Are cones over Grassmannianns of lines local complete intersections? Let $X_d^N\subset\mathbb{P}^N$ be a cone over the Grassmannian of lines 
$\mathbb{G}(1,d)\subset\mathbb{P}^{d(d+1)/2-1}\subset\mathbb{P}^N$ 
with vertex a linear space $L\subset\mathbb{P}^N$ of dimension $N-d(d+1)/2-2\geq 0$.
Question: Is $X_d^N$ a local complete intersection?
Of course, this is true when $d\leq 3$, but is it true, for example, when $d=4$?
Thanks in advance for any suggestions.
 A: No, that is not true.  Denote $\mathbb{G}=\mathbb{G}(1,d)$.  Denote $D=\frac{d(d+1)}{2} - 1$, so that the Plücker embedding of the Grassmannian is $$\mathbb{G} \hookrightarrow \mathbb{P}^D.$$  First consider the case that $N$ equals $D+1$, i.e., $X=X_d^{D+1}$ is the "usual" projective cone over $\mathbb{G}=\mathbb{G}(1,d)$ with vertex one point $p$.  Let $H$ be a hyperplane in $\mathbb{P}^N$ that does not contain $p$.  Up to $\text{Aut}(\mathbb{G_m},1) \cong \mathbb{Z}/2\mathbb{Z}$, there is a unique $\mathbb{G}_m$-action on $\mathbb{P}^N$ that has fixed locus $F=\{p\}\sqcup H$ and such that for every point $q$ in $V=\mathbb{P}^N\setminus F$, the induced map $$\mathbb{G}_m \to V, \ \ \lambda \mapsto \lambda\cdot q,$$ is a closed immersion.  
The open set $U = \mathbb{P}^N\setminus H$ is a $\mathbb{G}_m$-invariant open affine subset.  Thus the $k$-algebra $S=\mathcal{O}_U(U)$ has a natural $\mathbb{Z}$-grading.  As a $\mathbb{Z}$-graded $k$-algebra, and up to $\text{Aut}(\mathbb{G}_m,1)$, it is just the polynomial ring in $N$ variables with its usual grading.  Moreover, the maximal ideal $S_+\subset S$ generated by homogeneous elements of positive degree is simply the maximal ideal of the closed point $p$ of $U$.
The closed subscheme $X\subset \mathbb{P}^N$ is $\mathbb{G}_m$-invariant (as a closed subscheme, not pointwise).  Thus the ideal $$I=\mathcal{I}_{X/\mathbb{P}^N}(U),$$ is a homogeneous ideal of $S$.  There is a unique $\mathbb{Z}$-grading on $S/I$ such that the quotient homomorphism, $$S\to S/I,$$ is homogeneous.  In particular, $(S/I)_+ = S_+/I_+$ is the maximal ideal of $p\in X\cap U$.  The localization of $S/I$ with respect to this maximal ideal is $\mathcal{O}_{X,p}$, the stalk of $\mathcal{O}_X$ at $p$.
Consider the $S/I$-module $I/I^2$.  This is a finitely generated $S/I$-module.  The associated module $I/I^2\otimes_{S/I} \mathcal{O}_{X,p}$ is the stalk at $p$ of $\mathcal{I}_{X/\mathbb{P}^N}/\mathcal{I}^2_{X/\mathbb{P}^N}$.  If $X$ is a complete intersection of codimension $c$ in $\mathbb{P}^n$, then this stalk is generated by $c$ elements as an $\mathcal{O}_{X,p}$-module.  Since this module is generated by the image of $I/I^2$, and since the graded $S/I$-module $I/I^2$ is generated by homogeneous elements, there exist homogeneous elements $f_1,\dots,f_c$ of degrees $d_1,\dots,d_c$ that generated the stalk.  These graded elements give rise to a homomorphism of graded $S/I$-modules,
$$
\phi: (S/I)[-d_1]\oplus \dots \oplus (S/I)[-d_c] \to I/I^2, \ \ \phi(\mathbf{e}_i) = g_i.
$$ 
The cokernel of $\phi$ is a finitely generated graded $(S/I)$-module.  The support of $\text{Coker}(\phi)$ is a closed subscheme of $X\cap U$ that is $\mathbb{G}_m$-invariant, since $\text{Coker}(\phi)$ is graded.  By construction, this closed subset does not contain $p$.  The only $\mathbb{G}_m$-invariant closed subset of $U$ that does not contain $p$ is the empty set.  Thus, $\text{Coker}(\phi)$ is the zero module, i.e., $\phi$ is surjective.  
This is absurd.  The ring $S/I$ is the usual homogeneous coordinate ring of the Grassmannian with respect to the Plücker embedding.  The module $I/I^2$ is generated in degree $2$ by the usual Plücker quadratic relations.  The number of these relations is $>c$, except in the special cases that you already identified.  Therefore $X$ is not a local complete intersection near $p$.  In the general case when the vertex set has dimension $e$, intersect $X_d^N$ with a projective linear subspace $\mathbb{P}^{D+1}\subset \mathbb{P}^N$ whose intersection with the vertex set is a singleton $\{p\}$.  If $X_d^N$ were a local complete intersection near $p$, then also $X=\mathbb{P}^{D+1}\cap X_d^N$ would also be a local complete intersection near $p$.   
