How to determine explicit description for a projective variety?

Question

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?

Answer 1

Your two questions are actually very different. If you already have a map to projective space, all you want is to find the relations among the functions giving the map. If the functions are given in some explicit way, e.g. power series, then you can turn finding all relations of a given degree into a linear algebra problem and from there find all relations by, e.g. dimension considerations.

The hard question is to exhibit functions on your variety. For Riemann surfaces, we know they are there by the Riemann existence theorem (and Riemann-Roch). Exhibiting them is not automatic. For the case of quotients of the upper-half plane, there is a huge classical literature for example for constructing a Hauptmodul (when the surface has genus zero). So, e.g. writing down the j-function from scratch is not obvious. There are other techniques (Poincare series, ...). Maybe somebody who knows more about this will give a better answer.

Finally, there is no magic bullet to constructing functions, unless you specify how you are given the variety in the first place. You need to make your question more precise.

Answer 2

As Felipe says, these questions are very different, and I will only address #1. As Jack commented, for the specific example you gave, the answer is well-known. There's a general reason why the equations defining the image of a map of schemes should be difficult: the image may not be a scheme. For example, take $(x,y) \mapsto (x,xy)$, whose image is the plane minus a line but plus a point.

That said, there's a well-defined way to find the equations that define the closure of the image, and the key phrase is "elimination theory". One place to read about it very concretely is in Gr\"obner basis books, like Cox-Little-O'Shea.

Answer 3

In general, this is a very, very hard problem, honestly. For instance, Mumford's papers on the equations defining abelian varieties. On the other hand, somewhere else on this site, there's a citation of a theorem that every projective scheme has an embedding cut out by quadrics, which is the other extreme. Your two cases, however, are much simpler.

As Felipe said, once you have a map to projective space, you can just find the relations by your method of choice, he said linear algebra on power series, which sounds good to me. For the Grassmannian, you have your map, for a curve, take the tricanonical embedding, which is an embedding for any curve (this is not the smallest embedding, by any means! Every curve embeds in $\mathbb{P}^3$) consisting of a basis for the cubic differentials on the curve, which in many cases can be written down fairly explicitly.

Another approach for curves is that hyperelliptic curves can all be written as plane curves with singularities at infinity by writing them as $y^2=\prod (x-\lambda_i)$ where $\lambda_i$ are the finite branch points, and nonhyperelliptic curves have a canonical embedding, given straight by the differentials, which is a degree $2g-2$ curve in $g-1$ dimensional space, and using Riemann-Roch you can say a lot about the equations cutting it out, or just compute them explicitly if that's what you need by the power-series-and-linear-algebra approach.

Answer 4

In the context of Kaehler geometry, we have the Kodaira embedding theorem. If you google "Kodaira embedding" you can get lots of good references. See also Huybrechts and Griffiths-Harris.