Projections from flag varieties

Question

Let $K$ be a field and $k<n$. Denote by $G(k,n)$ the Grassmannian of $k$-dimensional vecors subspaces of $K^n$. I'm trying to check that the projection map

$$G(k+1,n)\times G(k,n)\longrightarrow G(k,n)$$ is open.

I need this in order to check that the projection map from the flag variety $\mathcal{F\ell}(K^n;k,k+1)\longrightarrow G(k,n)$ is open.

Answer 1

Flat morphisms locally of finite presentation are universally open (EGA IV2, Théorème 2.4.6), and thus your morphism is open.