It seems that Robin Hartshorne’s Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about the modern AG. But is it possible for someone who is out of the Academia and has not much background, except a typical Undergraduate Alegebra and Some Analysis, to just go through the book, page by page? If not, what is the proper rout for entering into a serious Algebraic Geometry book, like Hartshorne’s?
Hartshorne’s book is an edulcorated version of Grothendieck and Dieudonné’s EGA, which changed algebraic geometry forever.
EGA was so notoriously difficult that essentially nobody outside of Grothendieck’s first circle (roughly those who attended his seminars) could (or wanted to) understand it, not even luminaries like Weil or Néron .
Things began to change with the appearance of Mumford’s mimeographed notes in the 1960’s, the celebrated Red Book, which allowed the man in the street (well, at least the streets near Harvard ) to be introduced to scheme theory.
Then, in 1977, Hartshorne’s revolutionary textbook was published.
With it one could really study scheme theory systematically, in a splendid textbook, chock-full of pictures, motivation, exercises and technical tools like sheaves and their cohomology.
However the book remains quite difficult and is not suitable for a first contact with algebraic geometry: its Chapter I is a sort of reminder of the classical vision but you should first acquaint yourself with that material in another book.
There are many such books nowadays but my favourite is probably Basic Algebraic Geometry, volume 1 by Shafarevich, a great Russian geometer.
Another suggestion is Milne’s excellent lecture notes, which you can legally and freely download from the Internet.
The most elementary introduction to algebraic geometry is Miles Reid’s aptly named Undergraduate Algebraic Geometry, of which you can read the first chapter here .
Miles Reid ends his book with a most interesting and opinionated postface on the recent history and sociology of algebraic geometry: it is extremely profound and funny at the same time, in the best tradition of English humour.