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Chapter 2: Problem 5

Quadric Hypersurfaces. Let char \(k \neq 2,\) and let \(X\) be the affine quadric hypersurface \(\operatorname{Spec} k\left[x_{0}, \ldots, x_{n}\right] /\left(x_{0}^{2}+x_{1}^{2}+\ldots+x_{r}^{2}\right)-\) cf. \((I, E x .5 .12)\) (a) Show that \(X\) is normal if \(r \geqslant 2(\text { use }(\mathrm{Ex} .6 .4))\) (b) Show by a suitable linear change of coordinates that the equation of \(X\) could be written as \(x_{0} x_{1}=x_{2}^{2}+\ldots+x_{r}^{2} .\) Now imitate the method of \((6.5 .2)\) to show that: (1) If \(r=2,\) then \(\mathrm{Cl} X \cong \mathbf{Z} / 2 \mathbf{Z}\) (2) If \(r=3,\) then \(\mathrm{Cl} X \cong \mathrm{Z}\) (use \((6.6 .1)\) and \((\mathrm{Ex} .6 .3)\) above) (3) If \(r \geqslant 4\) then \(\mathrm{Cl} X=0\) (c) Now let \(Q\) be the projective quadric hypersurface in \(\mathbf{P}^{n}\) defined by the same equation. Show that: (1) If \(r=2, \mathrm{Cl} Q \cong \mathrm{Z},\) and the class of a hyperplane section \(Q . \mathrm{H}\) is twice the generator; (2) If \(r=3, \mathrm{Cl} Q \cong \mathrm{Z} \oplus \mathrm{Z}\) (3) If \(r \geqslant 4, \mathrm{Cl} Q \cong \mathrm{Z},\) generated by \(Q \cdot H\) (d) Prove Klein's theorem, which says that if \(r \geqslant 4\), and if \(Y\) is an irreducible subvariety of codimension 1 on \(Q\). then there is an irreducible hypersurface \(V \subseteq \mathbf{P}^{n}\) such that \(V \cap Q=Y\). with multiplicity one. In other words. \(Y\) is a complete intersection. (First show that for \(r \geqslant 4\). the homogeneous coordinate ring \(S(Q)=k\left[x_{0} \ldots \ldots x_{n}\right]\left(x_{0}^{2}+\ldots+x_{r}^{2}\right)\) is a UFD.

Short Answer

Expert verified
The class of the hypersurface X behaves differently for different values of \(r\). If \(r=2\), \(Cl(X) \cong Z/2Z\); if \(r=3\), \(Cl(X) \cong Z\); if \(r \geq 4\), \(Cl(X) = 0\). The class of projective hypersurface \(Q\), behaves on the other hand as follows. If \(r=2\), \(Cl(Q) \cong Z\); if \(r=3\), \(Cl(Q) \cong Z \oplus Z\); if \(r \geq 4\), \(Cl(Q) \cong Z\). Klein's Theorem holds, which means that for \(r \geq 4\), an irreducible subvariety \(Y\) of codimension 1 on \(Q\) is a complete intersection.

Step by step solution

01

Determine if X is normal

We must use Ex.6.4 to show that \(X\) is normal if \(r \geq 2\). Ex.6.4 states: An affine scheme \(X = \operatorname{Spec} A\) is normal if and only if the ring \(A\) is integrally closed. Hence, to show \(X\) is normal we need to prove that the quotient ring \(k[x_0, ..., x_n]/(x_0^2 + ... + x_r^2)\) is integrally closed.
02

Rewrite the equation of X by a suitable linear change of coordinates

The idea is to transform the given equation into a form where the quadratic terms are isolated on one side of the equation so that the equation takes the form \(x_0x_1 = x_2^2 + ... + x_r^2\). This step proposes a simplification of the hypersurface which eases subsequent computation.
03

Determine the class group of X

Depends on the value of \(r\), the class group \(Cl(X)\) varies. Use techniques in (6.5.2) and other clues to show that: (1) If \(r=2\), then \(Cl(X) \cong Z/2Z\); (2) If \(r=3\), then \(Cl(X) \cong Z\); (3) If \(r \geq 4\) then \(Cl(X) = 0\).
04

Determine the class group of Q

Now let \(Q\) be the projective quadric hypersurface defined by the same equation, determine the class group \(Cl(Q)\). (1) If \(r=2\), \(Cl(Q) \cong Z\), and the class of a hyperplane section \(Q.H\) is twice the generator; (2) If \(r=3\), \(Cl(Q) \cong Z \oplus Z\); (3) If \(r \geq 4\), \(Cl(Q) \cong Z\), generated by \(Q.H\).
05

Prove Klein's Theorem

Assume \(r \geq 4\), and \(Y\) is an irreducible subvariety of codimension 1 on \(Q\). We need to show that there is an irreducible hypersurface \(V \subseteq P^n\) such that \(V \cap Q=Y\) with multiplicity one. First, show that the homogeneous coordinate ring \(S(Q)=k[x_0,...,x_n]/(x_0^2 + ... + x_r^2)\) is a Unique Factorization Domain (UFD). This concludes \(Y\) is a complete intersection.

