Hey everybody,
I'd like to know if there is a difference between proper intersection and being in general position on a nonsingular variety. More precisely, suppose that $X$ is a nonsingular quasi-projective variety over an algebraically closed field $k$. (See Chapter I of Hartshorne.) Let $Y$ and $Z$ be closed subvarieties. (This means they are closed subsets which are varieties. In particular, they are reduced and irreducible.)
Now, one can define two notions for when $Y$ and $Z$ "intersect in a good way".
1. We say that $Y$ and $Z$ intersect properly if $$\codim C = \codim Y + \codim Z$$ for any irreducible component of the scheme-theoretic intersection $Y\cap Z$.
2. We say that $Y$ and $Z$ are in general position if, for any $x\in Y\cap Z$, there is an open affine neighborhood $U \subset X$ of $x$ such that the ideal sheaf $J_Y(U) = (f_1,\ldots,f_r)$ is generated by a regular sequence, the ideal sheaf $J_Z(U) = (g_1,\ldots,g_s)$ is generated by a regular sequence and $J_{Y\cap Z}=(f_1,\ldots,f_r,g_1,\ldots,g_s)$ is a regular sequence. (A sequence $(x_1,\ldots,x_n)$ in a ring $A$ is said to be regular if $x_i$ is a nonzero divisor in $A/(x_1,\ldots,x_{i-1})$.)
So, those are the definitions. The geometric interpretation of the second definition is not clear immediately.
I think I can show that 2 implies 1. (The nonsingularity of $X$ is essential here.)
Does 1 imply 2?
My apologies for making it so unreadable. I didn't see how to LateX this properly.
Thank you in advance.
Hairpee