I am starting to read 3264 by Eisenbud and Harris and I consistently cannot tell whether by $$A \cap B$$ they mean the scheme-theoretic or set-theoretic intersection.
For instance, in their definition of rational equivalence (pg. 16), are the two slices interpreted as schemes or sets?
And when they write $$[A \cap B]$$ in Theorem-Definition 1.5 (pg. 19), do they mean the cycle $$\sum_i C_i$$ for $C_i$ the irreducible components of the intersection (as varieties), or is it supposed to be the cycle associated to the scheme-theoretic intersection?
Thank you for your help.
They do mean the scheme-theoretic intersection! If you interpret them as sets in their defintion of rational equivalence, then you'll find that two points on $\mathbb{P}^1$ are rationally equivalent to a point on $\mathbb{P}^1$.
Note that for Theorem-Definition 1.5 it actually doesn't matter, since the assumption that $A$ and $B$ are generically transverse already implies that their intersection is reduced (Proposition 1.28).