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If [1:0:....:0] is an s-fold singularity of a degree $r$ hypersurface $F$ in $\mathbb{P}^n$ then the hypersurface can be written as $F=x_0^{r-s}g_s(x_1,...,x_n) + x_0^{r-s-1}g_{s+1}(x_1,...,x_n) + ... +g_r(x_1,...x_n)$. After we dehomogenize it is known that the initial term $g_s(x_1,...x_n)$ is the tangent cone at the singularity in $\mathbb{C}^n$. My question is known about the higher order terms such as $g_{s+1}(x_1,..,x_n)$? Do they admit a some geometric interpretation?

I know that the common locus of $g_s=g_{s+1}=...=g_{s+h}=0$ give the set of points whose line through the origin has intersection multiplicity s+h+1 with the hypersurface. I would like to find a geometric interpretation of just the hypersurface $g_{s+1}=0$.

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It seems unlikely that there is something nice. An interpretation should preferably be invariant under linear coordinate transformations and a homogeneous component itself isn't, it is only invariant modulo the ones of lower order.

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As implied by Torsten's answer, one can get information from $g_{s+1}$ only together with $g_s$. The common locus of $g_s$ and $g_{s+1}$ mentioned in the question is an example. If you blow up the origin, and p is a point of the strict transform X on the exceptional divisor, then one can read smoothness of X at p (and the tangent hyperplane $T_p X$) from $g_s$ and $g_{s+1}$ together. Under some genericity conditions one should be able to read the tangent cones also, I guess. –  quim May 7 '10 at 10:59
    
Thanks! I never thought about it in terms of blowups. –  Chirag Lakhani May 9 '10 at 20:34

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