I have to compute the dimension of : $$ X=V(xz-y^2,xz-xw,yz-yw,z^2-zw) \subset \mathbb{P}^3(\mathbb{C}) $$ My idea was to compute irreducible components of this algebraic sets and intersect with some easy hypersurface like $V(x)$ and compute the dimension of an irreducible component which shall be dimX-1.
It could be a good idea or there is a smarter method? Thanks for suggestions.