I am an aspiring student interested in doing research in extremal combinatorics, (hyper)graph theory, additive number theory (as in Tao and Vu's book), and theoretical computer science.
I took "Classical Algebraic Geometry (AG)" in my third year, and we covered things up to like projective varieties, coherent sheaves, divisors, and so on. I understand that for someone working on (extremal) combinatorics, one does not necessarily need that “fancy machinery” to solve problems; even if one needs it, usually it's sufficient to use polynomial methods and other (linear) algebraic methods. I know that some theorems in the field were proven with things like real varieties, flag algebra, and such.
So, I think my central question is, would I be "wasting" my time on brushing up my knowledge of AG from, e.g., Gathmann's and Fulton's notes (if I eventually in (extremal/arithmetic) combinatorics)? What do you think of the future of applying AG to discrete side of mathematics? I am aware that a lot of tools from AG seem to work in algebraic/multiplicative number theory, but not additive part of the story (where Fourier analysis shines)? I mentioned only varieties because scheme theory seem to be too general...
Many thanks!