This quote is from Matsumura's Commutative Ring Theory, page 100: "The principal ideal theorem corresponds to the familiar and obvious-looking proposition of geometrical and physical intuition (which is strictly speaking not always true) that 'adding one equation can decrease the dimension of the space of solutions by at most one'."
One consequence of the principal ideal theorem is that if $P$ is a prime ideal of height $r$ in a Noetherian ring $A$, then $P$ is the minimal prime divisor of some ideal $I=(a_1,\cdots,a_r)$ and that the height of $P/(a_1,\cdots,a_i)$ is precisely equal to $r-i$.
Question 1: How can we make a connection between the quote from Matsumura and the above consequence of the principal ideal theorem? Which is the solution space?
Question 2: Are there any examples where adding one equation drops the dimension by more than 1?