Reference Request: Algorithms for computation on curves (more generally varieties)

Question

I'm looking for a book explaining how algorithms allowing the computations of

• poles and zeros,
• singular points,
• divisors,
• genera,
• residues,
• etc,

on algebraic curves work. I expect most of those actually can work in higher dimensions: if so, an appropriate reference would be welcome. Probably this starts with something like Gröbner bases, but I don't know.

Thanks!

Answer 1

Probably you want Using Algebraic Geometry by O'Shea, Little, and Cox.