the projective space ${\mathbb{P}}^n$. In those articles, codimension $q$ foliations were defined locally by $1$-forms $\omega_1,\dots,\omega_q$ satisfying Frobenius integrability equations: $d\omega_i\wedge\omega_1\wedge\dots\wedge\omega_q=0$ for $i=1,\dots,q$. Later, de Medeiros observed that this definition is not general enough for singular foliations of codimension $q$, as singular foliations by curves in dimension $n\geq 3$ cannot be given by $n-1$ forms even locally, see [@deMed1] and Example \[exampleCodimension2inA3\]. The correct definition is given by a $q$-form verifying the Plücker relations and Frobenius integrability (see below for definitions).

As for why many results were stated with ${\mathbb{P}}^n$ as ambient variety, notice that working in ${\mathbb{P}}^n$ allows the use of homogeneous coordinates and so one can define a codimension $1$ foliation with an integrable polynomial $1$-form $\omega=\sum_i f_i(x) dx_i$, that is a $1$-form verifying $\omega\wedge d\omega=0$ and $\sum_i x_i f_i(x)=0$. Such a setting can give concrete examples of foliations which may be hard to produce and study in more general contexts, see for instance the book [@omegarlibro].

Going beyond isolated examples there is the problem of establishing irreducible components for the space $\mathcal{F}ol(X)$ parameterizing all integrable forms on a variety $X$. The existence of this space for a projective variety $X$ follows from the existence of the $\mathrm{Quot}_X(\Omega^1_X)$ 