I am having some trouble finding and/or understanding a general definition of subvarieties intersecting transversally. Assume that $Z_1,\ldots,Z_k$ are closed, irreducible subvarieties of a nonsingular algebraic variety $Y$.
Intuitively, I would say that these subvarieties intersect transversally if all varieties have pairwise intersection multiplicity one, i.e. $i(W,Z_i\cdot Z_j; Y)=1$ along any component $W$ of $Z_i\cap Z_j$ for $i\ne j$.
In our scenario, we should have $i(W,Z_i\cdot Z_j;Y)=\mathop{\mathrm{length}}_{\mathscr{O}_{Y,W}}\left(\mathscr{O}_{Z_i\cap Z_j, W}\right)$, if I am not mistaken.
Another intuitive definition would be that the tangent sheaves $\mathscr{T}_{Z_i}$ form a direct sum inside $\mathscr{T}_{Y}$. This seems to agree with the definition in this paper (in 5.1.2), but the author gives an equivalent definition which looks interesting:
For any $y\in Y$, there exists
- a system of parameters $x_1,\ldots,x_n$ on $Y$ at $y$ that are regular on an affine neighborhood $U$ of $y$ such that $y$ is defined by the maximal ideal $(x_1,\ldots,x_n)$ as well as
- integers $0=r_0 \le r_1 \le \cdots \le r_k \le n$ such that the subvariety $Z_i$ is defined by the ideal $I_i=\left(x_{r_{i-1}+1},\ldots,x_{r_i}\right)$ for all $1\le i\le k$.
Just for the record, what precisely does "defined by" mean here? I assume it means that $U\cap Z_i = Z(I_i)$ … or do we actually get that $I_i=I(Z_i\cap U)$?
I would like to know if and how these three definitions are equivalent – there is no (general) treatment of this in Hartshorne or even in Fulton's book on Intersection Theory, which befuddled me greatly.
Best Answer
This naive answer is strictly intended to provoke a knowledgeable answer from someone. Transversal intersection at p should mean that all subvarieties are smooth at p, all contain p, and the codimension of the intersection of the tangent spaces equals the sum of the codimensions of the individual tangent spaces. ????