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Chapter 2: Problem 2

V corieties in Projective Space. Let \(h\) be an algebraically closed field, and let \(X\) be a closed subvariety of \(\mathbf{P}_{k}^{n}\) which is nonsingular in codimension one (hence satisfies \((*)\) ). For any divisor \(D=\sum n_{1} Y_{i}\) on \(X\), we define the degree of \(D\) to be \(\sum n_{1}\) deg \(Y_{i},\) where deg \(Y_{i}\) is the degree of \(Y_{1},\) considered as a projective variety itself (I, \(\$ 7\) ) (a) Let \(V\) be an irreducible hypersurface in \(\mathbf{P}^{n}\) which does not contain \(X,\) and let \(Y_{i}\) be the irreducible components of \(V \cap X\). They all have codimension 1 by (I, Ex. 1.8 ). For each \(i\), let \(f_{1}\) be a local equation for \(V\) on some open set \(U\), of \(\mathbf{P}^{n}\) for which \(Y_{i} \cap U_{1} \neq \varnothing,\) and let \(n_{1}=c_{Y},\left(\bar{f}_{1}\right),\) where \(\bar{f}_{1}\) is the restriction of \(f_{i}\) to \(U_{i} \cap X .\) Then we define the dicisor \(V . X\) to be \(\sum n_{i} Y_{i} .\) Extend by linearity, and show that this gives a well-defined homomorphism from the subgroup of Div \(\mathbf{P}^{n}\) consisting of divisors, none of whose components contain \(X,\) to Div \(X\) (b) If \(D\) is a principal divisor on \(\mathbf{P}^{\prime \prime}\), for which \(D . X\) is defined as in (a). show that \(D . X\) is principal on \(X\). Thus we get a homomorphism \(\mathrm{Cl} \mathbf{P}^{n} \rightarrow \mathrm{Cl} X\) (c) Show that the integer \(n_{i}\) defined in (a) is the same as the intersection multiplicity \(i\left(X, V ; Y_{t}\right)\) defined in \((\mathrm{I}, \$ 7) .\) Then use the generalized Bezout theorem (I, 7.7) to show that for any divisor \(D\) on \(P^{\prime \prime}\), none of whose components contain \(X\) \\[ \operatorname{deg}(D . X)=(\operatorname{deg} D) \cdot(\operatorname{deg} X) \\] (d) If \(D\) is a principal divisor on \(X\), show that there is a rational function \(f\) on \(\mathbf{P}^{n}\) such that \(D=(f) . X .\) Conclude that deg \(D=0 .\) Thus the degree function defines a homomorphism deg:Cl \(X \rightarrow\) Z. (This gives another proof of (6.10) since any complete nonsingular curve is projective.) Finally, there is a commutative diagram and in particular. we see that the map \(\mathrm{Cl} \mathrm{P}^{n} \rightarrow \mathrm{Cl} X\) is injective.

Short Answer

Expert verified
First the divisor multiplication \(V . X\) and degree of this divisor are defined, then with these definitions, it's shown that this procedure creates a homomorphism from Div \(\mathbf{P}^{n}\) to Div \(X\). Furthermore, Bezout's theorem is applied to show the homogeneity property for the degree of divisors multiplication. Finally, it's proven that if a divisor is principal on \(X\), it's degree must be zero, which means the degree function is a homomorphism from \(Cl X\) to integers.

Step by step solution

01

Set up the notations

The irreducible variety \(V\) is a hypersurface in projective space \(\mathbf{P}^{n}\), \(X\) is a closed subvariety of \(\mathbf{P}_{k}^{n}\), and \(Y_{i}\) are the irreducible components of the intersection \(V \cap X\). The degree of a divisor \(D=\sum n_{i} Y_{i}\) is defined to be \(\sum n_{i}\) deg \(Y_{i}\). The divisor \(V . X\) is defined to be \(\sum n_{i} Y_{i}\). The aim is to prove that it gives a well-defined homomorphism from divisors of \(\mathbf{P}^{n}\) to divisors on \(X\).
02

Prove the well-definedness of the homomorphism

Introduce a local equation \(f_{1}\) for \(V\) on some open set \(U\) of \(\mathbf{P}^{n}\) such that \(Y_{i} \cap U_{1}\neq\varnothing\), let \(\bar{f}_{1}\) be the restriction of \(f_{i}\) to \(U_{i} \cap X\). For each \(i\), define \(n_{1}=c_{Y},\left(\bar{f}_{1}\right)\). By the properties of divisors and local equations, we can see this procedure can be extended by linearity which gives a well-defined homomorphism.
03

Show that intersection multiplicity equals to the coefficient in divisor multiplication

Argue that intersection multiplicity \(i\left(X, V ; Y_{t}\right)\) equals to \(n_{i}\). This is done by using the properties of intersection multiplicities and the definition of divisors.
04

Apply Bezout's theorem

Apply the generalized Bezout theorem \(\operatorname{deg}(D . X)=(\operatorname{deg} D) \cdot(\operatorname{deg} X)\) to divisors \(D\) on \(\mathbf{P}^{n}\), not containing \(X\). This shows the degree homogeneity property of divisors multiplication.
05

Further prove for principal divisor

Discuss the case when \(D\) is a principal divisor on \(X\). Demonstrate that there exists a rational function \(f\) on \(\mathbf{P}^{n}\) so that \(D=(f) . X\), leading to the conclusion that degree function is a homomorphism that maps the linear system \(Cl X\) to integers (deg: \(Cl X \rightarrow Z\)). This gives another proof of the fundamental theorem on the degree of principal divisors.

