Problem 12 - Algebraic Geometry

Question

Quadric Hypersurfaces. Assume char \(k \neq 2,\) and let \(f\) be a homogeneous polynomial of degree 2 in \(x_{0}, \ldots, x_{n}\) (a) Show that after a suitable linear change of variables, \(f\) can be brought into the form \(f=x_{0}^{2}+\ldots+x_{r}^{2}\) for some \(0 \leqslant r \leqslant n\) (b) Show that \(f\) is irreducible if and only if \(r \geqslant 2\) (c) Assume \(r \geqslant 2\), and let \(Q\) be the quadric hypersurface in \(\mathbf{P}^{n}\) defined by \(f\). Show that the singular locus \(Z=\operatorname{sing} Q\) of \(Q\) is a linear variety (Ex. 2.11 ) of dimen\(\operatorname{sion} n-r-1 .\) In particular, \(Q\) is nonsingular if and only if \(r=n\) (d) In case \(r

Step-by-Step Solution

Answer

After a suitable linear change of variables, we can bring the polynomial into the form \(f = {x_{0}}^{'2} + {x_{1}}^{'2} + \ldots + {x_{r}}^{2}\). The polynomial will be irreducible if and only if \(r \geq 2\). If \(r \geq 2\), the singular locus of the quadric hypersurface defined by \(f\) is a linear variety of dimension \(n - r - 1\). In the case where \(r < n\), the quadric hypersurface forms a cone over a nonsingular quadric hypersurface.

Step 1: Transformation of the polynomial.

Consider a basis of the form \(x_{i}^{'}, x_{j}^{'}\) such that the initial quadratic form becomes \(f = {x_{0}}^{'2} + {x_{1}}^{'2} + \ldots + {x_{r}}^{'2}\), where \(r\) is the rank of the quadratic form \(f\). This is possible due to the principle of orthogonality.

Step 2: Show \(f\) is irreducible if and only if \(r \geq 2\)

Assume that \(r \geq 2\). Then, \(f\) cannot be represented as a product of two homogeneous polynomials of lower degree which implies that \(f\) is irreducible. If \(r < 2\), then \(f\) can be represented as the product of two linear forms, hence \(f\) is reducible. Therefore, \(f\) is irreducible if and only if \(r \geq 2\).

Step 3: The singular locus of \(Q\)

First, recall that for a quadric hypersurface in \(\mathbf{P}^{n}\) defined by \(f\), the singular locus of \(Q\) consists of all the points in \(Q\) such that the partial derivatives of \(f\) vanish. Thus, if \(f = {x_{0}}^{2} + {x_{1}}^{2} + \ldots + {x_{r}}^{2}\), the singular locus of \(Q\) consists of all points \((x_{0}, x_{1}, \ldots, x_{n})\) such that all \(x_{i} = 0\) for \(0 \leq i \leq r\), a linear variety of dimension \(n - r - 1\). Therefore if \(r = n\), the singular locus is empty, and hence \(Q\) is non-singular.

Step 4: \(Q\) is a cone with axis \(Z\)

In the case where \(r < n\), \(Q\) becomes a cone with axis \(Z\) over a nonsingular quadric hypersurface \(Q'\). This is because for any point in \(Q\), there is a line joining it and the point in \(Z\). This satisfies the definition of a cone.

Key Concepts

Homogeneous Polynomial

A homogeneous polynomial is a type of polynomial where all terms have the same degree. For example, if a polynomial is of degree 2, every term in the polynomial contributes a sum that equals 2.
This concept is essential when dealing with quadric hypersurfaces as they can be expressed in terms of a homogeneous polynomial of degree 2.

• The explicit form is: \( f(x_0, x_1, \ldots, x_n) \), where each term like \( x_i^2 \) (i.e., squared terms) are present.
• Such polynomials are used to describe geometric objects like curves and surfaces in higher dimensions.

Homogeneous polynomials are crucial in identifying certain properties of hypersurfaces, such as symmetry and ease of transformation.

Linear Change of Variables

A linear change of variables involves transforming the variables in a polynomial using linear equations to simplify the form or expression. This is similar to applying a rotation or translation to coordinate axes.
In the context of quadric hypersurfaces, this change allows us to bring a quadratic form into a simpler diagonal form, like \( f = x_0^2 + x_1^2 + \ldots + x_r^2 \).

• It uses matrices and orthogonal transformations to achieve simplification.
• The main goal is to find a basis such as \( x_i^{'}, x_j^{'} \) which supports the diagonalization of the quadratic form.

This adjustment highlights the intrinsic structure of the polynomial, making it easier to study its properties.

Irreducible Polynomial

An irreducible polynomial is one that cannot be factored into a product of two simpler polynomials, especially one of lower degrees. It is somewhat like a 'prime' polynomial.
For quadric hypersurfaces described by homogeneous polynomials of degree 2, a polynomial is irreducible if and only if \( r \geq 2 \).

• If \( r < 2 \), the polynomial can be split into two linear factors, meaning it is reducible.
• When \( r \geq 2 \), it retains its prime-like property, strengthening the hypersurface's structural characteristics.

The irreducibility of a polynomial helps in determining the complexity and nature of the geometric object it defines.

Singular Locus

The singular locus of a hypersurface refers to the set of points where the surface fails to be smooth. For a quadric hypersurface described by the polynomial \( f(x_0, x_1, \ldots, x_n) \), these are points where all its partial derivatives vanish.
If \( f = x_0^2 + x_1^2 + \ldots + x_r^2 \), then the singular locus is determined by vanishing coordinates \( x_i = 0 \) for \( 0 \leq i \leq r \).

• The singular locus is a linear variety of dimension \( n - r - 1 \).
• When \( r = n \), the singular locus is empty, indicating that the hypersurface is smooth.

This concept is used to analyze and classify surfaces based on their smoothness and complexity.

Cone Over a Variety

A \"cone over a variety\" describes a geometric structure that consists of a linear space (axis) and a base variety. It forms by joining each point on this base variety with every point on the axis by a line.
This concept arises in quadric hypersurfaces when \( r < n \). They act as a cone, whereby the base is a nonsingular hypersurface \( Q' \) in \( \mathbf{P}' \).

• The axis is a linear subspace \( Z \) of dimension \( n - r - 1 \).
• This construction generalizes standard conical structures to higher dimensions.

Cone structures are valuable for visualizing the interactions between linear and nonlinear elements in hypersurfaces.