I'm looking to work with a professor who has a research problem for students in Algebraic Geometry. Unfortunately, given my unconventional background, I haven't had a course in Algebra as of yet. I was wondering if anyone could tell me the textbooks I should read in the next few months to get myself apace to the point where I can start learning Algebraic Geometry. I'm looking for recommendations that'll not only help me understand the basics but also aid in my goal of starting with Algebraic Geometry soon.
Here's the paper a former student of the professor sent to me: https://arxiv.org/pdf/math/0410281.pdf. He said the problem was loosely based around this paper, though I should get in touch with the most recent student of the professor to get more details on the problem.
It'd be great if someone could recommend a path through algebra to algebraic geometry by taking the content of this paper into consideration.