So I'm interested in working on the Abundance conjecture, which states that for every projective variety $X$ with Kawamata log terminal singularities over a field $k$, if the canonical bundle $K_{X}$ is nef, then $K_{X}$ is semi-ample.
I have a decent understanding of the conjecture, but, like all mathematicians, I want to improve it... Are you guys able to provide a really intuitive interpretation of what exactly this conjecture is really saying, without too much math jargon? Not an "explain like I'm five thing"... More like an "explain like I'm a student who is just starting algebraic geometry"...
Also, what are the main implications of this conjecture being true? And what are the implications of it being false?
Thank you in advance!
(I post an answer only because the links were too long for a comment, I'm only a master student doing a thesis on birational geometry, so wait a better answer from a more experienced mathematician, this count as a remark).
Dear Brandon, here are some MO links you could visit talking about Abundance's conjecture: let me just suggest you that if you've just started algebraic geometry you should first gain familiarity with birational geometry, like Debarre's, Kollàr and Moris's texts, before even understanding the statement of the conjecture. Anyway you could see
https://mathoverflow.net/questions/31605/does-negative-kodaira-dimension-imply-uniruled
https://mathoverflow.net/questions/115001/open-problems-in-birational-geometry-after-bchm/115005#115005
https://mathoverflow.net/questions/42763/abundance-for-algebraic-surfaces/42767#42767
Hope this help, good luck!