Let X be the union of two planes in $\mathbb{A}^4$ touching at origin. Blow up X at the origin. Call it $\overline{X}$. It has two disjoint copies of $\mathbb{A}^2$ blown up at the origins. Their exceptional divisors are $E_1$ and $E_2$ say. It is clear that $E_1$ and $E_2$ are isomorphic to $\mathbb{P}^1$. Choose an isomorphism between them and glue $\overline{X}$ via this isomorphism. My question is : Is this Glued scheme quasi-projective?

I think the ample cartier divisor $\mathcal{O}(-E_1-E_2)$ descends to an ample cartier divisor which makes the Glued scheme quasi projective.

Let $X$ is an irreducible surface got by identifying two smooth points of a smooth irreducible surface $Y$. The completion of the local ring at the only singular point of $X$ is the above ring i.e, two planes in $\mathbb{A}^4$ touching at a point. Then we do the same construction. That is Blow up X at the singular point, identify two exceptional divisors and get a new scheme Z. Is it quasi projective??