By applied algebraic geometry, I don't mean applications of algebraic geometry to pure mathematics or super-pure theoretical physics. Not number theory, representation theory, algebraic topology,differential geometry, string theory etc. However, since the border between pure math and applied math is kind of vague, if you are uncertain whether your answer is really about the applied algebraic geometry (for example, you know a direction that is half-pure and half-applied), please don't hesitate to add you answer or leave a comment.
To be specific, I wonder if the modern theory of schemes (and coherent sheaves, if applicable) has any applications outside of pure math while the classical theory of varieties won't be sufficient. I know there is an area called statistical algebraic geometry, but I think so far it still only uses classical algebraic geometry (no need to know schemes and sheaves). I hope to find an applied algebraic geometry area in which a background as strong as finishing most of the Hartshorne's exercises is not wasted.
Updates:
As per some of the comments below, I want to clarify some points:
(1) By "…while the classical theory of varieties won't be sufficient", you don't have to demonstrated that the research work in applied algebraic geometry you have in mind (that involves modern algebraic geometry notions) can't be translated to classical languages. I think as long as the author chooses modern language to write an applied algebraic geometry paper, there should be a reason behind it and we will find out why.
(2) As for "applicable" vs "potentially-applicable", I think nowadays it is clear many (if not most) applied math and statistics papers are only "potentially-applicable" for the time being (Just look at SIAM journals and conferences). Hence, I think "potentially-applicable" answers are welcome. If your answer is "too pure to be even potentially-applicable", someone would leave a comment below…
Best Answer
I will ignore the issue of what is "applicable" and what is only "potentially applicable", and the issue of whether something could be translated into classical language, and simply offer an example that I came across recently:
Max Lieblich, Lucas Van Meter: Two Hilbert Schemes in Computer Vision.
As the title suggests, this is a paper on the geometry of computer vision that uses Hilbert schemes, not to mention Artin stacks.( But really it's not as fancy as all that: in the end they turn out just to be algebraic spaces.)