The book is Fulton, Intersection Theory. My question pertains to Examples 6.1.4 and 6.1.5. In 6.1.4, we are looking at effective Cartier divisors $A,B$ and $D$ on a nonsingular surface $X$ with $A$ and $B$ assumed relatively prime. Define $A'=A+D$, $B'=B+D$. Then we compute $A'\times B'\cdot\Delta$ inside $X\times X$, arising from the fiber square $$\require{AMScd} \begin{CD} A'\cap B'@>>>X\\ @VVV@VVV\\ A'\times B'@>>> X\times X \end{CD} $$ where the right downwards arrow is the diagonal imbedding.
I see that the distinguished varieties for this intersection are the irreducible components of $D$, and the points of $A\cap B$. Let $E$ denote an irreducible component of $D$ with geometric multiplicity $m$; I wish to compute the equivalence of $E$ for the intersection class. Let $C$ denote the irreducible component of the normal cone $C_W X$ which projects to $E$, where $W=A'\cap B'$, and $N$ the restriction to $W$ of the normal bundle corresponding to the regular imbedding of $A'\times B'\subset X\times X$. Then we have that the equivalence of $E$ is given by $$m\{c(N)\frown s(C)\}_0.$$ As $C$ is 2-dimensional, and by the Whitney sum formula, this equals \begin{align*} &m\{(1+c_1(A+D)+c_1(B+D))\frown p_\ast(P(C)+c_1(\mathscr O(1))\frown [P(C)])\}_0 \\ &=m((A+B+2D)\cdot p_\ast([P(C)])-p_\ast(c_1(\mathscr O(-1))\frown [P(C)])) \end{align*} Identifying the blowup of $X$ along $A\cap B$ with the blowup along $A'\cap B'$, we find $p_\ast([P(C)])=[E]$, while $\mathscr O(-1)$ coincides with the restriction of $\mathscr O(\widetilde{A'\cap B'})$ to $\widetilde{A'\cap B'}$, the exceptional divisor of the blowup of $X$ along $A'\cap B'$. This exceptional divisor is the sum of a divisor which maps birationally to $D$, plus one $P^1$ for each point of $A\cap B$, each with multiplicity 1.
Therefore I find $$p_\ast(c_1(\mathscr O(-1))\frown [P(C)]) =D\cdot E+\sum_{P\in A\cap B\cap E} e_P E\;[P],$$ where $e_PE$ denotes the multiplicity of $E$ at $P$.
So my final formula for the equivalence of $E$ is $$m\Big(A\cdot E+B\cdot E+D\cdot E-\sum_{P\in A\cap B\cap E} e_P E\;[P]\Big).$$
This differs slightly from what Fulton writes down. In particular, he writes down $\operatorname{ord}_P(D)$ at $P\in A\cap B$. I don't believe he defines the order of a Cartier divisor at a subvariety of strictly less dimension (than that of the divisor), but even if you take it to mean the multiplicity, why do the other components of $D$ besides $E$ factor in to the equivalence of $E$?
When we take this to the next example, which asks to compute $V_1\times V_2\cdot \Delta$ where \begin{align*} V_1=V(z^3-xy(y-2x),w)\\ V_2=V(w^3-yx(x-2y),z) \end{align*} in $\mathbf P^4$, I wish to compute the equivalence of the line $x=z=w=0$ in $\mathbf P^4$. Fulton says it should be a 0-cycle of degree 3. I believe it should be a 0-cycle of degree equal to the degree of the 0-cycle given by the equivalence of $V(x)$ in the intersection $A'\times B'\cdot \Delta$, where $A'=A+D$, $B'=B+D$ where $A=V(y-2x)$, $B=(x-2y)$, $D=V(xy)$ in $\mathbf P^2$, and $\Delta=\mathbf P^2$ embedded diagonally in $\mathbf P^2\times\mathbf P^2$. If this is true, then as $V(x)$ has geometric multiplicity 1 in $V(xy)$, Fulton's formula $$A\cdot E+B\cdot E+D\cdot E-\sum_{P\in A\cap B}\operatorname{ord}_P(D)\;[P]$$ gives a 0-cycle of degree 1+1+2-2=2, while my formula gives 1+1+2-1=3.
Is my formula for the equivalence of $E$ correct, or have I made multiple errors in the above that cancel each other out?
