I'm rather confused by Exercise 14.13 in Harris's Algebraic Geometry, which asks the reader to prove that the Veronese and Segre varieties are smooth. The suggestion is that, 'given Exercise 14.3', this follows without calculation from the homogeneity of the defining equations. But Exercise 14.3 asks the reader to prove that the singular points of any variety form a proper subvariety.
Surely this is a mistake? Exercise 14.3 seems unrelated, and the homogeneity of the defining equations does not seem sufficient by itself to prove smoothness.
The word "homogeneity" in the hint does not refer to the fact that the defining equations are homogeneous, but rather that the Veronese and Segre varieties are homogeneous varieties, in the sense that there is an algebraic group $G$ acting transitively on the variety.
Exercise 14.3 says that any variety has at least 1 nonsingular point, but then homogeneity allows you to conclude the same for every point.