Space curves that intersect often

In intersection theory one tries to understand X {n-ary intersection} Y in terms of information about how X and Y lie in an ambient variety Z. When the sum of the codimensions of X and Y in Z exceeds the dimension of Z not much is known in this direction. The purpose of this note is to provide some results in perhaps the simplest nontrivial case of this-that of curves in P³ (protective three space). A weaker result for Pⁿ is also obtained. We work over any fixed algebraically closed field of arbitrary characteristic.