UDC 517.5 WHEN A CHAIN OF BLOWUPS

DEFINES AN AUTOMORPHISM OF C^{2}.

S.Yu. Orevkov

1. Introduction. In this note we give a new proof of a theorem of A.G. Vitushkin [1]. The proof is based on a formula for the canonical class of an algebraic compactification of C^{2} (formula (5) below). This formula was used in some author’s papers (see, e.g., formula (3) in [2]), however, its detailed proof is published here the first time. Similar formulae written in other terms appeared in different authors’ papers.

Let L be a line on CP^{2}. We shall consider it as the infinite line of the affine plane C^{2}, i.e. L = CP^{2} \ C^{2}. Let σ_{1} : V → CP^{2} be a birational morphism whose restriction onto σ_{1}^{−1}(C^{2}) is an isomorphism, i.e. σ_{1} is a composition (a chain) of blowups “at infinity”. Let E_{1}, …, E_{n} be the irreducible components of the curve E = σ_{1}^{−1}(L) where E_{1} is the proper transform of L. We say that the chain of blowups σ_{1} defines an automorphism of C^{2} if the last glued curve (denote it by E_{2}) admets a birational morphism σ_{2} : V → CP^{2} such that σ_{2}|_{σ_{2}^{−1}(C^{2})} is an isomorphism, σ_{2}^{−1}(L) = E, and E_{2} is the proper transform of L (in this case, σ_{2}^{−1}σ_{1} is an automorphism of C^{2}).

Following [1], let us define a test surface for the chain of blowups σ_{1} as a homology class S ∈ H_{2}(V; Z) such that

S · E_{2} = 1,  S · E_{i} = 0 for i ≠ 2  (1)

(as above, E_{2} is the last glued curve). Since E_{1}, …, E_{n} is a base in H_{2}(V; Z), the conditions (1) uniquely define the class S. If a chain of blowups σ_{1} defines an automorphism of C^{2} then S = σ_{2}^{−1}(l) where l is a generic line on CP^{2}. Hence, by the adjunction formula, we have

S^{2} = 1,  S · K_{V} = −3,  (2)

where K_{V} = −c_{1}(V) is the canonical class of V. A.G. Vitushkin proved that (2) is not only necessary but also a sufficient condition for a chain of blowups σ_{1} to define an automorphism C^{2}:

Theorem. (see [1]) A chain of blowups σ_{1} defines an automorphism of C^{2} if and only if the test surface S satisfies (2).

2. Discriminant of the intersection form. Let D = D_{1} + ⋯ + D_{n} be a curve on a smooth projective surface V, (D_{1}, …, D_{n} are its irreducible components). Let A_{D} = (D_{i} · D_{j})_{ij} be the intersection matrix. Let us define the discriminant d(D) of D as det(−A_{D}). The sign minus provides the equality d(σ^{−1}(D)) = d(D) for a blowup σ : Ṽ → V. If D_{i} · D_{j} ≤ 1 for i ≠ j then the graph of D is the graph Γ_{D} whose vertices are D_{1}, …, D_{n} and whose edges correspond to pairs (D_{i}, D_{j}) with D_{i} · D_{j} = 1.

Typeset by AMS-TEX

Lemma 1. (Mumford [3]) Suppose Γ_{D} has the form of a linear chain −∘− ⋯ −∘−. Then

a). If D_{i}^{2} ≤ −2 for all i then d(D) ≥ 2.

b). If A_{D} is negatively definite, D_{i}^{2} ≤ −1 for all i, and d(D) = 1 then D can be blown down to a smooth point.

3. Proof of Vitushkin’s theorem. Let σ_{1} : V → CP^{2} be a birational morphism and E = E_{1} + ⋯ + E_{n} = σ_{1}^{−1}(L). Then E_{1}, …, E_{n} is a base of the vector space H_{2}(V, Q). Let A = (E_{i} · E_{j}) be the intersection matrix and let B = A^{−1}. The graph of E is a tree. Since d(E) = −1, Cramer’s rule easily implies

Lemma 2. b_{ij} = d(E − [ij]) where [ij] is the minimal connected set of the form E_{i_{1}} ∪ ⋯ ∪ E_{i_{k}} which contains E_{i} ∪ E_{j}. □

Lemma 3. For any C_{1}, C_{2} ∈ H_{2}(V, Q), one has C_{1} ·C_{2} = ∑_{i,j} b_{ij}(C_{1} ·E_{i})(C_{2} ·E_{j}).

