# A question on "Ample subvarieties of algebraic varieties"

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?

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 $$X$$.)