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.