Algebraic Geometry Reading Course

Reddit!

My university doesn't run a course in Algebraic Geometry. I'm thinking of trying to organise a reading group for myself (an MSc research student) and other research students (mainly PhDs) who are interested. I'm a number theorist, but I do a lot with elliptic curves and have found myself wanting to properly learn algebraic geometry.

I'm looking for recommendations on approaches to take and notes to read. Ideally it should be something which is not too in-depth or technical, but which serves as a good, modern introduction, suitable for people to extend with further study as they wish. I am aware that I think Hartshorne is the canonical text; the whole book is rather large, though, and this would be for but a term, so specific recommendations are useful.

In terms of background, I can't really guarantee anything. I suppose people would be first-year UK PhD students in the main; work from that.

Interested to hear your thoughts.