I'm looking at the material in Shafarevich's "Basic Algebraic Geometry I".
Theorem: If $ X $ is a nonsingular variety and $ \varphi : X \rightarrow \mathbb{P}^{n} $ a rational map to projective space, then the set of points at which $ \varphi $ is not regular has codimension $ \geq 2. $
There are two corollaries to this theorem stated in the text. I'm having trouble understanding why they follow from the theorem.
Corollary 1: Any rational map of a nonsingular curve to projective space is regular.
I feel silly for being confused about this one, but I'm not seeing directly how the theorem implies this.
Corollary 2: If two nonsingular projective curves are birational then they are isomorphic.
Any help would be much appreciated.