Research Projects

The following consequence of David Mumford's result on 2-dimensional normal singular points using basic covering space theory is well-known. ``Let f : (C²;O) → (V; p) be a finite surjective complex analytic map, where (V; p) is a normal complex analytic germ. Then there is a factorization of f as a composite of finite surjective complex analytic maps (C²;O) → (C²;O) → (V; p); where the second map is unramified outside p.' I would like to take up the following algebraic analogue of the above problem. ` Let k be a field of characteristic 0 and f(X_1,...,X_n) a sum of Powers of X_i's of index ≥2. Let R:=k[[X_1,...,X_n]]/(f(X_1,...,X_n)). Let S be a power series ring over k in (n-1)-indeterminates. Then does there exist an k-algebra embedding from R to S such that S is integral over R?' A problem of independent interest is as follows. Let C be the field of complex numbers and D a C-linear derivation of of B, the polynomial ring in two variables over C. Then Cerveau asserted that D must be of the form D=d/dx+c.d/dy for a pair of variables x,y in B and some constant c in C. However, there is an error in the proof of Cerveau. I would like to rectify it and generalize the result to an arbitrary field of characteristic 0.