I am reading the book algebraic geometry: a first course by Joe Harris. I have a question about codimension of a variety.
Usually, if we have a subspace $W$ of dim $k$ in a space $V$ of dim $n$, then $W$ is of codimension $n-k$. But I cannot compute the codimension on line -4 above Exercise 11.43 on page 149 of the book. It is said that codim of $\Psi$ in $\mathbb{G}(k,n)$ is $(k+1)*(k-(m+l-n))$. I think that $dim(\mathbb{G}(k,n))=(k+1)*(n-k)$. Since it is shown that $dim(\Psi)=(k+1)(m-k)+(l-k)(n-l)$, codimension of $\Psi$ in $\mathbb{G}(k,n)$ should be $(k+1)*(n-k)-((k+1)(m-k)+(l-k)(n-l))$. But this is not $(k+1)*(k-(m+l-n))$.
Thank you very much.
Edit: here $\Psi=\{(\Gamma, \Theta): \Theta \subset \Gamma\} \subset \mathbb{G}(k,\Lambda) \times \mathbb{G}(l,n)$. $\Lambda \subset \mathbb{P}^n$ is a fixed $m$-plane. We need to compute the dimension of $\Sigma_{k}(\Lambda)=\{\Gamma: dim(\Gamma \cap \Lambda) \geq k\} \subset \mathbb{G}(l,n)$. Why introduce $\Psi$ to compute the dimension? Here $\mathbb{G}(k,n)$ is the set of all $k$-plane in $\mathbb{P}^n$.