For all of the downvoters, can you please explain why you gave me these votes? It seems like many Hartshorne level questions go above math.stackexchange and are more appropriately written here.

Yes, exactly. Sorry I did not put that into the question.

Sure, there is a hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(n)) = \text{Proj}(\mathbb{C}[s,t][x,y,z]/(s^ny - t^nz))$ from mathoverflow.net/questions/122952/on-a-hirzebruch-surface, but I am not sure how to find such an algebra in general.

For example, $\text{Spec}(\text{Sym}(I/I^2))$ for $I = (xy,xz) \subset \mathbb{C}[x,y,z] = R$ is the scheme $\text{Spec}(R[a,b]/(az-by))$

No, I want to get an actual algebra presentation for this construction.

You may want to try and look in the characteristic classes chapter at the end of the book "Complex Topological K-theory" by Efton Park. In it he discusses an approach for computing chern classes using connections which has a distinctly algebraic flavor.

Oh, why is that true? Do you have a reference?

Will $\mathcal{D}_{\mathbb{P}^n}(\mathbb{P}^n) = \mathbb{C}\left[\frac{\partial}{\partial x_0}, \ldots, \frac{\partial}{\partial x_n}\right]$?

Okay cool, this works for the example I was thinking about: attaching a 2-cell to a torus by the inclusion of a circle homotopic whose attaching map is homotopic to a point. This would give the fundamental groupoid $\Pi(X) \cong \Pi(S^2 \vee (S^1 \times S^1))$