My problem
Let $Z$ be a projective variety. I have to prove that $\Lambda = \{ (\text{Pl}(L), p) \in \text{Pl}(\text{Gr}(k,n))) \times Z \ | \ p \in L\}$ is a projective variety. Here $\text{Gr}(k,n)$ is the Grassmannian and $\text{Pl}: \text{Gr}(k,n) \rightarrow \mathbb{P}^{\binom{n}{k} -1}$ the plucker embedding.
My (flawed) attempt:
I know that $\text{PL}(\text{Gr}(k,n))$ is a variety. So it is described by finitely many polynomials $\{ f_i \}$. Since $Z$ is also a projective variety, it is described by $\{ g_j \}$. We can think of the Grassmanian set of all $(k-1)$ -dimensional linear subvarieties of the projective space $\mathbb{P}^{n-1}$. Thus the Plucker coordinates correspond to the minors of a $k \times n$ matrix. Take some vector space $X \in \text{Gr}(k,n)$, spanned by vectors $x_1, \ldots, x_k$. Now $y \in X$ iff it can be written as linear combination of $x_1, \ldots, x_k$. Then all $k+1$- minors of the matrix with rows $x_1, \ldots, x_k,y$ are zero.
I got some inspiration from this other question, however it's hard to see what applies to my problem and what doens't. Incidence correspondence of Grassmannian is a projective variety
My question
Any hints on how to continue are welcome. If I made some mistakes, please let me know. Thanks!