Proof. For k = 1, 2, let us set X_{k} = (x_{k}^{1}, …, x_{k}^{n}), Y_{k} = (y_{k}^{1}, …, y_{k}^{n}), where C_{k} = ∑ x_{k}^{i}E_{i} and y_{k}^{i} = C_{k} · E_{i}. Then Y_{k} = AX_{k}, hence X_{k} = BY_{k} and C_{1} · C_{2} = ⟨X_{1}, AX_{2}⟩ = ⟨BY_{1}, Y_{2}⟩. □

Let, as in Sect. 1, E_{2} be the last glued curve and S be the test surface. It follows from Lemmas 2,3 and (1) that

S^{2} = b_{22} = d(E − E_{2}).  (3)

Let us denote ν_{i} = E_{i}·(E−E_{i}). The adjunction formula for E_{i} yeilds (K_{V} +E)·E_{i} = ν_{i} − 2, hence by (1) and Lemma 3, we have

(K_{V} + E) · S = ∑_{i=1}^{n} b_{i2}(ν_{i} − 2).  (4)

∘^{E_{1}}− − −^{p_{1}} ∘_{|}_{|}_{|}−_{q_{1}} − −^{p_{2}} ∘_{|}_{|}_{|}−_{q_{2}} … −^{p_{m}} ∘_{|}_{|}_{|}−_{q_{m}}− −∘^{E_{2}}  ∘^{E_{1}}− − −^{p_{1}} ∘_{|}_{|}_{|}−_{q_{1}} − −^{p_{2}} ∘_{|}_{|}_{|}−_{q_{2}} … −^{p_{m}} ∘_{|}_{|}_{|}^{E_{2}}_{q_{m}} Fig. 1  Fig. 2

Suppose that (2) holds. Without loss of generality, we may assume that the curve E is minimal in the following sense. There does not exist j ≥ 3 such that E_{j}^{2} = −1 and ν_{j} ≤ 2. This means that we blow up each time a point of the exceptional curve of the previouse blowup. Therefore, the induction with respect to the number of blowups easily shows that Γ_{E} is either as in Fig. 1 or as in Fig. 2 where the dashed lines denote linear chains of vertices. For curves E_{j} such that ν_{j} = 3, let us denote the discriminants of the connected components of E − E_{j} in accordance with Figures 1 and 2. Lemma 1 and the minimality of E imply that q_{j} ≥ 2 for all j. Hence, (2) and (3) imply that Fig. 2 is impossible. Applying (4) and Lemma 2 to Fig. 1 and using the fact that ES = 1, we obtain

K_{V} · S + 1 = ∑_{ν_{i}=3} b_{i2} − ∑_{ν_{i}=1} b_{i2} = ∑_{j=1}^{m} p_{j}q_{j}r_{j} − r_{0} − ∑_{j=1}^{m} p_{j}r_{j} − b_{22} = −b_{22} − 1 + s (5)

where r_{j} = q_{j+1} … q_{m} and s = ∑(p_{j} − 1)(q_{j} − 1)r_{j}. Substituting (2) and (3) into (5), we obtain s = 0.

WHEN A CHAIN OF BLOWUPS DEFINES AN AUTOMORPHISM OF C^{2}. 3

Lemma 4. Let q be the restriction of a nondegenerate quadratic form of the signature (−, +, …, +) onto a subspace. If the discriminant of q is positive then q is positively definite. □

By (2) and (3), we have d(E − E_{2}) = S^{2} = 1 > 0. Hence, by Lemma 4, the minus intersection form is positively definite on E − E_{2}. In particular, all q_{j} and p_{j} are positive, hence, all the summands in the sum s (see (5)) are non-negative. Using the fact that all q_{j} > 1, the equality s = 0 implies p_{1} = ⋯ = p_{m} = 1. Hence, by Lemma 1b, the leftmost linear branch of Γ_{E} can be blown down. As the result, we obtain another graph of the same form which has m − 1 triple vertices. The discriminant of the leftmost linear branch of the new graph is p_{2}. Contunuing this process, we blow down all the components of E except E_{2}. The Theorem is proved.

4. Remark. It is proved in [1] that the conditions (2) are necessary and sufficient for a chain of blowups to be a composition of so-called triangular chains (see the definition in [1]). Explicitely writing down the blowups in coordinates, it is not difficult to see that triangular chains are those and only those which define triangular (i.e. of the form (x, y) ↦ (x + f(y), y) ) transformations of C^{2} up to linear changes of coordinates. Hence, the arguments from [1] prove more than the above Theorem. They provide also a proof of Jung’s theorem which claims that any automorphism of C^{2} is decomposable into a product of affine and triangular transformations. The author is grateful to A.G. Vitushkin for useful discussions

References

1. A.G. Vitushkin, Criterion for a chain of σ-processes to be a composition of triangular chains, Math. Notes (to appear).

2. A.V. Domrina, S.Yu. Orevkov, On four-sheeted polynomial mappings of C^{2}. I. The case of an irreducible ramification curve, Math. Notes 64 (1998), ??–??.

3. D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. I.H.E.S. 9 (1961), 229–246.

Steklov Math.Inst. Russian Acad.Sci.