TITLE: Research in applied algebraic geometry that essentially needs a background of modern algebraic geometry at Hartshorne's level
QUESTION [23 upvotes]: 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...

REPLY [3 votes]: Maybe research at the intersection of algebraic geometry, representation theory, and deep learning might be of interest to you.
Double framed moduli spaces of quiver representationshttps://arxiv.org/abs/2109.14589
This builds on previous work on the representation theory of neural networks, giving category-theoretic formulations too!
This one also has the same flavor; authors define a Kähler metric to perform gradient descent on the moduli space of quivers.
Kähler Geometry of Quiver Varieties and Machine Learning
For an even more abstract/categorical approach. Try this by Yuri Manin and Matilde Marcolli. It's Algebraic topology and category theory, not algebraic geometry though.
HOMOTOPY THEORETIC AND CATEGORICAL MODELS OF NEURAL INFORMATION NETWORKS
Finally, the paper below cites the work by all of the authors above.
But it bridges the abstract approach with the more concrete goal of developing better optimization techniques of neural networks derived from the geometry and symmetries of their parameter spaces.
THE QR DECOMPOSITION FOR RADIAL NEURAL NETWORKS
The following quote of the authors encapsulates their more pragmatic goal.
"Manin and Marcolli advance the
study of neural networks using homotopy theory, and the “partly oriented graphs” appearing in their work are generalizations of quivers. In comparison to the two aforementioned works, our approach is inspired by similar algebro-geometric and categorical perspectives, but our emphasis is on practical consequences for optimization techniques at the core of machine learning."