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.



Are cones over Grassmannianns of lines local complete intersections?
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 $p$. 


