Segre Embedding And Heights

Product

The Segre embedding allows us to define the product of projective varieties reasonably, and we will discuss it right now. To begin with we consider the product of an . In this section the ground field is an arbitrary algebraically closed field.

Definition 1. The Segre embedding is defined as follows:

Clearly, . The image on the right hand side has ordered lexicographically.

First of all we make sure that this function is well-defined, otherwise our work will be useless.

Proposition 1. The Segre embedding is a well-defined injective map.

Proof. Assume and for some , then

Next suppose that . Without loss of generality we can assume that and so that we can put them to . Then by looking at first elements we can identify with . Then using we can immediately identify and . Likewise other components are identified.

Next we study the image further using linear algebra

Image of the Segre Embedding And Rank of A Matrix

We can write elements in as a matrix, which can make things easier:

Therefore the image of is given by . Through an elementary observation, we see the matrix

has rank . The question is, is the converse true? For this reason we study the set

Note is the determinant of all submatrices the matrix . This contains all with rank . To show the converse, we consider the standard affine cover. Let and put . Then is the standard affine cover of as we all know. Then likewise we use the affine open subset and , to obtain

This is indeed the inverse map of on . Hence the converse is true.

Therefore, the image of the Segre embedding is a projective variety. As a classic example, the image of is determined by the polynomial .

Height

In this section we offer a way to understand the Segre embedding in number fields. To begin with, we need some definition.

Height is computed by absolute values on a field so we first normalise all absolute values on . Recall that two absolute values and are equivalent if there exists some such that . The question is, which should we pick.

The ordinary absolute value has nothing to worry about. But for -adic absolute values , instead of using other equivalent ones, we have a restriction that . All these absolute values will be denoted by

Likewise we define , where throughout will always be a number field. consists of the ordinary absolute value and extensions of -adic ones defined as follows:

In particular, satisfies the product formula:

for all . This restriction allows us to work fine with projective spaces, as we will see later.

Definition 2. The (absolute logarithmic) height of , with coordinates , is defined by

Actually, the height function can show the "algebraic complication" of , and is well-defined in many senses.

Proposition 2. The height is independent of the choice of .

Sketch of the proof. Let be another number field containing , then we can assume that , then is a finite separable extension. But in this case,

which implies that

Therefore

gives what we want.

Proposition 3. is well-defined on .

Proof. It remains to show that is independent of the choice of coordinates. For , we see

Note because o the product formula.

To highlight the ability of height to measure algebraic complication, let's mention the following theorem of Kronecker.

Theorem 1 (Kronecker). The height of is if and only if is a root of unity.

One direction is straightforward. To prove the converse, one may need some combinatorics, symmetric functions and Dirichlet's pigeon-hole principle. See theorem 2.4 of this note for a proof.

Segre Embedding And Height of Polynomials

Now let's invite the Segre embedding into the party:

Using the fact that , we see immediately that

The Segre embedding is immediately used after introducing the height of a polynomial.

Definition 3. For , we write

Then the height of is defined to be

where

Likewise, it can show the complication of in some way, but we are not interesting in it at this moment. Notice that products of multivariable polynomials can be understood as tensor products and therefore we have the following fact

Proposition 4. Let and be polynomials in different sets of variables, then