Question

Let \(X\) be a variety, \(f \in \Gamma(X)\). Let \(\varphi: X \rightarrow \mathbb{A}^{1}\) be the mapping defined by \(\varphi(P)=\) \(f(P)\) for \(P \in X\). (a) Show that for \(\lambda \in k, \varphi^{-1}(\lambda)\) is the pole set of \(z=1 /(f-\lambda)\). (b) Show that \(\varphi\) is a morphism of varieties.

Short answer

Question: Show that for a variety X and a function f in its coordinate ring, the map φ from X to the affine line defined by φ(P) = f(P) has the preimage of a point λ in the ground field k as the pole set of the rational function z = 1 / (f - λ), and that φ is a morphism of varieties. Answer: We have shown that the preimage of a point λ in the ground field k under the map φ is the pole set of the rational function z = 1 / (f - λ) by comparing the domain of z and the inverse image of φ. Moreover, we proved that φ is continuous and respects regular functions, which implies that φ is a morphism of varieties.

Step-by-step solution

Find the domain of z

Find the domain of the rational function z = 1 / (f - λ). To do this, we will determine where the denominator, f - λ, is not equal to 0. Since f - λ is a regular function, its zero set Z(f - λ) is a closed subset of X. The domain of z is the complement of this zero set, i.e., X - Z(f - λ).

Find the inverse image of φ

Now we need to find the inverse image of φ for the point λ in the ground field k. This means finding the set of points P in X such that φ(P) = λ, or equivalently, f(P) = λ. By definition, this is the set of points where the function f - λ vanishes, which is precisely the zero set Z(f - λ). Thus, the inverse image of φ for λ is Z(f - λ).

Compare the domains and inverse image

We showed in Step 1 that the domain of z is X - Z(f - λ), and in Step 2, we found that the inverse image of φ for λ is Z(f - λ). Since the domain of z is the complement of the inverse image, it follows that the preimage of λ under φ is the pole set of the rational function z = 1 / (f - λ). This proves part (a) of the exercise.

Show φ is continuous

To show that φ is continuous, we need to prove that the preimage of any closed set under φ is closed in X. Let C be a closed subset of the affine line A^1, and let D = φ^(-1)(C) be its preimage in X. Since C is the zero set of some polynomial g(t) in A^1, the set D is the zero set of the composite function g(f(P)). By definition, this is a closed subset of X. Thus, φ is continuous.

Show φ respects regular functions

To show that φ respects regular functions, we need to prove that the pullback of a regular function under φ is regular in X. Let h(t) be a regular function on A^1, and let g = h(f(P)) be its pullback under φ. Since h is regular, it can be represented as a polynomial in t. Thus, g = h(f(P)) is a polynomial in f(P), which is a regular function on X. This shows that the pullback of a regular function under φ is regular. Now that we have shown that φ is continuous and respects regular functions, we can conclude that φ is a morphism of varieties. This completes the proof of part (b) of the exercise.

Key concepts

Variety in Algebraic Geometry

In algebraic geometry, a variety is one of the fundamental concepts, serving as the central object of study. It describes a set of solutions to a system of polynomial equations over a field, such as the real or complex numbers. These solutions form a geometric shape, which is the variety itself.

For instance, consider the polynomial equation \(x^2 + y^2 = 1\) over the real numbers. The solutions to this equation form a circle, which is a simple example of a variety in the two-dimensional plane. In general, varieties can be much more complex, and they need not be restricted to curves; they can also be surfaces, solids, or even higher-dimensional objects.

Algebraic varieties come in several types, including affine varieties, which are defined in affine space, and projective varieties, which are defined in projective space. Properties of varieties are the subject of many important theorems and principles in algebraic geometry, such as Bezout's Theorem, which relates the degrees of curves and the number of their intersections.

Morphism of Varieties

A morphism of varieties, often simply called a morphism, is a function between two varieties that preserves the algebraic structure of these spaces. It's analogous to the concept of a 'function' or a 'map' in other areas of mathematics but with important additional properties that respect the geometrical and algebraic features of varieties.

For a map \(\varphi: X \rightarrow Y\) to be considered a morphism, it must satisfy two key conditions. Firstly, \(\varphi\) must be continuous; meaning the preimage of any closed set in \(\mathbb{A}^{1}\) is a closed set in \(\mathbb{A}^{1}\). Secondly, \(\varphi\) must respect regular functions; meaning if \(\mathbb{A}^{1}\) is equipped with a regular function, then this function's composition with \(\varphi\) must produce a regular function on \(\mathbb{A}^{1}\).

These properties are crucial because they ensure the map aligns with the rich structure that algebraic varieties possess. Morphisms are the pathways that allow one to study algebraic geometry with a category-theoretic approach, linking individual objects (varieties) in a highly structured environment (the category of varieties).

Regular Function

The concept of a regular function is an essential one in the study of algebraic geometry. Regular functions on a variety are similar to polynomials over a field and can be seen as the 'allowed' functions when working with varieties.

A regular function on an algebraic variety \(\mathbb{A}^{1}\) can be thought of as a function that is defined by the ratio of two polynomials, as long as the denominator is non-zero on the variety in question. This ensures that the function is well-defined everywhere on the variety.

For example, the function \(\mathbb{f}(x) = x^2\) is a regular function on the affine variety given by the real line, while \(\mathbb{f}(x) = 1/x\) is not regular on the whole real line, because it is not defined at \(\mathbb{0}\). Moreover, regular functions respect the operations of addition and multiplication, leading to the concept of the ring of regular functions on a variety, which provides a powerful tool for studying and understanding the structure of varieties in algebraic geometry.

Regular functions play a pivotal role in defining morphisms between varieties, as a morphism can equivalently be described as a map that pulls back regular functions on the target to regular functions on the source. This serves as a bridge between the geometric and algebraic aspects of the varieties, allowing for a deeper understanding of their properties.