Variety in $\mathbb{P}^{n_1}\times ...\times \mathbb{P}^{n_r}\times\mathbb{A}^{m} $

Question

I need help with defining variety in $\mathbb{P}^{n_1}\times ...\times \mathbb{P}^{n_r}\times\mathbb{A}^{m} $.

I am reading Fulton's book, and in Chapter 6, this definition is used, but it is not made by the author. It is left as a task for the reader, and I can not do it.

I also accept an indication of a book where that definition is made.

THANK YOU.