2
$\begingroup$

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.

$\endgroup$

1 Answer 1

4
$\begingroup$

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.

$\endgroup$
4
  • 1
    $\begingroup$ 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
    Apr 26, 2015 at 13:40
  • 2
    $\begingroup$ @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
    Sep 10, 2021 at 17:11
  • 2
    $\begingroup$ 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
    Sep 10, 2021 at 17:17
  • 2
    $\begingroup$ @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
    Sep 11, 2021 at 17:56

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.