Let $X$ be a quasiprojective variety. Does there exist a projective variety $Y$ and an open dense immersion $X \to Y$ such that $Y \setminus X$ is smooth? What if we just require that $Y \setminus X$ be complete intersection? (note that a smooth variety is locally complete intersection, but being complete intersection is a stronger condition in the affine sense, and a much stronger condition in the projective sense.)
Of course, if you take "the" projective closure of $X$ relative to some embedding in $\mathbb P^n$, it need not have this property. I'm asking if this can be arranged by a suitable choice of embedding.
For example, it might be the case that you just need to take the embedding associated to a sufficiently high twist $\mathcal O(n)$.