On automorphisms of Danielewski surfaces

Abstract

Abstract. We develop techniques for computing the AK invariant of a domain with arbitrary characteristic. We use these techniques to describe for any field k the automorphism group of k[X, Y, Z]/(X n Y − Z 2 − h(X)Z), where h(0) = 0 and n ≥ 2, as well as the isomorphism classes of these algebras.

On a class of Danielewski surfaces in affine 3-space

“…For surfaces defined by equations x n z − P (y) = 0 in A 3 , the picture has been completed by L. Makar-Limanov [15] who gave explicit generators of their automorphism groups. Similar results have been obtained over arbitrary base fields by A. Crachiola [3] for surfaces defined by equations…”

“…In contrast, a surface defined by an equation xz − P (y) = 0 admits at least two distinct A 1 -fibrations over the affine line, due to the symmetry between the variables x and z. In his proof, L. Makar-Limanov used the correspondence between algebraic C + -actions on an affine surface S and locally nilpotent derivations of the algebra of regular functions on S. It turns out that the argument is essentially independent of the base field k, up to replacing locally nilpotent derivations by suitable systems of Hasse-Schmidt derivations when the characteristic of k is positive (see e.g., [3]). …”

Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces

“…• A. Crachiola in [1] obtained similar results for slightly different surfaces defined by the equations X n Z − Y 2 − σ(X)Y = 0, where n ≥ 2 and σ(0) = 0, defined over arbitrary base field.…”

“…Finally, for item (5), let a, b ∈ B such that ab ∈ S = K[x] \ {0}. Then ab is an unity of S −1 B = K(x) [1] and, thus, a, b are unities of…”

On the cancellation problem

“…There exist smooth affine varieties without CP [3,4,[6][7][8], all the examples given so far are essentially based on the so called Danielewski construction. It is an interesting and in general still open problem whether the affine space A n (k) has CP.…”