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:

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!!