Question

Let \(A=\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{n}, B=\mathbb{P}^{m_{1}} \times \cdots \times \mathbb{A}^{m}\). Let \(y \in B, V\) a closed subvariety of A. Show that \(V \times\\{y\\}=\\{(x, y) \in A \times B \mid x \in V\\}\) is a closed subvariety of \(A \times B\), and that the map \(V \rightarrow V \times\\{y\\}\) taking \(x\) to \((x, y)\) is an isomorphism.

Short answer

In conclusion, we have shown that the set \(V \times \{y\}\) is a closed subvariety of \(A \times B\), and the map from \(V\) to \(V \times \{y\}\) taking \(x\) to \((x, y)\) is an isomorphism. Therefore, the claim is proven.

Step-by-step solution

Review the definitions of variety and closed subvariety

A variety is a topological space that is locally an affine variety, i.e., it can be covered by open subsets that are isomorphic to affine varieties. In our case, \(A \times B\) is the product of two varieties, and they have the form of \(\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{n}\) and \(\mathbb{P}^{m_{1}} \times \cdots \times \mathbb{A}^{m}\), so \(A \times B\) is itself a variety. A closed subvariety is a subset of a variety that is closed in the Zariski topology and is a variety itself.

Verify that \(V \times \{y\}\) is a closed subvariety of \(A \times B\)

To show that \(V \times \{y\}\) is a closed subvariety of \(A \times B\), we must demonstrate that it is a closed set in the Zariski topology and that it is a variety. Consider the projection map \(\pi: A \times B \rightarrow A\) defined by \(\pi(x, y) = x\). Since \(\pi\) is a continuous map in the Zariski topology, the preimage of a closed set is a closed set. Since \(V \subseteq A\) is a closed subvariety, it is closed in the Zariski topology. Then, \(\pi^{-1}(V) = V \times B\) is closed in \(A \times B\). Now, notice that \(V \times \{y\} \subseteq V \times B\), so \(V \times \{y\}\) is a closed set in the induced topology on \(V \times B\). Now, we need to show that \(V \times \{y\}\) is itself a variety. Since \(V \times \{y\}\) is a closed subset of \(V \times B\), we can deduce that \(V \times \{y\}\) is a variety. This is because the product of varieties is a variety and closed subsets of varieties are varieties. Therefore, \(V \times \{y\}\) is a closed subvariety of \(A \times B\).

Show that the map \(V \rightarrow V \times \{y\}\) taking \(x\) to \((x, y)\) is an isomorphism.

To show that the map \(\varphi: V \rightarrow V \times \{y\}\), defined by \(\varphi(x) = (x, y)\), is an isomorphism, we need to demonstrate that it is a bijective homomorphism of varieties. First, we show that \(\varphi\) is bijective. It is clear that for each \(x \in V\), there is a unique point \((x, y) \in V \times \{y\}\), so \(\varphi\) is injective. To show surjectivity, notice that for any point \((x, y') \in V \times \{y\}\) with \(x \in V\) and \(y' = y\), we have \(\varphi(x) = (x, y) = (x, y')\), so there is a point in \(V\) mapped to each point in \(V \times \{y\}\), making \(\varphi\) surjective. Next, we show that \(\varphi\) is a homomorphism of varieties. This means that the inverse image of a closed set in \(V \times \{y\}\) under \(\varphi\) is a closed set in \(V\). Let \(C \subseteq V \times \{y\}\) be a closed set. Let \(D = \varphi^{-1}(C) \subseteq V\). Since \(\varphi\) is bijective, its inverse \(\varphi^{-1}\) is well-defined and continuous in the Zariski topology. Thus, the inverse image of a closed set is a closed set, so \(D\) is closed in \(V\). This shows that \(\varphi\) is a homomorphism of varieties. Since \(\varphi\) is a bijective homomorphism of varieties, it is an isomorphism. Thus, the map \(V \rightarrow V \times \{y\}\) taking \(x\) to \((x, y)\) is an isomorphism.

Key concepts

Variety

In algebraic geometry, a variety is a fundamental concept. It represents a geometric object defined as a solution set for a system of polynomial equations. Varieties encompass objects such as curves and surfaces in a three-dimensional space.
One can think of varieties as shapes that maintain certain algebraic relations. A variety is formally a subset of a space like \( \mathbb{A}^n \) or \( \mathbb{P}^n \) that can locally be described as the common zeroes of a collection of polynomials.
Varieties are essential because they provide a bridge between algebra and geometry. They allow geometric properties to be studied through algebraic equations. This simplicity can reveal intricate relationships between algebraic expressions and geometric shapes underlying them.

Subvariety

A subvariety is simply a subset of a variety that itself fulfills the definition of a variety. The idea here is to think of subvarieties as smaller pieces or components within a larger, more complex geometric structure.
Subvarieties inherit their algebraic structure from their parent varieties. For instance, if a larger variety is defined by a system of polynomial equations, a subvariety within it aligns with these equations while also potentially meeting additional conditions.
In the given problem, showing that \( V imes \{y\} \) is a closed subvariety of \( A imes B \) demonstrates that it not only forms a subset but also maintains the complex structure required to qualify as a variety on its own. This highlights the elegance and complexity of studying how structures in algebraic geometry relate to one another.

Zariski topology

The Zariski topology is a unique type of topology applied to varieties in algebraic geometry. It is characterized by defining closed sets as the solution sets of polynomial equations. This is rather different from the standard topology we might be familiar with.
In the Zariski topology, a topological space, such as a variety, has very few open sets and many closed sets. This leads to quite a different view of openness and closeness compared to the usual topology on real or complex numbers.
The use of the Zariski topology allows one to leverage algebraic tools to study geometric properties. In the problem, verifying that \( V imes \{y\} \) is closed involves confirming it remains closed under the Zariski topology, thus linking topological characteristics directly with algebraic properties.

Isomorphism

An isomorphism in the context of algebraic geometry is a bijective homomorphism between varieties that has a continuous inverse. It suggests that two varieties have the same form, meaning they are essentially equivalent from a geometric and algebraic standpoint.
The importance of an isomorphism lies in the fact that it lets us view different algebraic structures as identical through a special mapping. This is crucial for recognizing when two geometric objects are fundamentally the same even if they come wrapped in different mathematical formulations.
Within the problem, proving that the map \( V \rightarrow V \times \{y\} \) is an isomorphism ensures that we can translate between these two structures without losing information, confirming they share an identical geometric and algebraic configuration.