A′, so it is integral over A and there is an equation

\frac{v_{+}^{sm}}{x^{m}} + a_{1} \frac{v_{+}^{s(m−1)}}{x^{m−1}} + ⋯ + a_{m} = 0 ,

where m ≥ 0 and a_{i} ∈ A_{id_{+}}. Thus a_{i} = v_{+}^{si}q_{i} for some elements q_{i} ∈ A_{0}, whence dividing the equation above by v_{+}^{sm} we obtain that

\frac{1}{x^{m}} + q_{1} \frac{1}{x^{m−1}} + ⋯ + q_{m} = 0 .

As A′_{0} is integrally closed this is only possible if x ∈ A′_{0} contradicting the choice of x.

Thus v = v_{+} is an element satisfying the assumptions of Remark 3.7, and we compute with it the divisor D′_{+} as follows (the calculation for D′_{−} is analogous). If we consider the new grading of A′ by assigning to u′ the degree 1, then v_{+}^{k} becomes an element of degree dd_{+}. Moreover, if u^{d_{+}} = P_{+}v_{+} with P_{+} ∈ K_{0} then by Remark 3.7 D_{+} = \frac{div(P_{+})}{d_{+}}. Since

u^{′dd_{+}} = (u^{∗β}u^{δ})^{dd_{+}} = (tu^{b})^{βd_{+}}u^{δdd_{+}}
= t^{βd_{+}}u^{d_{+}(βb+δd)} = t^{βd_{+}}u^{d_{+}k}
= t^{βd_{+}}P_{+}^{k}v_{+}^{k}

we obtain again by Remark 3.7 that on C′

D′_{+} = \frac{div(t^{βd_{+}}P_{+}^{k})}{dd_{+}} = \frac{β}{d} div(t) + \frac{k}{d}p^{∗}(D_{+})

and (2) follows. □

Let us consider the following important example.

Example 4.14. With A_{0} := ℂ[t], suppose that D_{+} = −\frac{e}{d}[0] and that D_{−} is any ℚdivisor on 𝔸_{ℂ}^{1} = Spec A_{0}. Applying Proposition 4.13 to s := \sqrt[d]{t} (i.e. b = 0) we get that the normalization of A := A_{0}[D_{+}, D_{−}] in the field L := Frac(A)[s] is given by

A′ = A′_{0}[−e[0], D′_{−}] ⊆ ℂ(s)[u, u^{−1}] ,

where A′_{0} = ℂ[s] and D′_{−} = p^{∗}(D_{−}) (as before, p : Spec ℂ[s] → Spec ℂ[t] denotes the projection s ↦ s^{d}). The divisor D′_{+} = −e[0] being integral we have

A′ ≅ A′_{0}[0, D′_{+} + D′_{−}] ⊆ ℂ(s)[ũ, ũ^{−1}] ,

where ũ := s^{e}u.

More concretely, if k := d_{−}(A) and if we choose a unitary polynomial Q ∈ ℂ[t] with D_{−} = −\frac{div(Q)}{k} then D′_{+} + D′_{−} = −\frac{div(Q(s^{d})s^{ke})}{k}. By Example 4.11 A′ ≅ A_{k,P} is the normalization of

B_{k,P} = ℂ[s, ũ, v]/ (ũ^{k}v − P (s)) , where P (s) := Q(s^{d})s^{ke} . (11)

The field extension Frac(A) ⊆ Frac(A)[s] is Galois with Galois group ℤ_{d} = ⟨ζ⟩, where ζ.s = ζs. Thus

A ≅ (A_{k,P})^{ℤ_{d}} ,

and the action of ζ on ũ = s^{e}u is given by ζ.ũ = ζ^{e}ũ . Therefore, the group ℤ_{d} acts on A_{k,P} via

ζ.s = ζs, ζ.ũ = ζ^{e}ũ and ζ.v = v . (12)

Thus we obtain the following characterization.

Proposition 4.15. For an algebra A = A_{0}[D_{+}, D_{−}] with A_{0} = ℂ[s] the following conditions are equivalent.

