Good day to you all.
I'm currently an undergraduate student with quite a strong affinity for self studying certain topics which interest me. One area which has fascinated me for quite a while is algebraic geometry, especially the (apparently) incredibly abstract machinery used in the modern formulation. I've so far been able to ascertain that two good books on the subject are:
Algebraic Geometry - R. Hartshorne
Foundations of Algebraic Geometry - Ravi Vakil.
Having looked through the ToCs for these books, it appears that Vakil's book is much more comprehensive, but I cannot entirely say for sure. I think I want to set this book as a goal, a mountain to be climbed, so to speak. Of course, before I set such a goal I need to know whether or not it is at all realistic.
The introduction to the text seems to indicate that Aluffi's Algebra text, as well as some general topology and basic notions from 1st or 2nd year undergraduate, will suffice as prerequisites, and I'm wondering if anyone who has read the book could confirm or deny this.
What more would be needed for this book, or for Hartshorne's?
If you would like a rough idea of my current mathematical level, I am currently only in my first year of an undergraduate degree at Oxford, but have so far already worked through Introduction to Abstract Algebra by W. Nicholson (which covers the usual stuff up to Galois) as well as a few other topics such as some axiomatic set theory, basic real analysis (up to integration), and of course the linear algebra, geometry, etc. already covered in my course. I'm currently working through Algebra: Chapter 0 by Aluffi as a second Algebra text, as well as Introduction to Complex Analysis by H. Priestley, and Introduction to Metric and Topological Spaces by W. Sutherland.
By the time I've finished these books, will I likely have enough of a background to begin to tackle Vakil's book? If not, what more would be required?
I will understand if this is an incredibly naive question, and I apologise if this is so. I'm not entirely clear on the level of mathematical maturity required for the text, hopefully my current reading list will reveal mine. I welcome any suggestions of other places to look to sate my curiosity, though bear in mind one of the main attractions to me at the moment is the abstraction.
As always, my tags are guesswork.