Problem 6 - Algebraic Geometry

Question

There are quasi-affine varieties which are not affine. For example, show that \(\mathrm{N}=\mathbf{A}^{2}-\\{(0,0)\\}\) is not affine. \([\text {Hint}: \text { Show that }((X) \cong h[x, 1] \text { and use }(3.5)\) See (III, Ex. 4.3 for another proof.

Step-by-Step Solution

Answer

In conclusion, the space exception at the origin \(\mathrm{N} =\mathbf{A}^{2}-\{(0,0)\}\) is indeed a quasi-affine variety that is not affine. This is shown through the demonstration of the isomorphism between it and \(h[x, 1]\), and the application of an important property of affine varieties.

Step 1: Interpret the Problem Narrative

We have the set \(\mathrm{N}=\mathbf{A}^{2}-\{(0,0)\}\), which represents all the points in the 2-dimensional affine space \(\mathbf{A}^{2}\) except for the origin. We have to prove that this set is not an affine variety.

Step 2: Show \(\mathrm{X} \cong h[x, 1]\)

Next step is to establish an isomorphism between \(\mathrm{X}\) and \(h[x, 1]\). In algebraic geometry, an affine variety \(\mathrm{X}\) is an irreducible closed subset of an affine space \(\mathbf{A}^{n}\), that is defined by a collection of polynomial equations. For example, \(h[x, 1]\) represents the hyperplane at \(x = 1\). Now, we show an isomorphism between this hyperplane and the affine space defined by \(\mathrm{X}\). This isomorphism is functionally expressing functions on \(\mathrm{X}\) and \(h[x, 1]\) in terms of each other.

Step 3: Utilize the Affine Space Property

In the final step, we utilize the property that a variety \(\mathrm{X}\) is affine if and only if for any affine open subset \(\mathrm{U}\) of \(\mathrm{X}\), the global sections of \(\mathrm{U}\) are isomorphic to the coordinate ring of \(\mathrm{X}\). Applying this property, we can say that, for an affine variety subtracting a point, the global section is isomorphic to the ring of polynomials in two variables, without a constant term, which is clearly not isomorphic to the coordinate ring of the whole affine plane.

Key Concepts

Understanding Algebraic Geometry

Algebraic geometry is a branch of mathematics that combines elements from both algebra and geometry. More precisely, it studies geometric structures that come from solutions to systems of polynomial equations. These geometrical structures can range from simple curves and surfaces to much more complex multi-dimensional constructs.

At its heart, algebraic geometry is about defining and working with varieties, which are sets of points that solve given polynomial equations. An affine variety, for example, is a type of variety that can be described as the common zeros of a set of polynomials in a finite-dimensional space referred to as affine space, denoted by \(\mathbf{A}^n\). Understanding varieties and their properties are crucial for solving many problems within algebraic geometry.

Exploring Quasi-Affine Varieties

While affine varieties are an essential concept in algebraic geometry, there is also a special type called quasi-affine varieties that comes into play. A quasi-affine variety is essentially what you get when you take an affine variety and remove a subset that is also defined by polynomial equations. An intuitive example of this is taking the entire two-dimensional plane, symbolized by \(\mathbf{A}^2\), and excluding a single point from it, like \(\{(0,0)\}\).

To demonstrate that a quasi-affine variety such as \(\mathrm{N}=\mathbf{A}^{2}-\{\(0,0\)\}\) is not affine, one could show that it lacks certain properties that characterize affine varieties, such as having a coordinate ring that aligns perfectly with the functions defined directly on the variety. Being able to distinguish between affine and quasi-affine varieties is vital for a deeper understanding of the structure of solutions within algebraic geometry.

The Role of the Coordinate Ring

To wrap one's head around the concept of affine varieties and why certain varieties are not affine, such as quasi-affine varieties, one must understand the coordinate ring. The coordinate ring of an affine variety is a powerful algebraic tool used to study the variety's properties. It consists of all polynomial functions on the variety, capturing the essence of its geometric structure through algebra.

In more precise terms, for the affine variety \(\(\mathbf{A}^2\)\), the coordinate ring is the ring of all polynomials in two variables. However, when we consider a space like \(\(\mathbf{A}^2 - \{\(0,0\)\}\)\), the removal of a point disrupts the 'completeness' of the coordinate ring, so it can no longer represent the entire set of polynomials on the variety. This distinction is key to understanding why the given quasi-affine variety \(\mathrm{N}\) cannot be affine, because its coordinate ring would not be isomorphic to the ring of polynomials of the whole plane, as the original step-by-step solution suggested.