Your question on mental abilities is of course too vague to admit a definitive answer, but I'll try to give some reflections on the subject.
1) Essentially, I strongly believe that the differences in abilities necessary to tackle the different branches of mathematics are vastly exaggerated.
In my experience good mathematicians are good at any subject.
The difference between their choices results from mathematics having become so vast that it is very difficult or impossible to have expertise in several subjects, unless you are Serre or Tao.
But my conviction, formed by introspection and anecdotal evidence, is that the subject mathematicians end up with very much depends on chance: books found in a library when 16 years old, teachers had in high-school or university, admired friends,...
To be quite honest, some subjects like combinatorics seem to require special gifts and be a little isolated, but even that is changing: I'm thinking of combinatorists like Stanley who use quite sophisticated "mainstream" mathematics, commutative algebra for example.
2) As for algebraic geometry, it certainly requires no special gifts.
Its origin is Descartes's (and Fermat's) fantastic invention of coordinate geometry, which allows one to solve difficult geometric problems by algebra, in an essentially purely mechanical way (which by the way Jean-Jacques Rousseau didn't like: read the extract from his Confessions in the epigraph to Fulton's Algebraic Curves, page iii)
In the 19-th century and in the 20-th century up to about the 1960's hard algebraic geometry was taught (under the name "analytic geometry") in high schools and a Swiss friend of mine showed me problems he solved when 17 years old, which would baffle most Ph.D holders in algebraic geometry nowadays.
The problem is that, in order in particular to solve quite classical problems and also for arithmetic reasons, algebraic geometers like van der Waerden, Zariski, Weil, Serre, Grothendieck,... had to introduce quite sophisticated machinery, culminating in the notion of scheme.
The unfortunate consequence of those developments is that too many introductory courses spend a semester (say) setting up this wonderful modern machinery and have no time left for showing how to apply it to concrete problems.
The good news is that an antidote to this state of affairs exists: it is called math.stackexchange !
I am amazed at the quality, concreteness and pertinence of many questions and answers relating to algebraic geometry here, and I can only advise you to become a frequent user of our site: just use the tag [(algebraic-geometry)] and start reading, pen in hand!