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Key Concepts

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

Algebraic Geometry
Algebraic geometry is a branch of mathematics that studies solutions to algebraic equations and their geometric properties. At its heart lies the study of algebraic varieties, which are sets of points that satisfy a system of polynomial equations. One of the fundamental objects in algebraic geometry is a hypersurface, which is defined as the zero set of a single polynomial in a given space. A quadric hypersurface is a hypersurface represented by a quadratic polynomial.

These geometrical structures can be examined within different scopes, such as affine space (where our problem about affine quadric hypersurface falls into) and projective space, which involves the concept of points 'at infinity' and leads us to projective quadric hypersurfaces. Techniques in algebraic geometry often involve assessing the normality, singularity, and other intrinsic properties of these geometrical entities. Understanding these properties helps in analyzing the shape and the dimensionality of the solutions that the algebraic equations represent.
Class Group
In algebraic geometry, the class group is an algebraic invariant that categorizes the divisors on a variety up to linear equivalence. Divisors are an abstraction of subvarieties of codimension 1, such as curves on a surface or points on a curve. The class group is denoted as Cl(X) for a variety X and encapsulates information about the variety's overall structure.

As seen in the exercise, different quadric hypersurfaces have different class groups, which directly relate to their geometric, topological, and arithmetic properties. The class group can be trivial (zero), meaning every Weil divisor is principal, or it can have more complex structures such as being represented by additive groups like the integers or the integers modulo a number (for example, \( \mathbf{Z}/2\mathbf{Z} \)).
Affine Schemes
An affine scheme can be thought of as the algebraic counterpart of an 'open piece' of an algebraic variety. Technically, it is the spectrum of a commutative ring, denoted by Spec(A), where A is a ring. This ties into the concept of a 'scheme', which generalizes varieties to include a broader range of 'spaces' constructed algebraically.

The significance of affine schemes in the exercise comes from their role in defining what is normality for algebraic varieties. A scheme is normal if its corresponding ring is integrally closed, meaning it contains all of its integral elements. The affine quadric hypersurface discussed in the problem is an example of an affine scheme, and whether it's normal depends on the structure of the defining equation and the associated ring.
Projective Quadric Hypersurface
A projective quadric hypersurface extends the notion of an affine quadric hypersurface into projective space, which is a space that includes points at infinity. The projective nature allows for the study of properties that are invariant under projective transformations, hence providing a more complete understanding of the geometric object. Given a quadratic equation, its projective quadric hypersurface is defined in the projective space \( \mathbf{P}^n \) and represents all possible scalings of solutions to the equation.

This enriches the study of properties like the class group Cl(Q) of the hypersurface Q, as highlighted in the exercise. The behavior of class groups in the projective setting differs from the affine setting, reflecting the additional complexity introduced by the projective space's extra dimensions.
Unique Factorization Domain
A Unique Factorization Domain (UFD) is a special type of ring in which every element can be written uniquely as a product of prime elements, mirroring the fundamental theorem of arithmetic for the integers. This property is desired in algebraic geometry as it often simplifies the study of a variety's properties, such as its divisor class group.

In the context of the exercise, proving that the coordinate ring of a projective quadric hypersurface is a UFD is central to Klein's Theorem. The theorem itself connects to the idea that certain geometric entities (subvarieties) can be represented neatly by intersections with other geometric structures (hypersurfaces), which has profound implications for their geometric and algebraic classification.

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Most popular questions from this chapter

(a) Let \(\varphi: \overline{\mathscr{H}} \rightarrow \mathscr{S}\) be a morphism of presheaves such that \(\varphi(\mathcal{U}): \mathscr{F}(U) \rightarrow \mathscr{G}(U)\) is injective for each \(U\). Show that the induced \(\operatorname{map} \varphi^{+}: \overline{\mathscr{H}}^{+} \rightarrow \mathscr{S}^{+}\) of associated sheaves is injective. (b) Use part (a) to show that if \(\varphi: \bar{y} \rightarrow \mathscr{G}\) is a morphism of sheaves, then im \(\varphi\) can be naturally identified with a subsheaf of \(\mathscr{G}\). as mentioned in the text.

