# Criterion for a rational variety being the projective space

Let $$M$$ be a smooth projective complex variety. Assume there is an open subscheme $$U\subset M$$ and an open immersion $$U\hookrightarrow \mathbf{P}^n$$ such that the codimension of the complements of $$U$$ in both $$M$$ and $$\mathbf{P}^n$$ is bigger than one.

Question: Is it always true that $$M\cong \mathbf{P}^n$$?

• This is true for any smooth projective variety of Picard rank one. The Picard groups are isomorphic by S2 extension. By hypothesis, every invertible sheaf on each is either ample, trivial, or anti-ample. Computing the degree on a curve contained in the open, we see that these are the same for the two varieties. Thus, there is an invertible sheaf that is very ample for both varieties. Again using S2 extension, the homogeneous coordinate rings for this invertible sheaf are equal. As each variety is isomorphic to Proj, they are isomorphic to each other. Commented Aug 9 at 3:09

Your condition means that there is a birational map $$f: M\dashrightarrow \mathbb{P}^n$$ which is isomorphic in codimension $$1$$. This gives an isomorphism $$f_*: \text{Cl}(M) \to \text{Cl}(\mathbb{P}^n)$$ between class groups of Weil divisors.
In particular, the Picard rank of $$M$$ is $$1$$. This implies that $$f$$ is actually an isomorphism. Indeed, for an ample divisor $$D$$ on $$M$$, $$f_*D$$ is ample on $$\mathbb{P}^n$$. Moreover, since $$f$$ is isomorphic in codimension $$1$$, we have identifications between global sections: $$H^0(M, mD)\simeq H^0(\mathbb{P}^n, mf_*D)$$. So there is an isomorphism between the graded algebras $$\bigoplus_{m\geq 0}H^0(M, mD)\simeq \bigoplus_{m\geq 0}H^0(\mathbb{P}^n, mf_*D).$$ Taking Proj of those graded algebras, we get $$M\simeq \mathbb{P}^n$$ which is induced by $$f$$.
This argument still works if you replace $$\mathbb{P}^n$$ by a $$\mathbb{Q}$$-factorial normal variety of Picard number $$1$$, and replace $$M$$ by a $$\mathbb{Q}$$-factorial normal variety.