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Chapter 1: Problem 12

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \(r \geq 2\).
03

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.
04

\(Q\) 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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The d-Uple Embedding. For given \(n, d>0,\) let \(M_{0}, M_{1}, \ldots, M_{\mathrm{Y}}\) be all the monomials of degree \(d\) in the \(n+1\) variables \(x_{0}, \ldots, x_{n},\) where \(N=\left(\begin{array}{c}n+d \\\ n\end{array}\right)-1 . \mathrm{Wc}\) define a mapping \(\rho_{d}: \mathbf{P}^{n} \rightarrow \mathbf{P}^{\prime}\) by sending the point \(P=\left(a_{0}, \ldots, a_{n}\right)\) to the point \(\rho_{d}(P)=\left(M_{0}(a), \ldots, M_{\mathrm{v}}(a)\right)\) obtained by substituting the \(a_{i}\) in the monomials \(M_{j}\) This is called the \(d\) -uple embedding of \(\mathbf{P}^{n}\) in \(\mathbf{P}^{N}\). For example, if \(n=1, d=2\), then \(N=2,\) and the image \(Y\) of the 2 -uple cmbedding of \(\mathbf{P}^{1}\) in \(\mathbf{P}^{2}\) is a conic. (a) Let \(\theta: k\left[y_{0}, \ldots, y_{v}\right] \rightarrow k\left[x_{0}, \ldots, x_{n}\right]\) be the homomorphism defined by sending \(y_{i}\) to \(M_{t},\) and let a be the kernel of \(0 .\) Then a is a homogencous prime ideal, and so \(Z(a)\) is a projective variety in \(\mathbf{P}^{\prime}\) (b) Show that the image of \(\rho_{d}\) is exactly \(Z\) (a). (One inclusion is casy. The other will require some calculation.) (c) Now show that \(\rho_{d}\) is a homeomorphism of \(\mathbf{P}^{n}\) onto the projective variety \(Z\) (a). (d) Show that the twisted cubic curve in \(\mathbf{P}^{3}\) (Ex. 2.9 ) is equal to the 3 -uple embed\(\operatorname{ding}\) of \(\mathbf{P}^{1}\) in \(\mathbf{P}^{3},\) for suitable choice of coordinates.

The Quadric Surface in \(\mathbf{P}^{3}\) (Fig. 2 ). Consider the surface \(Q\) (a surfuce is a variety of dimension 2 ) in \(\mathbf{P}^{3}\) defined by the equation \(x y-z w=0\) (a) Show that \(Q\) is equal to the Segre embedding of \(\mathbf{P}^{1} \times \mathbf{P}^{1}\) in \(\mathbf{P}^{3}\). for suitable choice of coordinates. (b) Show that \(Q\) contains two families of lines (a line is a linear varicty of dimension \(11: L_{1} ; \ldots . M_{t}^{\prime} .\) each parametrized by \(t \in \mathbf{P}^{1} .\) with the properties that if \(L_{t} \neq L_{u} .\) then \(L_{t} \cap L_{u}=\varnothing:\) if \(M_{t} \neq M_{u} \cdot M_{t} \cap M_{u}=\varnothing \cdot\) and for all \(t . u\) \(L_{t} \cap M_{u}=\) one point (c) Show that \(Q\) contains other curves besides these lines, and deduce that the Zariski topology on \(Q\) is not homeomorphic via \(\psi\) to the product topology on \(\mathbf{P}^{1} \times \mathbf{P}^{1}\) (where each \(\mathbf{P}^{1}\) has its Zariski topology).

For a homogeneous ideal \(a \subseteq S\), show that the following conditions are equivalent: (i) \(Z(a)=\varnothing\) (the empty set): (ii) \(\sqrt{\mathfrak{a}}=\) either \(S\) or the ideal \(S_{+}=\bigoplus_{d>0} S_{d}\) (iii) \(a \geq S_{d}\) for some \(d>0\)

Recall that a curve is rational if it is birationally equivalent to \(\mathbf{P}^{1}\) (Ex. 4.4). Let \(Y\) be a nonsingular rational curve which is not isomorphic to \(\mathbf{P}^{1}\) (a) Show that \(Y\) is isomorphic to an open subset of \(\mathbf{A}^{1}\) (b) Show that \(Y\) is affine. (c) Show that \(A(Y)\) is a unique factorization domain.

(a) Let \(\varphi: X \rightarrow Y\) be a morphism. Then for each \(P \in X . \varphi\) induces a homomorphism of local rings \(\varphi_{p}^{*}: c_{m, p, 1} \rightarrow\left(p_{1}, 1\right.\) (b) Show that a morphism \(\varphi\) is an isomorphism if and only if \(\varphi\) is a homeomorphism, and the induced map \(\varphi_{P}^{*}\) on local rings is an isomorphism, for all \(P \in X\) (c) Show that if \(\varphi(X)\) is dense in \(Y\), then the map \(\varphi_{p}^{*}\) is injective for all \(P \in X\)
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