What is a generically reduced scheme?

Question

I am reading the book "3264 & All That Intersection Theory in Algebraic Geometry". In the following definition (see page 30)

Definition 1.22. Let $f:Y\rightarrow X$ be a morphism of smooth varieties. We say a subvariety $A \subset X$ is generically transverse to $f$ if the preimage $f^{-1}(A)$ is generically reduced and...

What is the definition "generically reduced"? I can not find this definition in any part of this book.

Answer 1

"$X$ is generically $\mathcal{P}$" means that there is a dense open subset $U\subset X$ where property $\mathcal{P}$ holds.