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I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's underlined in the below image:enter image description here

I don't understand why is it possible to find a point in the general fibre such the dimension of the tangent space at this point will be $\langle\alpha,\beta\rangle$. I don't know if it has got anything to do with generic smoothness. Even if it does, I don't understand how to use it here.

I would greatly appreciate your help!!

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    $\begingroup$ This is indeed just generic smoothness. The source of the map is reduced, so it has a smooth dense open subset and the intersection of a general fibre with this open set will be smooth (by generic smoothness). $\endgroup$
    – naf
    Commented Aug 21, 2022 at 4:36
  • $\begingroup$ @naf Thanks for your comment. As I mentioned in the post, I don't understand very well why I can use generic smoothness. I looked at the generic smoothness in David Eisenbud's Commutative Algebra book and it had some technical conditions that I don't understand why it is applicable here. Can you please elaborate why you think we can use generic smoothness here in this setting. I would really appreciate your help and patience!! $\endgroup$
    – It'sMe
    Commented Aug 21, 2022 at 4:59
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    $\begingroup$ I assume the base field is $\mathbb{C}$ in the paper, otherwise one cannot immediately use generic smoothness. Assuming this, we are in the $\mathrm{char} k = 0$ of Eisenbud's Cor 16.23. All local rings of a smooth variety are regular so the other assumption there will be satisfied once you restrict to a suitable open subset of the source. $\endgroup$
    – naf
    Commented Aug 22, 2022 at 7:53
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    $\begingroup$ For a discussion about Schofield's proof, and way to recover the result in positive characteristic, see W. Crawley-Boevey, Subrepresentations of general representations of quivers, Bull. London Math. Soc. 28 (1996), 363-366, mathscinet.ams.org/mathscinet-getitem?mr=1384823 $\endgroup$
    – wcb
    Commented Aug 29, 2022 at 14:04

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