Question

Let \(k=\mathbb{C}\). Give \(\mathbb{A}^{n}(\mathbb{C})=\mathbb{C}^{n}\) the usual topology (obtained by identifying \(\mathbb{C}\) with \(\mathbb{R}^{2}\), and hence \(\mathbb{C}^{n}\) with \(\mathbb{R}^{2 n}\) ). Recall that a topological space \(X\) is path-connected if for any \(P, Q \in X\), there is a continuous mapping \(\gamma:[0,1] \rightarrow X\) such that \(\gamma(0)=P, \gamma(1)=Q\). (a) Show that \(\mathbb{C} \backslash S\) is path-connected for any finite set \(S\). (b) Let \(V\) be an algebraic set in \(\mathbb{A}^{n}(\mathbb{C})\). Show that \(\mathbb{A}^{n}(\mathbb{C}) \backslash V\) is path-connected. (Hint:: If \(P, Q \in \mathbb{A}^{n}(\mathbb{C}) \backslash V\), let \(L\) be the line through \(P\) and \(Q\). Then \(L \cap V\) is finite, and \(L\) is isomorphic to \(A^{1}\) (C).)

Short answer

Question: Show that the punctured complex plane and the punctured affine space are path-connected. Short Answer: The punctured complex plane (\(\mathbb{C} \backslash S\)) and the punctured affine space (\(\mathbb{A}^{n}(\mathbb{C}) \backslash V\)) are path-connected because, for any two given points, there is always a continuous path that avoids the finite set \(S\) or algebraic set \(V\), thus connecting the two points within the respective spaces.

Step-by-step solution

Given Points and Set S

Let \(P, Q \in \mathbb{C} \backslash S\), where \(S\) is a finite set.

Choose a Line

Choose a line \(L\) that goes through \(P\) and \(Q\).

Show the Intersection is Finite

Since \(S\) is finite, the intersection \(L \cap S\) is also finite.

Form a Continuous Path

Since the intersection is finite, we can form a continuous path \(\gamma\) that avoids the intersection with \(S\) and goes from \(P\) to \(Q\). This shows that \(\mathbb{C} \backslash S\) is path-connected. Part (b) - Proving Path-Connectedness for Punctured Affine Space

Given Points, Algebraic Set V, and Line L

Let \(P, Q \in \mathbb{A}^{n}(\mathbb{C}) \backslash V\), where \(V\) is an algebraic set. Let \(L\) be a line through \(P\) and \(Q\).

Show the Intersection is Finite

According to the hint, the intersection \(L \cap V\) is finite.

Use the Isomorphism

Since \(L\) is isomorphic to \(A^{1}(C)\), we can construct a continuous path \(\gamma: [0,1] \rightarrow L\) between \(P\) and \(Q\).

Proving Path-Connectedness

We can show that this path \(\gamma\) also avoids \(V\), as the intersection \(L \cap V\) is finite. Since \(P\) and \(Q\) were arbitrary points in \(\mathbb{A}^{n}(\mathbb{C}) \backslash V\), this shows that \(\mathbb{A}^{n}(\mathbb{C}) \backslash V\) is path-connected.

Key concepts

Affine Space

Affine space is a fundamental concept in algebraic geometry, closely linked to how we intuitively think about geometric space. It describes a set of points that form a space similar to Euclidean space but without a fixed origin or coordinate axes. In the context of algebraic geometry, affine space over the complex numbers, denoted as \(\mathbb{A}^{n}(\mathbb{C})\), refers to an n-dimensional space where each point can be represented as an n-tuple of complex numbers.

An essential property of affine space is its 'flat' structure, which facilitates the study of polynomial equations. When dealing with affine space, it's easier to apply algebraic operations to geometric objects. For instance, in the exercise, when considering the continuous path between two points in affine space, we utilize its flat nature to construct a straight line (arguably the simplest path) between those points, which is key to proving path-connectedness.

Affine space sets the stage for exploring more complex geometric concepts like algebraic sets, which are essentially the solutions to polynomial equations within this space.

Algebraic Sets

Algebraic sets, also known as algebraic varieties, are collections of points in affine space that satisfy certain polynomial equations. These sets are a central concept in algebraic geometry because they provide a concrete way to visualize solutions to systems of polynomial equations.

In our exercise, the algebraic set \(V\) is defined in the n-dimensional complex affine space \(\mathbb{A}^{n}(\mathbb{C})\). Algebraic sets can be finite or infinite, and their complexity can range from a single point to intricate shapes. The exercise highlights a crucial aspect that algebraic sets intersecting with a line in affine space result in a finite number of points — this property is instrumental in proving the path-connectedness of the punctured affine space by showing that a continuous path can avoid this finite set of points.

Understanding algebraic sets allows us to analyze and solve geometric problems algebraically, bridging the gap between abstract mathematical concepts and tangible geometric understanding.

Topological Space

A topological space is an abstract mathematical notion that defines the foundational framework for topology. It is composed of a set of points along with a structure that categorizes how these points are grouped together — this structure is known as a topology.\

The topology gives us the ability to speak about concepts such as continuity, convergence, and connectedness, which are not defined in a purely algebraic context. In the provided exercise, we consider the usual topology on \(\mathbb{C}^{n}\), which is derived from viewing complex numbers as pairs of real numbers (thus dealing with \(\mathbb{R}^{2n}\)).

Path-connectedness is a property of a topological space that ensures for any two points in the space, there is a path — a continuous mapping — that connects them. This characteristic is crucial in the exercise to demonstrate that even after removing a finite set (or an algebraic set) from \(\mathbb{C}^{n}\), the space remains path-connected, meaning that any two points can be joined by a path that remains within the space.

Continuous Mapping

Continuous mapping is a cornerstone concept in topology and is deeply linked to the notion of path-connectedness. A continuous mapping between two topological spaces is a function that, intuitively speaking, does not 'tear' or 'glue together' the space. It preserves the closeness of points, ensuring that the image of a neighborhood around a point remains a neighborhood around the image point.

In our scenario, a continuous mapping, denoted \(\gamma\), is a function from the interval [0,1] to the complex affine space. It characterizes the path between two points \(P\) and \(Q\). The fact that \(\gamma(0) = P\) and \(\gamma(1) = Q\) tells us that the path starts at \(P\) and ends at \(Q\), thus proving the space's path-connectedness after a finite set or an algebraic set has been removed.

A solid grasp of continuous mappings is vital for understanding more complex topics in topology, such as homotopy and homeomorphism, which further explore the malleability and structure of spaces. It's through these continuous paths we can connect various points on a topological space, affirming the coherence and integrity of the space even after certain alterations, such as removals.