I have a general question, and then the specific version of that question I need for research. All vector spaces over $\mathbb{C}$.

## Grassmanians of planes

The $(2,n)$-Grassmannian, denoted $Gr(2,n)$, is the space of all 2 dimensional subspaces of $n$ dimensional space. This is naturally a complete complex variety. Its variety structure is usually realized through the **Plucker embedding**

$$Gr(2,n)\hookrightarrow \mathbb{P}^{\binom{n}{2}}$$

This starts with a space $V\subset \mathbb{C}^n$, chooses basis vectors $v_1,v_2$, and then maps $V$ to the point in $\mathbb{P}^{\binom{n}{2}}$ with homogenous coordinate given by all $2\times 2$-minors of the $2\times n$ matrix with rows $v_1,v_2$.

For $i,j\in [n]$, the **Plucker coordinates** $x_{ij}$ is the $\{i,j\}$th homogeneous coordinate on $\mathbb{P}^{\binom{n}{2}}$; therefore, the composition with the Plucker embedding corresponds to the $ij$th minor. The image of this map is defined by the homogeneous **Plucker relations**. For all, $i<j<k<l$, one has
$$ x_{ik}x_{jl}=x_{ij}x_{kl}+x_{il}x_{jk}$$
Let $\mathcal{O}Gr(2,n)$ denote the *homogeneous coordinate ring*, this is the graded algebra generated by the Plucker coordinates, modulo the Plucker relations.

The group $PGL_n(\mathbb{C})$ acts transitively on $Gr(2,n)$, by its action on $\mathbb{C}^n$. Therefore, $Gr(2,n)$ is a homogeneous space. From this, it follows immediately that $Gr(2,n)$ is a smooth, irreducible variety.

## Smoothness of hypersurfaces

Any homogeneous element $f$ in $\mathcal{O}Gr(2,n)$ defines a hypersurface $V_f$ in $Gr(2,n)$. My general question is:

How can one effectively check if $V_f$ is smooth?

As an example of the problem, the main general technique I know for showing a variety is smooth is to embed it in affine or projective space with codimension $c$, and then check the $c\times c$ minors of the Jacobian matrix of the defining ideal. The problem with this approach here is that the codimension of the Plucker embedding is quite large in general, and the minors rapidly become unwieldy.

## My specific concern

Outside curiosity, I am not interested in the problem for general hypersurfaces. The specific hypersurface I have in mind is defined by $f=x_{ij}-x_{kl}$, the difference of two Plucker coordinates. This is a frustratingly simple variety, but I am having trouble saying things about it. Is it smooth? Is it irreducible?

More generally, I can look at the subscheme defined by intersecting several hypersurfaces of this form. Is it smooth? Is it irreducible?