(i) \{−D_{+}\} = \frac{e}{d}[0], where 0 ≤ e < d and gcd(e, d) = 1.

(ii) A ≅ (A_{k,P})^{ℤ_{d}}, where A_{k,P} is the normalization of B_{k,P} in (11) and where ℤ_{d} = ⟨ζ⟩ acts via the formulas in (12).

Like in the parabolic case V may possess at most cyclic quotient singularities, as follows from Miyanishi’s Theorem (see [Miy, Lemma 1.4.4(1)]). The type of quotient singularities is determined from the divisors D_{+}, D_{−} by the following result. As before, C = Spec A_{0} is a smooth affine curve with function field K_{0} = Frac(A_{0}) and A := A_{0}[D_{+}, D_{−}] with ℚ-divisors D_{+} and D_{−} on C. Denote π : V = Spec A → C the canonical projection.

Theorem 4.16. (a) The set of singular points Sing V is contained in the fixed point

set F which is the zero locus F = V (I) of the ideal I := A_{+}A + A_{−}A of A.

(b) The map π|F : F → C is injective, and π(F) = {a ∈ C|D_{+}(a) + D_{−}(a) < 0}.

(c) For a point a′ ∈ F with image a := π(a′) ∈ C we write

D_{+}(a) = −\frac{e_{+}}{m_{+}}  and  D_{−}(a) = \frac{e_{−}}{m_{−}}

with the convention that

m_{+} > 0, m_{−} < 0,  gcd(e_{+}, m_{+}) = gcd(e_{−}, m_{−}) = 1  and
m_{+} = 1 if D_{+}(a) = 0, m_{−} = −1 if D_{−}(a) = 0 .

Let p, q ∈ ℤ with |{p \atop q} {e_{+} \atop m_{+}} | = 1. Then a′ ∈ F is a quotient singularity of type

(Δ(a), e), where Δ(a) := − | \begin{matrix}e_{+} & e_{−} \\ m_{+} & m_{−} \end{matrix} | and e ≡ | \begin{matrix}p & e_{−} \\ q & m_{−} \end{matrix} | mod Δ(a) .

In particular, a′ ∈ Sing V if and only if Δ(a) ≠ 1.

Proof. As in the proof of Proposition 3.8(b) we can reduce the statement to the case that A_{0} = ℂ[t] and |D_{+}| ∪ |D_{−}| is contained in the origin, so that D_{±} = ∓e_{±}/m_{±}[0].

(a) If v ∈ A, v ≠ 0, is a homogeneous element of non-zero degree then the localization A_{v} is smooth (see Remark 4.7(2)). Thus Sing V ⊆ V (I), as claimed.

(b) The map A_{0} → A/I is obviously surjective. Thus

π|F : F = Spec(A/I) → C

is a closed embedding. Moreover, F = ∅ if and only if 1 ∈ I if and only if 1 = a_{+}a_{−} for some homogeneous elements of A of opposite degrees, and the latter happens if and only if D_{+} + D_{−} = 0 by Remark 4.7(1).

(c) Notice first that the elements

t^{e_{+}}u^{m_{+}}, t^{e_{−}}u^{m_{−}} ∈ K_{0}[u, u^{−1}]

belong to A. By definition, the ideal I + tA of A (this is just the maximal ideal of the point a′ ∈ F) is generated by the monomials t^{e}u^{m} with (e, m) ∈ ℤ × ℤ, where (e, m) ≠ (0, 0) and

e + mD_{+}(0) ≥ 0 if m ≥ 0, e − mD_{−}(0) ≥ 0 if m ≤ 0 .

In other words, (e, m) is an element of the cone C := C ((e_{+}, m_{+}), (e_{−}, m_{−})) generated by the vectors (e_{±}, m_{±}) in the plane. Hence A is a toric algebra generated by the semigroup C ∩ ℤ^{2} and so is a quotient A_{d,e} for some d, e ≥ 0 (see Lemma 2.4). To determine d, e, we must find a basis of ℤ^{2} such that (e_{+}, m_{+}) is one of the basis vectors. This is done as follows.

