# Examples of projective manifold birational to intersection of two quadrics

Let $$X$$ be a smooth intersection of two quadrics in $$\mathbb{P}^{2n+1}$$.

Question: Does there exist a smooth projective variety $$Y$$ (different from $$X$$) of dimension $$2n-1$$ such that $$Y$$ is birational to $$X$$ and if $$U$$ is the maximal open subset of $$Y$$ which is isomophic to an open subset of $$X$$, then $$Y \setminus U$$ has co-dimension at least $$2$$ ?

When n>1, then $$Y$$ and $$X$$ have dimension at least 3. By the Lefschetz hyperplane theorem for $$X$$, we have $$\mathrm{Pic}(X) \cong \mathbb{Z}$$. As the codimension of $$Y\setminus U$$ is 2 we have $$\mathrm{Pic}(Y) \cong \mathrm{Pic}(U)$$ and the latter is a quotient of $$\mathrm{Pic}(X)$$ (by excision for the inclusion $$U\subset X$$). As $$Y$$ is projective, $$\mathrm{Pic}(Y) \cong \mathbb{Z}$$. As $$\mathrm{Pic}(U) \cong \mathbb{Z}$$, then again by excision the codimension of $$X\setminus U$$ in $$X$$ is at least 2.
The pullback of the ample generator ($$\mathcal{O}_X(1)$$) of $$\mathrm{Pic}(X)$$ to $$U$$ extends to the ample generator $$L$$ of $$\mathrm{Pic}(Y)$$. By Hartog's extension type results: $$H^0(Y,L^{\otimes m})\cong H^0(U,L^{\otimes m}|_U) \cong H^0(X,\mathcal{O}_X(m))$$. As some power is very ample, it follows that $$Y \cong X$$.
So for any $$n$$, the answer is no.