Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:

Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety of dimension at least three. Assume that $Y$ is a strict complete intersection in $X$ then the natural map $$ Pic(X)\rightarrow Pic(Y) $$ is an isomorphism.

Now, take $X = \mathbb{P}^1_{(x_0,x_1)}\times\mathbb{P}^n_{(y_0,\dots,y_n)}$ with $n\geq 3$, and $Y = \{x_0 = 0\}\subset X$. Then $Y\cong\mathbb{P}^n$ is a complete intersection in $X$ but $X$ has Picard rank $2$ while $Y$ has Picard rank $1$ so that $Pic(X)\rightarrow Pic(Y)$ can not be an isomorphism.

What am I misunderstanding in Hartshorne's statement?


1 Answer 1


I suspect that you are supposed to view the projective variety $X$ as being given with a chosen projective embedding $X\subset \mathbb P^n$, and therefore a distinguished ample divisor $\mathcal{O}_X(1) = \mathcal{O}_{\mathbb P^n}(1)|_X$. Then the complete intersection $Y$ should be cut out by sections of $\mathcal{O}_X(d)$, rather than any old line bundle on $X$.

(Indeed the problem with your example is that your choice of $Y$ is not an ample divisor on $X$.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.