Let $X$ be a quasi projective variety over $\mathbb{C}$. By the tangent cone of $X$ at a point $p \in X$, I mean the subvariety of the tangent space of $X$ at $p$ as it is defined in Harris' "Algebraic Geometry: A first course" (Lecture 20). In particular, the tangent cone is a *reduced* subscheme.


Now let $X$ be locally around $p$ a complete intersection. I wonder whether the tangent cone at $p$ is also a complete intersection. If this is really the case, I would be glad for a reference. Otherwise I would be grateful for a counterexample.



Tangent cone of a complete intersection
If you use the *wrong* definition of tangent cone, then certainly there are counterexamples. For instance, for the origin $p=(0,0,0)$ in $\mathbb{A}^3$, consider the curve $$X=\text{Zero}(\ s(t+u) + f(s,t,u),\ tu + g(s,t,u)\ ),$$ where $f$ and $g$ are sufficiently general polynomials of high degree. The tangent cone is the complete intersection $$\text{Zero}(\ s(t+u), \ tu \ ).$$ However, the underlying reduced scheme of this nonreduced complete intersection is $$\text{Zero}(\ tu,\ su, \ st \ ),$$ which is *not* a complete intersection.