If we choose p, q ∈ ℤ with |{p \atop q} {e_{+} \atop m_{+}} | = 1, then the vectors ẽ_{1} := (e_{+}, m_{+}) and ẽ_{2} := (p, q) form a basis of ℤ^{2}, and

(e_{−}, m_{−}) = Δ′ẽ_{1} + Δẽ_{2}, where Δ′ := |{p \atop q} {e_{−} \atop m_{−}} | and Δ := Δ(0) .

As ẽ_{1} and (e_{−}, m_{−}) form a basis of the cone C, it follows from Lemma 2.4 that A has a quotient singularity of type (Δ, e), where 0 ≤ e < d and e ≡ |{p \atop q} {e_{−} \atop m_{−}} | mod Δ. Note that Δ and Δ′ are coprime since so are e_{−} and m_{−}.

The determinant Δ has always positive sign as

D_{+}(0) + D_{−}(0) = \frac{Δ}{m_{+}m_{−}} ≤ 0 and m_{+} > 0, m_{−} < 0 ,

and so (c) follows. □

Corollary 4.17. If A_{d,P} is the normalization of the algebra

B_{d,P} = ℂ[t, u, v]/(u^{d}v − P (t)) ,

where P (t) = ∏_{i=1}^{k}(t−a_{i})^{r_{i}} with a_{i} ≠ a_{j} for i ≠ j (see Example 4.11) then the singular points of the surface V_{d,P} = Spec A_{d,P} are the points a′_{i} ∈ V_{d,P} (1 ≤ i ≤ k), where t = a_{i}, u = v = 0 and r_{i} ∤ d.

Proof. It was shown in Example 4.11 that D_{−} = 0 and D_{+}(a_{i}) = −\frac{r_{i}}{d}. Therefore, Δ(a_{i}) = e_{+} > 1 if and only if r_{i} ∤ d, which implies our assertion. □

In the sequel we use the following notation.

Definition 4.18. Let O = ℂ^{∗}z be the orbit through a point z ∈ V \F. Following [FiKa_{1}] we say that O is of type (d, q) if d is the order of the stabilizer

Stab_{z} = ker(ℂ^{∗} → Aut O) ⊆ ℂ^{∗}, so that Stab_{z} = ⟨ζ⟩ ≅ ℤ_{d} ,

and q (0 ≤ q < d) is determined from the tangent representation of Stab_{z} on the tangent plane T_{z}V via pseudo-reflections

Stab_{z} ∋ ζ ⟼ ( \begin{matrix}1 & 0 \\ 0 & ζ^{q} \end{matrix} ) .

The orbit O is called principal if d = 1 and exceptional otherwise (see [FiKa_{1}]-[FiKa_{3}] for a detailed description of the structure of V near the exceptional orbits).

In the next result we will characterize the orbit types of the surface V = Spec A with A := A_{0}[D_{+}, D_{−}], where D_{+} and D_{−} are ℚ-divisors on the smooth affine curve C = Spec A_{0}. Let π : V → C denote the projection. To examine the orbits over a point a ∈ C, we write

D_{+}(a) = −e_{+}/m_{+} and D_{−}(a) = e_{−}/m_{−}

with the conventions as in Theorem 4.16(c). Let q_{+} be defined by 0 ≤ q_{+} < m_{+} and q_{+}e_{+} ≡ −1 mod m_{+}, and similarly q_{−} by 0 ≤ q_{−} < −m_{−} and q_{−}e_{−} ≡ 1 mod m_{−}. With these notations the following result holds.

Theorem 4.19. The exceptional orbits of V are located over |D_{+}| ∪ |D_{−}|. The orbits over a given point a ∈ |D_{+}| ∪ |D_{−}| are as follows.

(a) If D_{+}(a) + D_{−}(a) = 0 then π^{−1}(a) consists of one orbit of type (m_{+}, q_{+}).

(b) If D_{+}(a) + D_{−}(a) < 0 then π^{−1}(a) contains two orbits O^{+} and O^{−} of types (m_{+}, q_{+}) and (−m_{−}, q_{−}), respectively. Moreover, their closures Ō^{±} intersect in the unique fixed point, say a′, of the fiber, and π^{−1}(a) = O^{+} ∪ {a′} ∪ O^{−}.

Proof. With the same reasoning as in the proof of Proposition 3.8(b), it is sufficient to treat the case that A_{0} = ℂ[t] and that D is supported on a = 0 ∈ 𝔸_{ℂ}^{1}, i.e. D_{±} = ∓e_{±}/m_{±}[0]. Note that in this case m_{+} = d(A_{≥0}) and m_{−} = −d(A_{≤0}).

Let us first assume that D_{+} +D_{−} = 0, so that e := e_{+} = −e_{−} and m := m_{+} = −m_{−}. The algebra A is then the semigroup algebra ℂ[C ∩ ℤ^{2}] of the real cone C generated by the vectors ±(e, m) and (1, 0). If we choose p, q ∈ ℤ with |{p \atop q} {e \atop m}| = 1, then

C ∩ ℤ^{2} = {(a, b)|(a, b) ∈ ℤ(e, m) + ℕ(p, q)}.

Hence A is the algebra of Laurent polynomials A = A_{0}[x, x^{−1}, y], where

x := t^{e}u^{m} ∈ A_{m} and y := t^{p}u^{q} ∈ A_{q}.

Clearly t = x^{−q}y^{m}. The action of ℂ^{∗} is given by λ.x = λ^{m}x and λ.y = λ^{q}y, whence there is only one orbit O over t = 0 and it is given by the equation y = 0. The stabilizer of any point of O is the group E_{m} ⊆ ℂ^{∗} of m-th roots of unity, and the type of the orbit is (m, q) = (m_{+}, e_{+}), as required.

Let us now treat the general case. Consider a generator v_{+} = fu^{m_{+}} of A_{m_{+}} as A_{0}- module so that D_{+} = div(f)/m_{+}. The localization A_{v_{+}} = A[f^{−1}u^{−m_{+}}] is the subring A_{0}[D_{+}, −D′_{−}] of Frac(A_{0})[u, u^{−1}] with D′_{−} := max(D_{−}, −D_{+}). (cf. Lemma 4.8). As D_{+} + D_{−} ≤ 0 we have D′_{−} = −D_{+}, so by the first case the open subset Spec A_{v_{+}} of V contains an orbit O^{+} of type (m_{+}, q_{+}). Similarly, if v_{−} = gu^{m_{−}} ∈ A_{m_{−}} is a generator, then Spec A_{v_{−}} contains an orbit O^{−} of type (−m_{−}, q_{−}). As the whole fiber is given by v_{+} · v_{−} = 0, the functions v_{+}, v_{−} must vanish on Ō^{−} and Ō^{+}, respectively. The intersection Ō^{+} ∩ Ō^{−} is given by v_{+} = v_{−} = 0 and so is the unique fixed point of the fiber. □

Example 4.20. In the example of the algebra A = A_{d,P} treated in Corollary 4.17 we have D_{+} = 0 and D_{−} = −div(P)/d = ∑_{i} −\frac{r_{i}}{d} [a_{i}] (see Example 4.11). The exceptional orbits are located over the points a_{i} ∈ 𝔸_{ℂ}^{1}, and π^{−1}(a_{i}) = O_{i}^{+} ∪ {z_{i}} ∪ O_{i}^{−}, where z_{i} is the unique fixed point of the fiber (located over the point (0, 0, a_{i}) of Spec B ⊆ ℂ^{3}). Applying Theorem 4.19, the orbit O_{i}^{+} is principal, and if we write r_{i}/d = e_{i}/m_{i} with gcd(e_{i}, m_{i}) = 1 then O_{i}^{−} is of type (m_{i}, q_{i}), where

q_{i}e_{i} ≡ −1 mod d with 0 ≤ q_{i} < m_{i} .

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Fakultät für Mathematik, Ruhr Universität Bochum, Geb. NA 2/72, Universitätsstraße 150, 44780 Bochum, Germany

E-mail address: Hubert.Flenner@ruhr-uni-bochum.de

Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères cédex, France

E-mail address: zaidenbe@ujf-grenoble.fr