Example of a normal cone, that is not irreducible For a pair of (irreducible) varieties $W \subset V$ over an algebraically closed field $k$, let $C = C_{W} V$ be the normal cone, i.e.
$$C = \operatorname{Spec}_W \bigoplus_{n\geq 0} I^n / I^{n+1},$$
where $I \subset \mathcal O_V$ is the ideal sheaf defining $W$.
What is an example where $C$ has different irreducible components $C_1, C_2$, both dominant over $W$? Clearly, $W \subset V$ cannot be a regular embedding, because in that case $C$ would be a vector bundle.
For context, I came upon this question when reading Chapter 6.1 of Fulton's Intersection Theory, where he says that the images of the $C_i$, called there distinguished varieties of some intersection, need not be distinct.
 A: Take $W$ the node on a nodal curve $V$. Then $C_WV$ has two components both supported over $W$, corresponding to the branches of the node. I believe $C_WV$ is actually isomorphic to two $\mathbb{A}^1$'s meeting at a point.
In general $\mathbb{P}(C_WV)$ is the exceptional divisor of $Bl_WV$, which is two points in the case above. If $W$ is nowhere dense in $V$ then $C_WV$ is accordingly the cone over the exceptional divisor. This is the case I usually picture, though for intersection theory the other case matters for excess intersections. Namely if $W$ contains any irreducible components of $V$ then $C_WV$ will have extra components that (I think) are just copies of those components, blown up where they meet the non-dense parts of $W$.
Edit: For the nodal cubic, I think the computation is: let
$$R = \frac{k[x,y]}{(y^2-x^2(x+1))}, \quad I = (x,y)R,$$
and take the Rees algebra
$$\mathrm{Rees}_I(R) = \bigoplus_{n \geq 0} I^n = R[It] \cong \frac{R[T,S]}{(xS-yT, yS - x(x+1)T, S^2 - (x+1)T^2)}$$
where $T = xt$ and $S = yt$. I'm not 100% certain I found all the relations in the presentation of $\mathrm{Rees}_I(R)$ here but I think I have. Now modding out by $I = (x,y)$ gives
$$R/(x,y) \otimes \mathrm{Rees}_I(R) = \frac{R[T,S]}{(x,y) + (\text{relations above})} = \frac{k[T,S]}{(S^2-T^2)},$$
which is two crossing lines.
