Now to specifics:


Let $V \subset \mathbb{A}^3$ be a reducible affine algebraic set defined by two irreducible polynomials $f,g \in K[X,Y, Z]$ of degree $d$ ($K$ algebraically closed). So, if $V$ is regular, is then its projective closure in $\mathbb{P}^3$ also regular?


I tend to think that this is not necessarily the case, because the ideal $I$, which generates projective closure of $V$ is not necessarily generated by $f^\*$ and $g^\*$ (homogenized generators of $V$), i.e. in general 


$\langle f^\*, g^\* \rangle \neq \langle f,g\rangle^\*$ 


So basically I can’t say anything about the rank of Jacobian matrix in projective case with my prior knowledge of rank of Jacobian in affine case. 


Does that make sense?



Is projective closure of a regular affine algebraic set also regular?
This can easily fail. Start backwards: take a projective variety which is not regular, say $X\subseteq \mathbb P^n$ and let $S\subseteq X$ be its singular set. Choose a hypersurface $H$ that contains $S$. Then $\mathbb P^n\setminus H$ is an affine variety and $X\setminus H$ is regular.


In your specific situation, choose $V$ to be two parallel lines in $\mathbb A^3$. Then the projective closure of $V$ is two intersecting projective lines, which is not regular at the intersection point. Alternatively, choose a singular projective curve in $\mathbb P^3$ with a single singular point and move that point to infinity. The corresponding affine curve will be regular, but its projective closure is singular. 


