# Two ways to compute the codimension of a subvariety

In the book Algebraic Geometry: a first course by Joe Harris, there is a method to compute the codimension of the subvariety $\Sigma_{k}(\Lambda)=\{\Gamma: \dim(\Gamma \cap \Lambda) \geq k\} \subset \mathbb{G}(l,n)$ and the codimension is $(k+1)(k-(m+l-n))$ page 149 of the book. Here $\mathbb{G}(k,n)$ is the set of all $k$-plane in $\mathbb{P}^n$; $\Lambda \subset \mathbb{P}^n$ is a fixed $m$-plane.

There is another way to compute codimension using the number of equations imposed on $\Sigma_k(\Lambda)$. If there are $r$ equations imposed on $\Sigma_k(\Lambda)$, then the codimension is $r$. Since $\dim(\Gamma \cap \Lambda) \geq \dim \Gamma + \dim \Lambda - \dim \mathbb{P}^n = m+l-n$, if $m+l-n \geq k$, then $\dim(\Gamma \cap \Lambda)\geq k$ always holds and so there is no equation imposed on $\Sigma_k(\Lambda)$ and the codimension of $\Sigma_k(\Lambda)$ is $0$. Now $k > m+l-n$. I think that we need $k-(m+l-n)$ equations such that $\dim(\Gamma \cap \Lambda)\geq k$ holds and hence the codimension is $k-(m+l-n)$. But why is the codimension $(k+1)(k-(m+l-n))$? Thank you very much.

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