Espace Etale of a Presheuf. (This exercise is included only to establish the connection between our definition of a sheaf and another definition often found in the literature. See for example Godement [1. Ch. II, \$1.2].) Given a presheaf \(\mathscr{F}\) un \(X\), we define a topological space Sp茅( \(\bar{y}\) ), called the espuce 茅ral茅 of \(\mathscr{F},\) as follows. As a set. Sp茅(. \(\overline{\mathscr{F}})=\cup_{P e}, x^{-} \overline{\mathscr{H}}_{P} .\) We define a projection map \(\pi: \operatorname{Sp茅}(\mathscr{F}) \rightarrow X\) by sending \(s \in \overline{\mathscr{H}}_{p}\) to \(P\). For each open set \(U \subseteq X\) and each section \(s \in \overline{\mathscr{F}}(\mathcal{L}),\) we obtain a \(\operatorname{map} \bar{\Im}: L \rightarrow \operatorname{Spei}(\mathscr{F})\) by sending \(P \mapsto s_{P},\) its germ at \(P .\) This map has the property that \(\pi \quad \bar{s}=\) id \(_{l},\) in other words, it is a "section" of \(\pi\) over \(U\). We now make Sp茅(. \(\overline{\mathscr{H}}\) ) into a topological space by giving it the strongest topology such that show that the sheaf \(\bar{y}^{+}\) associated to \(\bar{y}\) can be described as follows: for any open set \(l \subseteq X, \overline{\mathscr{H}}^{+}(U)\) is the set of continuous sections of \(\operatorname{Spei}(\mathscr{F})\) over \(U\). In particular, the original presheaf \(\mathscr{I}\) was a sheaf if and only if for each \(U, \mathscr{F}(U)\) is equal to the set of all continuous sections of Sp茅(. \(\overline{\mathcal{F}}\) ) over \(U\)

Closed Subschemes of Proj \(S\). (a) Let \(\varphi: S \rightarrow T\) be a surjective homomorphism of graded rings, preserving degrees. Show that the open set \(U\) of \((\mathrm{Ex} .2 .14)\) is equal to Proj \(T,\) and the morphism \(f: \operatorname{Proj} T \rightarrow\) Proj \(S\) is a closed immersion. (b) If \(I \subseteq S\) is a homogeneous ideal, take \(T=S / I\) and let \(Y\) be the closed subscheme of \(X=\) Proj \(S\) defined as image of the closed immersion \(\operatorname{Proj} S / I \rightarrow X\) Show that different homogeneous ideals can give rise to the same closed subscheme. For example, let \(d_{0}\) be an integer,and let \(I^{\prime}=\bigoplus_{d \geqslant d_{0}} I_{d} .\) Show that \(I\) and \(I^{\prime}\) determine the same closed subscheme. We will see later (5.16) that every closed subscheme of \(X\) comes from a homogeneous ideal \(I\) of \(S\) (at least in the case where \(S\) is a polynomial ring over \(S_{0}\) ).

Dimension of the Fibres of a Morphism. Let \(f: X \rightarrow Y\) be a dominant morphism of integral schemes of finite type over a field \(k\) (a) Let \(Y^{\prime}\) be a closed irreducible subset of \(Y\), whose generic point \(\eta^{\prime}\) is contained in \(f(X) .\) Let \(Z\) be any irreducible component of \(f^{-1}\left(Y^{\prime}\right),\) such that \(\eta^{\prime} \in f(Z)\) and show that \(\operatorname{codim}(Z, X) \leqslant \operatorname{codim}\left(Y^{\prime}, Y\right)\) (b) Let \(e=\operatorname{dim} X-\operatorname{dim} Y\) be the relative dimension of \(X\) over \(Y\). For any point \(y \in f(X),\) show that every irreducible component of the fibre \(X_{y}\) has dimension \(\geqslant e .\left[\text { Hint }: \text { Let } Y^{\prime}=\\{y\\}^{-},\) and use (a) and (Ex. 3.20b).] \right. (c) Show that there is a dense open subset \(U \subseteq X,\) such that for any \(y \in f(U)\) \(\operatorname{dim} U_{y}=e .[\text {Hint}: \text { First reduce to the case where } X \text { and } Y\) are affine, say \(X=\operatorname{Spec} A\) and \(Y=\operatorname{Spec} B .\) Then \(A\) is a finitely generated \(B\) -algebra. Take \(t_{1}, \ldots, t_{e} \in A\) which form a transcendence base of \(K(X)\) over \(K(Y),\) and let \(X_{1}=\operatorname{Spec} B\left[t_{1}, \ldots, t_{e}\right] .\) Then \(X_{1}\) is isomorphic to affine \(e\) -space over \(Y\) and the morphism \(X \rightarrow X_{1}\) is generically finite. Now use (Ex. 3.7) above.] (d) Going back to our original morphism \(f: X \rightarrow Y\), for any integer \(h\), let \(E_{h}\) be the set of points \(x \in X\) such that, letting \(y=f(x)\), there is an irreducible component \(Z\) of the fibre \(X_{y},\) containing \(x,\) and having \(\operatorname{dim} Z \geqslant h .\) Show that (1) \(E_{e}=X\) (use (b) above); (2) if \(h>e\), then \(E_{h}\) is not dense in \(X\) (use (c) above \(;\) and (3)\(E_{h}\) is closed, for all \(h\) (use induction on \(\operatorname{dim} X\) ). (e) Prove the following theorem of Chevalley-see Cartan and Chevalley [1 expos茅 \(8] .\) For each integer \(h,\) let \(C_{h}\) be the set of points \(y \in Y\) such that dim \(X_{y}=h .\) Then the subsets \(C_{h}\) are constructible, and \(C_{e}\) contains an open dense subset of \(Y\)

Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective.
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