Homogeneous locally nilpotent derivations and affine ML-surfaces

Abstract

Let B = k[X0, X1, X2] be the polynomial ring in three variables over an algebraically closed field k of characteristic zero. We consider the homogeneous case of the problem of describing locally nilpotent derivations of B. Given integers a0, a1, a 2 satisfying gcd{a0, a 1, a2} = 1, we define a Z -grading g on B by declaring that Xi is homogeneous of degree ai (for i = 0, 1, 2). In this thesis, we give an explicit description of the g -homogeneous locally nilpotent derivations of B when the integers a0, a1, a2 are not pairwise relatively prime. In the case where a0, a1, a 2 are pairwise relatively prime, we characterize the kernels of g -homogeneous locally nilpotent derivations of B among all subalgebras of B. Now assume that k is an arbitrary field of characteristic zero. In the remainder of this thesis, we study properties of affine k-surfaces which have trivial Makar-Limanov invariant. In particular, we prove that such surfaces have only finitely many singular points. As an application, we also prove that a complete intersection surface with trivial Makar-Limanov invariant is normal; in particular, any hypersurface of the affine space A3k with trivial Makar-Limanov invariant is normal.