Problem 3 - Algebraic Geometry

Question

(a) Let \(f\) be the rational function on \(\mathbf{P}^{2}\) given by \(f=x_{1} / x_{0} .\) Find the set of points where \(f\) is defined and describe the corresponding regular function. (b) Now think of this function as a rational map from \(\mathbf{P}^{2}\) to \(\mathbf{A}^{1}\). Embed \(\mathbf{A}^{1}\) in \(\mathbf{P}^{1}\), and let \(\varphi: \mathbf{P}^{2} \rightarrow \mathbf{P}^{1}\) be the resulting rational map. Find the set of points where \(\varphi\) is defined, and describe the corresponding morphism.

Step-by-Step Solution

Answer

The rational function \(f\) is defined at all points in \(\mathbf{P}^{2}\) except those on the line \([0: x_1: x_2]\), and the corresponding regular function is given by \(f = x_1 / x_0\). The rational map \(\varphi\) resulting from the embedding of \(\mathbf{A}^{1}\) in \(\mathbf{P}^{1}\) is defined everywhere in \(\mathbf{P}^{2}\) except at the point \([0:0:1]\), and its corresponding morphism is given by \(\varphi([x_0: x_1: x_2]) = [x_1: x_0]\).

Step 1: Determine where the function \(f\) is defined

A point \([x_0: x_1: x_2]\) in \(\mathbf{P}^{2}\) is undefined when \(x_0 = 0\). Hence, \(f\) is defined everywhere in \(\mathbf{P}^{2}\) except at the points for which \(x_0 = 0\), i.e., the points on the line \([0: x_1: x_2]\) in \(\mathbf{P}^{2}\).

Step 2: Describe the regular function corresponding to \(f\)

The regular function corresponding to the rational function \(f\) can be represented by the same formula \(f = x_1 / x_0\) in the open set where \(f\) is defined, i.e., \(\mathbf{P}^{2}\) - \([0: x_1: x_2]\). It associates with each point \([x_0: x_1: x_2]\) in this open set the value \(x_1 / x_0\).

Step 3: Interpret function \(f\) as a rational map from \(\mathbf{P}^{2}\) to \(\mathbf{A}^{1}\)

The rational function \(f\) assigns to each point \([x_0: x_1: x_2]\) in \(\mathbf{P}^{2}\) - \([0: x_1: x_2]\) the point in \(\mathbf{A}^{1}\) with coordinate \(x_1 / x_0\), hence it defines a rational map from \(\mathbf{P}^{2}\) to \(\mathbf{A}^{1}\).

Step 4: Embed \(\mathbf{A}^{1}\) in \(\mathbf{P}^{1}\)

The embedding of \(\mathbf{A}^{1}\) in \(\mathbf{P}^{1}\) is given by the mapping that associates with each point in \(\mathbf{A}^{1}\) with coordinate \(x\) the point in \(\mathbf{P}^{1}\) with homogeneous coordinates \([x: 1]\). Utilizing this embedding, the rational map from \(\mathbf{P}^{2}\) to \(\mathbf{A}^{1}\) defined by \(f\) becomes a rational map \(\varphi\) from \(\mathbf{P}^{2}\) to \(\mathbf{P}^{1}\), given by \(\varphi([x_0: x_1: x_2]) = [x_1: x_0]\).

Step 5: Determine where the rational map \(\varphi\) is defined

The rational map \(\varphi\) is undefined at the points in \(\mathbf{P}^{2}\) where both \(x_0\) and \(x_1\) are zero, i.e., the intersection of the line \([0: x_1: x_2]\) with the line \([x_0: 0: x_2]\), which is the point \([0:0:1]\). Hence, \(\varphi\) is defined everywhere in \(\mathbf{P}^{2}\) except at the point \([0:0:1]\).

Step 6: Describe the morphism corresponding to \(\varphi\)

The morphism corresponding to the rational map \(\varphi\) associates with each point \([x_0: x_1: x_2]\) in the open set \(\mathbf{P}^{2}\) - \([0:0:1]\) the point \([x_1: x_0]\) in \(\mathbf{P}^{1}\). It can be represented by the same formula \(\varphi([x_0: x_1: x_2]) = [x_1: x_0]\).

