Help me with this solution of the exercise 4.17 from Fulton's Algebraic Curves

Question

I'm studying Fulton's algebraic curves book and in order to prove the well-definiteness of the divisor $div(z)$ on page 97 I'm trying to understand this solution which I found online.

I didn't understand these points:

1. Why $J_z$ is an ideal containing $I(V)$? if we take $F\in I(V)$, $\overline F$ is identity in $\Gamma_h(V)=k[X_1,\ldots,X_{n+1}]/I(V)$, right? Then $\overline Fz$ is not necessarily in $\Gamma_h(V)$, because $z$ is in the fraction field $k(V)$.

2. Why the fact that $J_z$ is an homogeneous ideal containing $I(V)$ implies $J_z$ is algebraic subset of $V$?

I'm sorry for these basic questions, I really have problems with this subject. If anyone could help me with one of these questions or help me with a direct proof of the well-definiteness of $div(z)$, I would be grateful.

Thanks

Answer 1

1. If $F\in I(V)$, then $\overline{F} = 0$ in $\Gamma_h(V)$, so $\overline{F}z = 0$ too.

2. Because $J_z\supseteq I(V)$, we have $V(J_z)\subseteq V\bigl(I(V)\bigr)$. By the Nullstellensatz, $V\bigl(I(V)\bigr) = V$.