A singular algebraic variety is not a smooth manifold, but is it the union of a finite number of smooth manifolds?

Question

I have the solution set of a collection of homogenous polynomials with coefficients in $\mathbb{R}$, which is a projective variety. In general projective varieties are not smooth, they may have singularities, self intersections such as the system where $x=(x_1,x_2,x_3)$ satisfies $x_1x_2=0$ in $\mathbb{R}P^2$. However is it always the case that we can view a projective variety as a finite union of smooth manifolds embedded in projective space? It seems intuitively true to me but I'm neither a geometer or algebraist so I don't have the tools or the references to figure this out for myself.