Key Concepts

Projective Spaces

Projective spaces are fundamental objects in algebraic geometry offering a way to extend affine spaces to capture the notion of 'points at infinity'. Unlike affine spaces that correspond to our intuitive understanding of geometric space, projective spaces allow for parallel lines to meet at a point at infinity, thus they have no parallels.

Think of the projective space denoted by \( \mathbf{P}^n \) as the set of lines through the origin in the \( (n+1) \) -dimensional space \( \mathbf{A}^{n+1} \), but without the origin itself. Each point in \( \mathbf{P}^n \) actually represents a direction in \( \mathbf{A}^{n+1} \) and is identified with a set of proportional coordinates called homogeneous coordinates. For example, in the projective plane \( \mathbf{P}^2 \) (which was the setting of the original exercise), points have coordinates \( [x_0: x_1: x_2] \) where the \( x_i \) are not all zero and \( [x_0: x_1: x_2] \) represents the same point as \( [\lambda x_0: \lambda x_1: \lambda x_2] \) for any nonzero scalar \( \lambda \).

This concept is vital for understanding the original exercise and solution as it uses the properties of projective spaces to define functions and maps that would otherwise be undefined in an affine setting.

Morphisms in Algebraic Geometry

In algebraic geometry, morphisms are structure-preserving maps between algebraic varieties that respect the algebraic properties of these spaces. They are the algebraic analogue to the concept of function in calculus but with several restrictive properties that maintain the 'shape' of algebraic structures.

A morphism \( \varphi \) from one variety \( X \) to another \( Y \) takes points in \( X \) and maps them to points in \( Y \) according to algebraic rules that must hold over the entire domain where \( \varphi \) is defined. In the exercise, the map \( \varphi \) is a morphism because it maps points in projective space \( \mathbf{P}^2 \) to points in \( \mathbf{P}^1 \) in a way that is well-defined except at certain points where the map becomes undefined (singularities).

Understanding morphisms helps in grasping how algebraic geometry concerns itself with not just the study of shapes defined by equations, but also the ways these shapes can be transformed algebraically into each other.

Rational Functions

Rational functions in algebraic geometry are analogous to rational functions in basic algebra, but with an important distinction—instead of being defined on numerical fields, they are defined over algebraic varieties. A rational function can be expressed as the quotient of two polynomials and, just like rational functions in algebra, they are not always defined (specifically at points where the denominator is zero).

In the context of the exercise, the function \( f = x_1 / x_0 \) is a rational function on the projective plane \( \mathbf{P}^2 \), defined wherever \( x_0 \) is not zero. It captures the essence of projective space where the idea of 'division' must be carefully treated because the values of \( f \) are homogenized—\( f \) represents the same value even if we multiply both the numerator and denominator by the same nonzero scalar \( \lambda \).

The concept of a rational function is crucial in understanding algebraic varieties as it shows how we can construct functions on these spaces despite the intricacies caused by their algebraic nature.

Embedding of Affine Space into Projective Space

Embedding an affine space into a projective space involves creating a correspondence that maps affine points to projective points. In algebraic geometry, this allows us to treat affine spaces within the more general setting of projective spaces. An important consequence of this is that it provides a way to study affine varieties, which can seem limited by their lack of 'points at infinity', in the context of the richer structure provided by projective space. A key benefit is that we can extend many results and theorems from affine to projective settings.

For instance, in the original exercise, the affine line \( \mathbf{A}^1 \) is embedded into the projective line \( \mathbf{P}^1 \) by mapping a point with coordinate \( x \) in \( \mathbf{A}^1 \) to the point with homogeneous coordinates \( [x: 1] \) in \( \mathbf{P}^1 \)—effectively adding a 'point at infinity' to the affine line to create the projective line. This embedding is essential to understand how the rational function \( f \) on \( \mathbf{P}^2 \) gets transformed into the morphism \( \varphi \) into \( \mathbf{P}^1 \)—thus enabling the study of \( f \) within the broader projective context.