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I'm having trouble calculating singularities of varieties: when my professor covered it it was the last lecture of the course and a shade rushed.

I'm not sure if the definition I've been given is standard, so I will quote it to be safe: the tangent space of a projective variety $X$ at a point $a$ is $T_aX = a + \mbox{ker}(\mbox{Jac}(X))$.

A variety is smooth at $a \in X$ if $a$ lives in a unique irreducible component $x_i$ of $X$ and $\dim T_a(X) = \dim X_i$, where dimension of a variety has been defined to be the degree of the Hilbert polynomial of $X$. A projective variety is smooth if its affine cone is.

I tried to calculate a few examples and it all went very wrong.

Example: The Grassmannian $G(2,4)$ in its Plucker embedding is $V(X_{12} x_{34} - x_{13}x_{24}+ x_{14}x_{23}) \subset \mathbb{P}^5$

I calculated the Hilbert polynomial to be $\frac{1}{12}d^4+...$, so it has dimension 4 (as expected), but I get

$$\mbox{Jac}(G(2,4))= [x_{34}, -x_{24}, x_{23}, x_{14}, -x_{13}, x_{12}]$$

Which has rank 1 where $x \ne 0$, so nullity 5. So assumedly $\dim T_aX = \dim( a + \mbox{ker}(\mbox{Jac}(X))) = \dim \mbox{ker} \mbox{Jac}(X) = \mbox{nullity} (\mbox{Jac}(X))$.

Which isn't 4?

Which is a bit silly, as the Grassmannian is obviously smooth.

I'm probably going wrong somewhere, but I've gotten myself thoroughly confused. Any help would be greatly appreciated.


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up vote 0 down vote accepted

You are confusing $\mathbb P^5$ and $\mathbb A^6$.
The calculation you did is valid for the cone $C\subset \mathbb A^6$ with equation $ x_{12} x_{34} - x_{13}x_{24}+ x_{14}x_{23}=0$.
It is of codimension $1$ (hence of dimension $5$), and smooth outside of the origin, as your jacobian matrix shows.
The image $\mathbb G(2,4)\subset \mathbb P^5$ of the grassmannim under the Plücker embedding is $\mathbb P(C\setminus \lbrace 0\rbrace) \subset \mathbb P^5$
It is also smooth of codimension $1$, hence of dimension $4$ as expected.

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Thanks: knew it was something silly! – Tom H Apr 18 '12 at 16:20

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