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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.

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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.

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    $\begingroup$ thank you. so you are saying that this definition of tangent cone is "wrong". is there a "right" definition such that the answer to my question becomes yes? is the "wrong" tangent cone of a complete intersection perhaps always a set-theoretic complete intersection? $\endgroup$
    – Hans
    Commented Apr 26, 2015 at 13:40
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    $\begingroup$ @Hans: in this article, Keith Kendig shows: if the intersection of the tangent cones of two varieties is proper, then that intersection is the tangent cone of the intersection of the varieties. Is that what you want? eudml.org/doc/161952 $\endgroup$
    – roy smith
    Commented Sep 10, 2021 at 17:11
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    $\begingroup$ The point is that for your questions to have answer yes, one needs a hypothesis on the dimension of the intersection of the cones. Then it seems to be yes in both senses. $\endgroup$
    – roy smith
    Commented Sep 10, 2021 at 17:17
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    $\begingroup$ @Hans: The example given in this answer, from notes of Milne, seems to be an example of a complete intersection curve of two surfaces whose (correct) tangent cone is apparently not a complete intersection. Of course the hypothesis on the intersection of the tangent cones of the two surfaces being a curve, is violated. math.stackexchange.com/questions/3114128/… $\endgroup$
    – roy smith
    Commented Sep 11, 2021 at 17:56

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