For a form $f \in k[x_1, \cdots, x_n]$, where $k$ is a field of characteristic zero (or more specifically, a number field, and usually $\mathbb{Q}$). The Schmidt rank of $f$ (with respect to $k$), which we denote by $h_k(f)$, is defined as the smallest positive integer $l$ for which $f$ can be written in the form $$\displaystyle f = u_1 v_1 + \cdots + u_l v_l,$$ where $u_i, v_i$ are forms of degree at least one for $i = 1, \cdots, l$. A Schmidt rank of one is equivalent to $f$ being reducible over $k$. For more details, see W. Schmidt's paper "The density of integer points on homogeneous varieties", Acta Mathematica 1985, Volume 154, Issue 3-4, pp 243-296.

I am not sure what to think of the so-called Schmidt rank geometrically. Specifically, "low" Schmidt rank should be atypical. Indeed, the rank one case, as stated above, is equivalent to reducible; and it is well-known (and easy to show) that the typical form $f$ is irreducible over $k$ when $n \geq 3$ (the case $n = 2$ corresponds to binary forms, which are always reducible over an algebraically closed field). One way to think about this is as follows: Let $H_{k,n}(m)$ denote the Hilbert scheme of degree $m$ hypersurfaces in $n$ variables over the algebraic closure of $k$. Then there is a natural morphism from $H_{k,n}(m) \times H_{k,n}(l)$ to $H_{k,n}(m+l)$, induced by multiplication of forms of degrees $m$ and $l$ respectively. The image of this morphism in $H_{k,n}(m+l)$ is a closed set, and by taking unions, we see that the set of irreducible forms in $H_{k,n}(m+l)$ is a Zariski open set in $H_{k,n}(m+l)$, which is essentially the whole space as long as it is non-empty.

However, I do not know how to think about higher Schmidt ranks in this way.

More specifically, the Schmidt rank should go down by one typically when one restrict $f$ to a hyperplane. In particular, we should expect the typical $f$ to satisfy the following: $f(x_1, \cdots, x_{j-1}, 0, x_{j+1}, \cdots, x_n)$ has rank $h_k(f) - 1$ for $j = 1, \cdots, n$. Moreover, for any form $f$ and any $1 \leq m \leq n-1$, there exists an $m$-plane $P$ defined over $k$ for which $f$ restricted to $P$ has Schmidt rank equal to $h_k(f) + m - n$, as this is the "typical" behaviour.

Any insight would be appreciated.

For a form $f \in k[x_1, \cdots, x_n]$, where $k$ is a field of characteristic zero (or more specifically, a number field, and usually $\mathbb{Q}$). The Schmidt rank of $f$ (with respect to $k$), which we denote by $h_k(f)$, is defined as the smallest positive integer $l$ for which $f$ can be written in the form $$\displaystyle f = u_1 v_1 + \cdots + u_l v_l,$$ where $u_i, v_i$ are forms of degree at least one for $i = 1, \cdots, l$. A Schmidt rank of one is equivalent to $f$ being reducible over $k$. For more details, see W. Schmidt's paper "The density of integer points on homogeneous varieties", Acta Mathematica 1985, Volume 154, Issue 3-4, pp 243-296.

I am not sure what to think of the so-called Schmidt rank geometrically. Specifically, "low" Schmidt rank should be atypical. Indeed, the rank one case, as stated above, is equivalent to reducible; and it is well-known (and easy to show) that the typical form $f$ is irreducible over $k$ when $n \geq 3$ (the case $n = 2$ corresponds to binary forms, which are always reducible over an algebraically closed field). One way to think about this is as follows: Let $H_{k,n}(m)$ denote the Hilbert scheme of degree $m$ hypersurfaces in $n$ variables over the algebraic closure of $k$. Then there is a natural morphism from $H_{k,n}(m) \times H_{k,n}(l)$ to $H_{k,n}(m+l)$, induced by multiplication of forms of degrees $m$ and $l$ respectively. The image of this morphism in $H_{k,n}(m+l)$ is a closed set, and by taking unions, we see that the set of irreducible forms in $H_{k,n}(m+l)$ is a Zariski open set in $H_{k,n}(m+l)$, which is essentially the whole space as long as it is non-empty.

However, I do not know how to think about higher Schmidt ranks in this way.

One should expect that the Schmidt rank would go down by one typically when one restrict $f$ to a hyperplane. In particular, we should expect the typical $f$ to satisfy the following: $f(x_1, \cdots, x_{j-1}, 0, x_{j+1}, \cdots, x_n)$ has rank $h_k(f) - 1$ for $j = 1, \cdots, n$. Moreover, for any form $f$ and any $1 \leq m \leq n-1$, there should exist an $m$-plane $P$ defined over $k$ for which $f$ restricted to $P$ has Schmidt rank equal to $h_k(f) + m - n$, as this is the "typical" behaviour.

Any insight would be appreciated.