The Stacks project

Exercise 111.49.9. Example of a nonprojective proper variety. Let $k$ be a field. Let $L \subset \mathbf{P}^3_ k$ be a line and let $C \subset \mathbf{P}^3_ k$ be a nonsingular conic. Assume that $C \cap L = \emptyset $. Choose an isomorphism $\varphi : L \to C$. Let $X$ be the $k$-variety obtained by glueing $C$ to $L$ via $\varphi $. In other words there is a surjective proper birational morphism

and an open $U \subset X$ such that $\pi : \pi ^{-1}(U) \to U$ is an isomorphism, $\pi ^{-1}(U) = \mathbf{P}^3_ k \setminus (L \cup C)$ and such that $\pi |_ L = \pi |_ C \circ \varphi $. (These conditions do not yet uniquely define $X$. In order to do this you need to specify the structure sheaf of $X$ along points of $Z = X \setminus U$.) Show $X$ exists, is a proper variety, but is not projective. (Hint: For existence use the result of Exercise 111.37.9. For non-projectivity use that $\mathop{\mathrm{Pic}}\nolimits (\mathbf{P}^3_ k) = \mathbf{Z}$ to show that $X$ cannot have an ample invertible sheaf.)