Is there a way to use MAGMA to study surfaces defined over a weighted projective space (by "study" I mean computing e.g. invariants (e.g. $p_a$, $p_g$), singularities, etc)? For example, I was trying, as a trial test, to get MAGMA working on Chatelet surfaces with affine equation $y^2 - az^2 = P(x)$, but with no success. Any idea?
I take it you want birational invariants of projective desingularisations of a surface. Magma can't do very much at the moment for surfaces in weighted projective space but if you can find an embedding into ordinary projective space with singularities that aren't too bad (only simple singularities), it can practically handle surfaces in P^n for n reasonable large. It's also worth trying the functions that work with singular surfaces in P^3.
Your example of Chatelet surfaces doesn't really help give an idea of how complicated the types of surfaces you want to work with may be.
For a start, they are all rational so you know the birational invariants a priori. Also, if P is degree 3 or 4 (standard for Chatelet surfaces, I think), then the ordinary projective closure in P^3 (simply homogenising with one new variable) works quite well. If P has degree 3 and is separable, then the projective closure is just a cubic surface with 2 "A2" singularities. Magma can handle this easily via either the P^3 hypersurface functions or the general surface with simple singularities functions. It's just a weak/degenerate degree 3 Del Pezzo surface.
For P of degree 4, the projective closure in P^3 has a one-dimensional singular locus, but I think that the P^3 hypersurface code should handle most cases reasonable well still. If the coefficients of P aren't too huge, anyway.
Mike Harrison - Magma group