By explicit description I mean a description by explicit algebraic equations. What is the general method to find this in a situation in which you already know that the variety is projective by some other method, for instance by some major theorem whose internals you don't know, but took for granted?

I have two situations in mind:

$1$. The Grassmannian.

We have the Plucker embbedding:

$ \iota : \mathrm{Gr}_{k}(K^n) \rightarrow \mathbb{P}(\wedge^k K^n) $

$\mathrm{span}( v_1, \ldots, v_k ) \mapsto K( v_1 \wedge \cdots \wedge v_k )$

By this we know that the Grassmannian is a closed analytic subspace of the projective space. Therefore we know that it is actually algebraic, by Chow's lemma. But how to explicitly describe the Grassmannian by equations?

$2$. Compact Riemann surfaces

We have some Riemann surface of arbitrary genus, for example given as a quotient of the upper half plane by some group. By some abstract theorem(Riemann-Roch??), we know that a Riemann surface can be embedded in the projective space as a closed subspace. And then again by Chow's lemma, it is actually algebraic. But the question remains how to compute explicit equations for this. Fortunately for elliptic curves we know the answer from Weierstrass theory. But what to do in the more general case?

closed immersiongiven by Plucker embedding (i.e., gives "right scheme structure"). He doesn't say it, but does find defining ideal sheaf on standard affine opens, so if one knows what homogeneous Plucker relations are then one sees Grothendieck gets it. I prefer this to Harris' book because Grothendieck works functorially, Harris works with field-valued pts (but with general method...). $\endgroup$