The Principles of Algebraic Geometry is a great book with, IMHO, many typos and mistakes. Why don't we collaborate to write a full list of all of its typos, mistakes etc? My suggestions:

Page 10 top, definition of $\mathcal{O}_{n,z}$ is wrong (or at least written in a confusing way)

Page 15, change of coordinates given for the projective spaces only work when $i < j$. It states that the given transitions also work in the case when $j< i$.

Page 27, need to put a bar on the second entry of the $h_ij(z)$ operator defined. Also, shouldn't the title of this section be geometry of complex manifolds, instead of calculus on complex manifolds?

Page 35, definition of what is a sheaf is wrong. The gluing condition should be for any family of open sets, not just for pairs of open sets! (I've seem PhD students presenting this definition of sheaf on pg seminars...)

Page 74, writes $D(\psi \wedge e)$, but $\psi$ and $e$ are in two different vector spaces, and one cannot wedge vectors in different vector spaces... I guess they mean tensor product.

Page 130, definition of divisor: it says it's a linear combination of codim 1 of irreducible subvarieties. By linear it means over $\mathbb{Z}$ not over the complex numbers (better should say, like Hartshorne, that $Div$ is the free abelian group generated by the irreducible subvarieties).

Page 180, equation (*) has target a direct sum of line bundles, not tensor.

Page 366, when it says "supported smooth functions over $\mathbb{R}^n$, are these complex valued or real valued functions?

Page 440 top equation. Is it really correct?

Page 445 Second phrase of hypercohomology section; it says sheaves of abelian sheaves. Probably means set of abelian sheaves.