Does $\mathbb{V}(\mathcal{E})$ remain the same if I twist by a line bundle?

@AdamP.Goucher Apparently having an unpopular opinion on the usage of downvotes will lead to lot's of downvotes from users on this site.

Please don't link wikipedia as an answer.

Unfortunately, this is not what I am looking for, but, I appreciate your answer. I just want to get a better understanding of how to compute residues by hand just for the sake of it.

@ChrisGerig Would you like to expand this comment into an answer with examples?

@SamGunningham I was looking at if I can get an analogue of the Serre intersection formula for constructible sheaves.

@DonuArapura Without explicit examples it is difficult for the newcomer to understand the subject. I've found getting an idea about how to do the computations in conjunction with suggestions for improving the level of generality of the computations works best for pedagogical purposes.

Do you know how to construct a variety from the embedding of $\mathbb{P}^1$ in $\mathbb{P}^3$ using $\mathcal{O}(3)$?

Hmm, with no common factors then.

and non-equal (or maybe even projectively equivalent) polynomials