It is explained that the following two problems are equivalent: (i) describing all ane rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k(X;Y;Z). Then the solution of (i) is sketched. (Outline of our joint work with Peter Russell.) Introduction. An \ane ruling" of an algebraic surface X is a mor- phism p : U ! where is a curve, U is a nonempty open subset of X isomorphic to A 1 and p is the projection A 1 ! (note that we… Expand

where Xs denotes the smooth locus of X, consider: Problem 1. Find all affine rulings of X. Problem 2. Find all pairs of curves C1, C2 on X such that X n (C1 [ C2) is isomorphic to P2 minus two lines.… Expand

Let k be a field. To any pair (x,y) ≠ (0,0) of elements in tk[[t]], and hence to any irreducible and residually rational power series f ∈ k[[X,Y]], we associate a matrix, the Hamburger-Noether… Expand