# Computation of normal cones

I am reading Fulton's intersection theory but I have a poor intuition about the normal cone. I know the cone $C_{Y/X}$ is the normal bundle if $Y\subset X$ is smooth. I would appreciate it if someone could show what the normal cone of the following schemes of $\mathbb{A}^2$.

1. An intersection of two lines $k[x,y]/(xy)$.
2. A fat line $k[x,y]/(x^k)$
3. A fat point $k[x,y]/(x,y)^k$
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By definition, $C_{Y/X}=\operatorname{Spec}\left(\oplus_l I^l/I^{l+1}\right),$ where $Y\subseteq X$ is defined by the ideal $I\subseteq k[X]$ in the affine coordinate ring of $X.$ In the first case, we have $I=(xy)\subseteq k[x,y]=k[X].$ Thus, $C_{Y/X}$ has coordinate ring $\oplus_l(xy)^l/(xy)^{l+1}.$
Noticing that $(xy)^0/(xy)^1=k[x,y]/(xy),$ we see that there is a canonical surjection $\left(k[x,y]/(xy)\right)[t]\to\oplus_l(xy)^l/(xy)^{l+1}$ taking $t\mapsto xy\pmod {(xy)^2}$. Moreover, according to Fulton p.12-13, this map is an isomorphism, since $xy$ is a nonzerodivisor in $k[x,y]$ and $k[x,y]/(xy)$ is nonzero (i.e. $xy$ is a regular sequence). Thus, there is an isomorphism $C_{Y/X}\to Y\times\Bbb A^1=\operatorname{Spec}\left(k[x,y,t]/(xy)\right)$ is a cartesian product of a union of two coordinate axes with a line.
For the next example, all this analysis should work, since $x^n$ is a regular sequence of $k[x,y].$ Thus we should find that $C_{Y/X}=Y\times\Bbb A^1=\operatorname{Spec}\left(k[x,y,t]/(x^n)\right)$ is a cartesian product of a fat line with a reduced line.
For the third example, $I=(x,y)^n$ is not generated by a regular sequence, so it is not so obvious to me what the geometry is. We now have a surjection $k[x,y,t_1,\ldots, t_N]/(x,y)^n\to\oplus_l(x,y)^{nl}/(x,y)^{n(l+1)}$ taking $t_1\mapsto x^n,t_2\mapsto x^{n-1}y,\ldots,t_N\mapsto y^n,$ giving a closed embedding of $C_{Y/X}$ in $Y\times\Bbb A^N$ with $N={n+2\choose 2}.$ I would be happy to see further elaborations of the techniques to be pursued here.
Thank you for the detailed answer, Andrew! It is quite surprising that nothing interesting happens at the origin in example (1) & (2), as I expected that something non-trivial would occur at the singular point (such as non-reduced fiber over the origin). The last example seems very interesting; $n=3$ yields the twisted cubic . I would like to take a closer look at this. Again, many thanks! Your answer is really helpful. – M. K. Sep 12 '12 at 17:19