I.3.15ac_ERRATA
5/15/2021
Ahhhh, the smell of errata in the morning. 'Tis been a while since I've casually
stated a false statement and not gotten away with it. Let's take a look.
Since

Z_{i} is clearly generated by polynomials that have no Y terms, and these
polynomials determine X_{i}.

(i.e. analogous to before, plug in y for the y_{i} coordinates and leave the x_{i}
coordinates as indeterminates, giving us a polynomial in k[{x_{i}}]. We do this for
every y ∈ Y ). Then
x  ∈ X_{i}  
⇐⇒x × Y  ⊂ X_{i} × Y  
⇐⇒∀y ∈ Y : x × y  ∈ Z_{i}  
⇐⇒∀y ∈ Y,∀f ∈ I(Z_{i}) : f(x × y)  = 0  
⇐⇒∀f ∈ I(Z_{i}),∀y ∈ Y : f(x × y)  = 0  
⇐⇒∀f ∈ I(Z_{i}),∀g ∈ S_{f} : g(x)  = 0  
⇐⇒∀g ∈⋃ _{f∈I(Zi)}S_{f} : g(x)  = 0  
⇐⇒x  ∈ Z(⋃ _{f∈I(Zi)}S_{f})  