Notes: 1) I know next to nothing about algebraic geometry, although I am greatly interested in the field. 2) I realize that "constructive" might be a technical term, here I am using it only in an informal manner.
I hope that this question belongs on this site, since it is not strictly research-level.
As an autodidact (I have an ongoing formal education in physics, but the amount of math we learn here is abysmal, so most of my mathematics knowledge is self-taught), I have noted that algebraic geometry seems to be really impenetrable for somebody who has no formal education in the field, unlike, say, differential geometry or functional analysis, which are areas I can effectively learn on my own.
Pretty much every time I encounter AG-related stuff on sites such as this one or math.se, I see layers and layers of abstractation on top of one another to the point where it makes me wonder, is this field of mathematics constructive, in the sense that can it be used to actually calculate anything or have any use outside mathematics?
The point I am trying to make is that, using differential geometry as an example, is is constructive. No matter how abstractly do I define manifolds, tensor fields, differential forms, connections, etc, they are always resoluble into component functions in some local trivializations, with which one can actually calculate stuff.
Every time I used DG to calculate stellar equilibrium or cosmological evolution or geodesics of some model spacetime, I get actual, direct, palpable, realizable results in terms of real numbers.
I can use differential forms to calculate the volumes of geometric shapes, and every time I use a Lagrangian or Hamiltonian formalism to calculate trajectories for classical mechanical systems, I make use of differential geometry to obtain palpable results.
On top of that, I know that DG is useful outside physics too, I have heard of uses in economy, music theory etc.
I am curious if there is any real-world application of AG where one can use AG to obtain palpable results. I am not curious (for the purpose of this question) about uses to mathematics itself, I know they are numerous. But every time I try to read about AG I get lost in the infinitude of sheaves, stacks, schemes, functors and other highly abstract objects, which often seem so impossible to me to be resolved into calculable numbers.
The final point is, I would like to hear about some interesting applications of algebraic geometry outside mathematics, if there are any.