Thank you so much Ulrich. If you turn these 3 comments into an answer I can give you the bounty points! Again thanks.

I was pretty sure it was incorrect since it wasn't providing the correct answer :). Any thoughts on how to show it correctly?

I certainly apologize Angelo. In the future I will try to include more background to my question to make it self-contained. Thank you for pointing to those specific theorems in your notes. I of course had read those theorems when learning about descent, but the general use of the phrase "by descent theory" threw me off as to what statements were being used and how, as S. Carnahan pointed out above. Thanks for the response and sorry again.

Yeah, in my head the only question that was bothering me was the smoothness. It seemed clear to me that the exceptional locus would just be $\mathbb P^1$'s which wouldn't contribute to the discrepancy. As usual you have explained things very clearly.

I have given a full proof of this with all of the deformation theoretic details in Section 7 of my paper "On (2,4) Complete Intersection threefolds containing an Enriques Surface" arxiv.org/abs/1210.1903.

Yes. I didn't think there were other T^i modules out there, but in case there were, I wanted to clarify that I meant the ones discussed in deformation theory, and first defined in the paper of L-S. Would you like me to edit the question?

Actually we don't have $H^1(X,TX)\cong Ext^1(\Omega_X^1,\mathcal O_X)$. The deformation theory exact sequence (aka local-to-global spectral sequence) gives $0\rightarrow H^1(X,TX)\rightarrow Ext^1(\Omega_X^1,\mathcal O_X)\rightarrow H^0(X,\mathcal T^1)$. Since $\mathcal T^1$ is a skyscraper sheaf support at the ODPs and of dimension 1 at each of them, we find that $H^0(X,\mathcal T^1)=\mathbb C^k$, where $k$ is the number of ODPs. Moreover, this last map is nonzero, so those two v. spaces are not isomorphic. Moreover this last space is $T^1_loc$ as it is the deformation space of the ODPs.

Do you have a reference for this theorem? Also, when you say the fibers are irreducible, are you referring to the trivial family $\mathbb P^2_A$? THanks

@J.C. Ottem: The second part is in fact an exercise there, but I'm asking how to finish it. I just don't know how to get the unlocalized claim from what I've explained so far. As for the first question, do you know on what page it's explained? From another exercise there, I know that the global deformation functor T^1 fits in the same spot in a 4-term exact sequence (see p. 42, exercise 5.8) as Ext^1(\Omega,O_X) does in the s.e.s. coming from the local-to-global spectral sequence. But does this necessarily mean they're isomorphic? A priori the maps might be different. Let me know, thanks