3
$\begingroup$

Is it possible for a reduced, equidimensional germ of complex analytic singularity to have a tangent cone with embedded components but without multiple irreducible components? If it is, how can you build an example?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

If I am doing the computation properly, then one example arises from the tangent cone at the origin of the closed subset $C\subset \mathbb{A}^4$ with defining ideal $$I = \langle x,y\rangle \cap \langle z,w^2-x\rangle = \langle xz,yz,xw^2-x^2,yw^2-xy \rangle,$$ where $(x,y,z,w)$ are coordinates on $\mathbb{A}^4$. The tangent cone should have defining ideal, $J=\langle xz,yz,x^2,xy \rangle$, which is the equation of the union $Z(x,y)\cup Z(x,z)$, but with embedded structure on the intersection, $Z(x,y,z)$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .