Confusion about pulling back homogeneous forms.

Consider the map $i: P^1_{[s:t]} \to P^2_{[x:y:z]}$ that sends $[s:t]$ to $[s:t:1]$, and consider the polynomial $f = y^2z - x^3$. I would like to compute the intersections of $X = Z(f)$ with the line at infinity, and to do so I would like to pull back $f$ to $P^1$ by $i$ somehow. However, just making a naive substitution leaves me with the polynomial $y^2 = x^3$, which is not homogeneous, and has no well defined homogenenization (either $y^3 = x^3$ or $y^2 x = x^3$ seem reasonable.)

I know that in principal such a pullback should be possible, because $i^* O(3) = i^{-1} (O(3)) \otimes_{i^{-1} O_{P^2}} O_{P^1}$ must be some line bundle on $P^1$, and ought to be $O(3)$ since $i$ was was a degree 1 embedding.

Is the only way to see what section of $O(3)$ I get by computing it in affine charts, since there the pullback is (I think) given by substitution. But maybe this is the incorrect approach also (though I don't understand why), since even in charts I get a very confusing answer, since the two equations I get by substitution don't agree on the overlap.

Thanks for helping!

• You mean $\mathbb{P}^2$ – Ben Nov 21 '15 at 19:43
• @Ben Yes, thanks. – Lorenzo Najt Nov 21 '15 at 19:44
• Your map $i$ is not well defined, it must send $[s:t]$ to $[s:t:0]$. This explained why you didn't get something homogeneous after. – Roland Nov 21 '15 at 19:45
• @Roland Oh, of course, that makes sense. I will think through this again. Thanks! – Lorenzo Najt Nov 21 '15 at 19:47