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I want to compute the dimension of Severi varieties. In order to do it, I have show nodes impose independent conditions. I found this on the book "intersection theory" by Fulton. But I got some problems.

First, how does the author get (b)?

For me $L$ is a subspace of $H^0(\mathbb{P}^2,\mathscr{O}_{\mathbb{P}^2}(n-3))$. Why in (c) the author claim $L$ is a subspace of that $H^0(X,N_f\otimes f^*\mathscr{O}(-3))$?

How to show the canonical divisor is the same as that one?

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