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??



Gluing Schemes along subschemes
Let $Y$ be the total space of the vector bundle $O(-1)\oplus O(-1) \to \mathbb{P}^1$. Then your glued up scheme is isomorphic to the subscheme of $Y$ given by the union of the two closed subschemes given by the total spaces of the two line bundle $O(-1)\to \mathbb{P}^1$. So yes, your scheme is quasi-projective. 


