Projective morphisms

by ayushkhaitan3437

This post is about the morphisms between projective varieties. There are some aspects of such morphisms that I’m troubled about. The development will closely follow that in Karen Smith’s “Invitation to Algebraic Geometry”.

First, say we have a morphism \phi:\Bbb{P}^1\to\Bbb{P}^2 such that [s,t]\to[s^2,st,t^2]. We will try and analyze this map.

This map has to be homogeneous: in that each coordinate has to be homogeneous, and the homogeneity has to be of the same degree. This is the only way that such a map between projective varieties can be well-defined.

Now let us talk about the mappings from affine charts. Essentially, the affine charts cover the projective space, and hence every projective variety that lives in that space. When we talk about a particular affine chart, we can reduce the number of variables by 1. Because the value of one variable is always 1: hence it can be neglected. However, is the image also an affine chart? That depends. In this case of [s,t]\to [s^2,st,t^2], the image of an affine chart will be an affine chart. This is because s=1\implies s^2=1. Similarly, t=1\implies t^2=1.

We’ve covered all the possible points in the domain by picking out the affine charts. Hence, we have fully described the map.

A map f between projective varieties is a projective morphism if for each p\in V, where V is the domain, there exists an neighbourhood N(p) such that f|_{N(p)} is a homogeneous polynomial map. Is an affine chart an open set? Yes. If it is the zth affine chart it is the complement of the algebraic set z=0 in \Bbb{P}^n.

Let us now consider a different map: consider V(xz-y^2)\in\Bbb{P}^2. Let us call this curve C. Now consider the map C\to \Bbb{P}^1, defined as [x,y,z]\to [x,y] if x\neq 0 and [x,y,z]\to [y,z] if z\neq 0. What does this mean?

First of all, why is the option y\neq 0 not included? If y\neq 0, both x,z\neq 0 is implied. Hence, this case is a subcase of the two cases considered earlier. Secondly, what does it mean to map to a projective space of a lower dimension? The curve is one-dimensional. Is that the reason why we can embed it in \Bbb{P}^1? Probably. Note that we haven’t yet proven that this mapping is an embedding. However, this will indeed turn out to be the case.

Is this map consistent? In other words, are the two maps the same in the intersection of the open sets x\neq 0 and y\neq 0? Let us see. [x,y]\to [xz,yz]\to[y^2,yz]\to [y,z]. Hence, when x,z\neq 0, this map is consistent.

Why do we have to have such a broken up map? Why not one consistent map? First of all, mapping from affine charts seems like a systematic way to map. You can always ensure that at least one coordinate is non-zero; both in the domain and range. That is really all there is to it. Sometimes on restricting to affine charts, you write affine maps, like in the precious example. In other cases, including this one, you write a projective map. Defining the various projective maps, whether they change with affine charts or not, is of paramount importance. The affine map part is just an observation which may or may not be made.

How to map a curve f(x_0,x_1,x_2,\dots,x_n) to \Bbb{P}^1 in general? This seems to be a very difficult question. [This](http://web.stanford.edu/~tonyfeng/Brill_Noether1.pdf) suggests that every smooth projective curve can be embedded in \Bbb{P}^3. That seems to be the best we can do.

For completeness, I would like to mention that the two maps given above are inverse to each other, although this is unrelated to the motivation for this article.

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