\^\* : {

  ----- ------------------------
   $u$   $x_1 ^{k_1} U_1 (x_1)$
   $v$   $x_1 ^{k_1} V_1 (x_1)$
  ----- ------------------------

. f\_2 \^\* : {

  ----- ------------------------
   $u$   $x_2 ^{k_2} U_2 (x_2)$
   $v$   $x_2 ^{k_2} V_2 (x_2)$
  ----- ------------------------

. $$ and $\bigl( U_1 (0), V_1 (0) \bigr) , \bigl( U_2 (0), V_2 (0) \bigr) \neq (0,0)$. We call a [*tangent vector to $C_i$ at $p_i$*]{} any non-zero vector in ${\mathcal T}_q X$ proportional to $\bigl( U_i (0), V_i (0) \bigr)$, and [*tangent direction to $C_i$ at $p_i$*]{} the point in ${\mathbb P}\left( {\mathcal T}_q X \right)$ determined by a tangent vector to $C_i$ at $p_i$. Geometrically, we may easily associate to each smooth point of $f_i (C_i)$ a tangent vector in the same way we did above, and then the tangent direction at any point is simply the limiting position of the tangent directions at the smooth points.

We say that [*$C_1$ and $C_2$ are transverse at the point $q = f_i (p_i) \in X$*]{} if their respective tangent directions at $p_1$ and $p_2$ are distinct and we will say that [*$C_1$ and $C_2$ are not transverse at the point $q \in X$*]{} if the tangent directions coincide.

Finally, we say that 