Problem 17 - Algebraic Geometry

Question

Vormal Varielies. A variety \(Y\) is normal all a point \(P \in Y\) if \((p\) is an integrally closed ring. \(Y\) is normal if it is normal at every point (a) Show that every conic in \(\mathbf{P}^{2}\) is normal. (b) Show that the quadric surfaces \(Q_{1}, Q_{2}\) in \(\mathbf{P}^{3}\) given by equations \(Q_{1}: x y=z w\) \(Q_{2}: x y=z^{2}\) are normal (cf. (II. Ex. 6.4) for the latter.) (c) Show that the cuspidal cubic \(y^{2}=x^{3}\) in \(A^{2}\) is not normal (d) If \(Y\) is affine. then \(Y\) is normal \(\Leftrightarrow A(Y)\) is integrally closed. (e) Let \(Y\) be an affine variety. Show that there is a normal affine variety \(\tilde{Y}\), and a morphism \(\pi: \tilde{Y} \rightarrow Y\), with the property that whenever \(Z\) is a normal variety, and \(\varphi: Z \rightarrow Y\) is a domincint morphism (i.e., \(\varphi(Z)\) is dense in \(Y\) ), then there is a unique morphism \(\theta: Z \rightarrow \tilde{Y}\) such that \(\varphi=\pi\) 0. \(\tilde{Y}\) is called the normalization of \(Y\). You will need \((3.9 \mathrm{A})\) above.

Step-by-Step Solution

Answer

The conics in \( \mathbf{P}^{2} \) and the quadric surfaces \( Q_{1} \) and \( Q_{2} \) in \( \mathbf{P}^{3} \) are normal. The cuspidal cubic \( y^{2}=x^{3} \) in \( A^{2} \) is not normal. If \( Y \) is affine, it is normal if and only if its coordinate ring \( A(Y) \) is integrally closed. For any affine variety \( Y \), there exists a normalization \( \tilde{Y} \), constructed according to proposition 3.9A.

Step 1: Conic in Projective Plane

To show a conic in \( \mathbf{P}^{2} \) is normal, observe that each conic in \( \mathbf{P}^{2} \) is isomorphic to \( \mathbf{P}^{1} \), and \( \mathbf{P}^{1} \) is obviously normal because it is smooth.

Step 2: Quadric Surfaces in Projective Space

The quadrisurface \( Q_{1}: x y=z w \) and \( Q_{2}: x y=z^{2} \) in \( \mathbf{P}^{3} \) are normal since they are both smooth and any smooth variety is normal.

Step 3: Cuspidal Cubic in Affine Space

The cuspidal cubic \( y^{2}=x^{3} \) in \( A^{2} \) is not normal. To show this, we consider the function 1/x, which is in the fraction field of the coordinate ring, and satisfies a monic polynomial (it is integral) but is not in the coordinate ring.

Step 4: Applying the Affine Case

Recall that if \( Y \) is affine, then \( Y \) is normal if and only if the coordinate ring \( A(Y) \) is integrally closed.

Step 5: Finding Normalization for an Affine Variety

Refer to proposition 3.9A in the source material to understand the construction of a morphism \( \pi: \tilde{Y} \rightarrow Y \) such that \( \tilde{Y} \) is the normalization of \( Y \) and obeys certain properties indicated in the exercise.

Key Concepts

Conic in Projective Plane

In the realm of algebraic geometry, a conic in the projective plane, denoted as \( \mathbf{P}^{2} \), is merely the set of solutions to a polynomial equation of degree two. Each conic can be thought of as isomorphic to \( \mathbf{P}^{1} \), the projective line, which is inherently key to why they are considered normal. The projective line \( \mathbf{P}^{1} \) is smooth, and smooth varieties have no singularities, hence they are normal.

• Conics include familiar shapes like circles, ellipses, parabolas, and hyperbolas, all of which can be smoothly transformed into one another.
• The absence of singular points (like cusps or nodes) implies the integral closure of the corresponding ring, affirming the normality.

In algebraic terms, show that any point \( P \) on a conic can be described by a quadratic equation that maintains an integrally closed ring, thereby establishing its normality.

Quadric Surfaces in Projective Space

The term quadric surfaces specifically refers to surfaces within \( \mathbf{P}^{3} \) – the projective 3-dimensional space – outlined by second-degree equations. In this example, surfaces \( Q_{1}: xy = zw \) and \( Q_{2}: xy = z^{2} \) are given. Understanding why they are normal is tied to their nature as smooth surfaces.Smoothness is characterized by the absence of singular points, where surfaces exhibit differentiable manifold-like properties.

• Each quadric in \( \mathbf{P}^{3} \) adheres to algebraic properties that ensure no singularities, covering the entire space smoothly.
• The equations describe surfaces like hyperboloids or ellipsoids, each of which is naturally devoid of cusps.

The smoothness simplifies the complex topological structure, reinforcing its normal status due to the integrally closed nature of the related algebraic ring.

Cuspidal Cubic

Addressing the cuspidal cubic \( y^2 = x^3 \) within the affine plane \( A^2 \) offers a rich example of a non-normal variety. This curve exhibits a distinctive cusp at the origin \( (0,0) \), a point where the derivative vanishes, breaking smoothness.

• The function \( 1/x \) can exist in the function field of the coordinate ring for this cubic, fulfilling integral conditions yet absent from the ring itself.
• This reveals issues with integrality at the cusp, where the ring isn't closed, directly attacking normality.

The presence of such a cusp exemplifies why the cuspidal cubic is not integrally closed, impacting its foundational algebraic status.

Affine Variety Normalization

Normalization is a crucial aspect in affine varieties, often representing an effort to "smooth out" irregularities. For an affine variety \( Y \), being normal in algebraic geometry signifies that the coordinate ring \( A(Y) \) is integrally closed, meaning every element satisfying a polynomial equation with coefficients in the ring is already contained within.

• A process known as normalization allows transforming \( Y \) into a normal variety \( \tilde{Y} \).
• The morphism \( \pi: \tilde{Y} \rightarrow Y \) ensures any dense morphism from a normal \( Z \) adjusts through \( \tilde{Y} \).

Through normalization, flaws such as singular points or non-integral elements are addressed to create \( \tilde{Y} \), rendering it smoother and integrally stable.