I am a bit confused and am trying to clarify some notions. First consider the following well-known statement.
A dominant map $f:X\rightarrow Y$ between regular varieties is flat if and only if it is equidimensional.
Question 1. Doesn't a regular variety mean that it is nonsingular?
Question 2. I am not sure what equidimensional means in this context. Does it mean that the fibers $f^{-1}(c)$ for each $c\in Y$ are equidimensional?
Question 3. What are some examples of a variety that is equidimensional but not a complete intersection?
Thank you.