1
$\begingroup$

What is an example of a non-smooth intersection (pullback) of two smooth projective schemes over a field?

I suppose that such two schemes exist but this is totally against my intuition. In my (defective) geometric picture, a singularity of the intersection would be in the big varieties, too.

$\endgroup$
6
$\begingroup$

If you believe in schemes, take any smooth curve $C\subset \mathbb P^2$ in the projective plane and intersect it with its tangent line $T_P \subset \mathbb P^2$ at some arbitrary point $P\in C$.
The intersection $T_P\cap C$ will not be smooth at $P$ , since it is not reduced there.

If you only want to hear about good old varieties, consider the smooth surface $S\subset \mathbb P^3$ defined by the equation $ZT-XY=0$ and intersect it with the (obviously smooth) plane $P\subset \mathbb P^3$ given by the equation $Z=0$.
The intersection $I=P\cap S $ is the variety given by the two equations $Z=0$ and $XY=0$.
It consists of the union of the two lines $Z=X=0$ and $Z=Y=0$ .
This intersection variety $I$ is singular at the intersection $[0:0:0:1]$ of the two lines.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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