a b s t r a c tLet B be the polynomial ring in three variables over a field k of characteristic zero. A k-derivation D : B → B is said to be triangular if there exists a triple (X, Y , Z ) of elements of B satisfying B = k[X , Y , Z ], DX ∈ k, DY ∈ k[X ] and DZ ∈ k[X , Y ]. We give a new characterization of triangular derivations.Let B = k[X 1 , X 2 , X 3 ] be the polynomial ring in three variables over a field k of characteristic zero. Recall that a k-derivation D :Because of our lack of understanding of the group of automorphisms of B, it is a nontrivial problem to decide whether a given derivation is triangular. As triangular derivations are in particular locally nilpotent, one usually seeks criteria for deciding whether a given locally nilpotent derivation D : B → B is triangular. This problem was considered by several authors (cf. [1,[16][17][18]10,3]). In [3], the problem was reduced to the case where D is irreducible and a criterion was given for that case. Section 5 of the present paper gives a new criterion for that irreducible case. (Refer to 2.7 for the definition of irreducibility.)Let us say a few words about the method of proof. Consider any pair (D, s) where D : B → B is an irreducible locally nilpotent derivation and s is a preslice of D (i.e., s is an element of B satisfying D(s) = 0 and D 2 (s) = 0). It is known that ker(D) is a polynomial ring in two variables over k, so the inclusion ker(D) → B determines a morphism of algebraic varieties Q : A 3 → A 2 . For each λ ∈ k, let S λ ⊂ A 3 be the hypersurface given by the equation s = λ and let f λ : S λ → A 2 be the composition S λ → A 3 Q − → A 2 , so the pair (D, s) determines the family (f λ ) λ∈k of morphisms. In Section 3 we show that, for general λ ∈ k, f λ is a birational morphism whose missing curves and fundamental points satisfy certain constraints (cf. Section 1 for definitions). The hope, then, is to use the theory of birational morphisms of surfaces for understanding the relation between the geometric properties of the surfaces S λ and the algebraic properties of the derivation D. That analysis turns out to be feasible for the cases that we consider in Sections 4 and 5. Note that, although Q : A 3 → A 2 is a wellstudied morphism, it appears to be the first time that a systematic analysis of the geometric properties of preslices leads to answering an algebraic question about derivations.Section 4 considers an interesting subset of k[X , Y , Z ] whose elements we call the ''weakly basic'' polynomials and which includes in particular all variables of k[X , Y , Z ]. Theorem 4.2 describes what happens when a weakly basic polynomial is a preslice of a locally nilpotent derivation. In the special case where the weakly basic preslice is a variable, one obtains the results of Section 5 on triangular derivations.Conventions. All rings are commutative and have a unity. The set of units of a ring R is denoted R * . If r ∈ R, we denote by R r the localization S −1 R where S = {1, r, r 2 , . . . }. If R is an integral domain, Frac R is its field of ...
