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

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!

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

• Their Ideals, Varieties, and Algorithms is probably a good bet too. Commented Nov 27, 2020 at 22:21
• While this certainly is a book focused on computation in algebraic geometry, I don't think it treats most of the OP's desired topics. As far as I can see, there is basically nothing on divisors, genera, or residues (or differentials forms at all) in that book. There is a section on computing in local rings, which at least addresses the computation of zeroes and poles. Commented Nov 28, 2020 at 0:14
• My understanding is that Ideals, Varieties, and Algorithms is more of an intro to AG from a computational perspective, written by the authors specifically to get their undergraduate REU participants up to speed on basics, while Using Algebraic Geometry is the book for someone who already knows a good amount of the theory, which is the impression I'm getting from the question. But for people who come across this answer in the future it is indeed worth mentioning that both books exist. Commented Nov 28, 2020 at 0:19
• @RichardD.James I would think that working with divisors and genera (and in general questions relating to sheaf cohomology) would be in some way covered, maybe not explicitly given that the reader is expected to already know something about making the necessarily translations, in the chapters on modules and free resolutions (specifically Section 6.4, Hilbert Polynomials and Geometric Applications), with complications introduced by singularities covered by the chapter on local rings. Commented Nov 28, 2020 at 0:27
• Ah, I knew about Ideals, Varieties, and Algorithms which didn't seem to cover the topics I listed, but I didn't know about this other book; thanks! It seems like a good reference, but I'm yet hopeful for one with a more explicit focus on the algorithms themselves! Commented Nov 28, 2020 at 5:57