Reference that the blow-up of a smooth variety along a smooth subvariety is smooth.

Question

Accoding to this post:

The blow-up of a non-singular variety along a non-singular subvariety is well-known to be non-singular

Is that difficult, or rather trivial? In the latter case, what is a hint to solve it? If its difficult, what is a reference? I scanned Hartshorne, Chapter II 7, where the general blow-up is defined, but I didn't find anything.

Answer 1

This is Theorem II 8.24 in Hartshorne's Algebraic Geometry.