Some answers at Math Overflow as (former) user122276 and hm2020:

https://mathoverflow.net/questions/486781/universal-enveloping-algebras-as-operators-on-jets/497977#497977 --
https://mathoverflow.net/questions/391931/which-representations-of-mathfraksl2-are-homomorphic-images-of-the-tensor/392578#comment1025085_392578 -- https://mathoverflow.net/questions/385099/when-do-flat-holomorphic-connections-exist -- https://mathoverflow.net/questions/500481/lie-algebra-of-a-real-complex-algebraic-matrix-group-in-terms-of-power-series#comment1305776_500481

MSE: What is a "moduli space" in math/physics?

What is the definition of moduli space, in math vs in physics?

Some answers to "bountied" questions:

The order and the structure of $R = \mathbb F_2[x]/(x^4 + x^2 +1)$ -- Intersection pairing of weighted projective plane -- A formula for the radical of $\mathbb{Z}/n\mathbb{Z}$. -- Motivating (iso)morphism of varieties -- Why do we use divisors in algebraic geometry? -- Is there a theory that describes eigenspaces of matrix functions near degeneracies? -- https://math.stackexchange.com/users/858083/hm2020?tab=bounties

Some simple proofs and constructions in algebraic geometry:

Transition maps of vector bundle $\mathbb{C}^{n+1}\backslash\{0\}\times\mathbb{C}/\sim$ over $\mathbb{C}P^n$. -- A description of line bundles on projective spaces, $\mathcal{O}_{\mathbb{P}^n}(m)$ defined using a character of $\mathbb{C}^*$. -- Why is the projection map $\mathbb{A}^{n+1}_k\setminus \{0\} \to \mathbb{P}^n_k$ a morphism of schemes? -- Distributions of a group scheme as differential operators. -- Grassmannian $\hbox{Gr}(2,\mathbb{C}^5)$ is homeomorphic to $\hbox{Gr}(3,\mathbb{C}^5)$ -- What are the algebraic features and the geometric interpretation of the symmetric algebra? -- Grothendieck point of view of algebraic geometry -- Quasi coherent module over affine schemes -- Inverse image of the sheaf associated to a module (Hartshorne, Proposition II.5.2) -- Explicit Isomorphism from $\operatorname{Spec}(\operatorname{Sym}^*_{\mathcal{O}_X}(\mathcal{O}_{\Bbb P^2}(-1)^*))$ to an Incidence Variety