0
$\begingroup$

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 $g_{s+1}=0$.

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – quim
    May 7, 2010 at 10:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.