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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 profound and rich branch of mathematics, which studies solutions to algebraic equations and their geometrical properties. It connects algebra, specifically polynomial algebra, with geometry through the use of curves, surfaces, and higher-dimensional analogues called varieties. Within this field, projective space plays an important role because it allows us to study polynomial equations without the exceptional case of 'infinity'.

In the context of the given exercise, algebraic geometry is used to study divisors鈥攁 formal sum of subvarieties鈥攁nd their degrees on a given projective variety. The relationship between divisors and projective space is a central aspect of their intersection theory, which is vital for understanding many properties about the geometric structure of varieties.
Intersection Multiplicity
Intersection multiplicity is a fundamental concept in algebraic geometry, measuring the 'complexity' of the intersection between two varieties. In simple terms, it quantifies how many times a variety intersects another in a particular region, accounting for both geometric and algebraic considerations.

In our exercise, the intersection multiplicity is denoted by the integer ni, which appears in the divisor V . X. Essentially, it tells us how 'tangled' the varieties X and V are around each component Yi. It helps identify the extent to which V intersects with X at each component, thus determining the coefficients in the divisor.
Generalized Bezout's Theorem
Generalized Bezout's Theorem is a powerful statement in algebraic geometry, extending the classical Bezout's Theorem to higher dimensions. While the classical theorem concerns the number of intersection points of two curves in the plane, the generalized version pertains to the intersection of higher dimensional varieties.

In our problem, it is used to establish the degree homogeneity of the divisor V . X, indicating that the degree of this intersection divisor is the product of the degrees of D and X. This is a cornerstone result that connects the algebraic degree of varieties to the geometry of their intersections, demonstrating the power of algebraic geometry to generalize results from classical geometry.
Homomorphism in Algebraic Geometry
In algebraic geometry, a homomorphism typically refers to a structure-preserving map between algebraic objects, such as groups or rings. The maps they describe enable us to transfer problems from a complex setting to a simpler one where solutions might be more straightforward to find.

In relation to our exercise, we examine a homomorphism from the group of divisors on \(\textbf{P}^{n}\) that do not fully contain X, to the group of divisors on X. The well-defined homomorphism signifies that the algebraic operation of intersecting a fixed variety with the divisors of \(\textbf{P}^{n}\) maintains the group structure when considered as divisors on X.
Principal Divisor
In the realm of algebraic geometry, a principal divisor is associated with a rational function on a variety. It's a formal sum of codimension one subvarieties鈥攍ike curves on surfaces or points on curves鈥攚eighed by orders of vanishing or poles of the function at those subvarieties.

Within our exercise, we are asked to show that if D is a principal divisor on \(\textbf{P}^{n}\), then D . X is also principal on X. This bears significant implications: it implies that the intersection process with X respects the principal nature of divisors. It also shows that the degree of a principal divisor on X is always zero, establishing a homomorphism between the group of divisors on X modulo principal divisors鈥攊ts class group鈥攁nd the integer group under addition.

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Most popular questions from this chapter

Let \(\left(X, C_{X}\right)\) be a ringed space, and let \(\delta\) be a locally free \(C_{X}\) -module of finite rank. We define the dual of \(\mathscr{E}\), denoted \(\tilde{\delta}\), to be the sheaf \(\Varangle\) om \(_{e x}\left(\mathscr{E}, \mathcal{O}_{x}\right)\) (a) Show that \((\tilde{\delta})^{\sim} \cong \mathcal{E}\). (b) For any \(C_{x}\) -module \(\mathscr{F}, \mathscr{H}\) om \(_{e_{X}}(\mathscr{E}, \mathscr{F}) \cong \check{\mathscr{E}} \otimes_{\text {ox }} \mathscr{F}\). (c) For any \(C_{x}\) -modules \(\mathscr{F} . \mathscr{S}, \operatorname{Hom}_{\epsilon_{x}}(\mathscr{E} \otimes \mathscr{F}, \mathscr{S}) \cong \operatorname{Hom}_{\mathscr{C}_{X}}\left(\mathscr{F}, \mathscr{H}_{O} m_{e_{x}}(\mathscr{E}, \mathscr{S})\right)\). (d) (Projection Formula). If \(f:\left(X, O_{X}\right) \rightarrow\left(Y, O_{Y}\right)\) is a morphism of ringed spaces, if \(\mathscr{F}\) is an \(\mathscr{O}_{X}\) -module, and if \(\mathscr{E}\) is a locally free \(\mathscr{O}_{Y}\) -module of finite rank, then there is a natural isomorphism \(f_{*}\left(\mathscr{F} \otimes_{\mathcal{O}_{\mathbf{X}}} f^{*} \mathscr{E}\right) \cong f_{*}(\mathscr{F}) \otimes_{\mathcal{O}_{\mathbf{Y}}} \mathcal{E}\).

Let \(A\) be a ring, let \(X=\operatorname{Spec} A\). let \(f \in A\) and let \(D(f) \subseteq X\) be the open complement of \(V\) ( \((f)\) ). Show that the locally ringed space \(\left(D(f),\left.C_{X}\right|_{D u_{1}}\right)\) is isomorphic to Spec \(A_{f}\).

(a) Let \(\mathscr{F}^{\prime}\) be a subsheaf of a sheaf \(\mathscr{F}\). Show that the natural map of \(\mathscr{F}\) to the quotient sheaf \(\mathscr{F}^{\prime} \mathscr{F}^{\prime}\) is surjective, and has kernel \(\mathscr{F}\) : Thus there is an exact sequence $$0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{H} / \mathscr{F}^{\prime} \rightarrow 0$$ (b) Conversely, if \(0 \rightarrow \overline{\mathscr{F}}^{\prime} \rightarrow \widetilde{\mathscr{F}} \rightarrow \widetilde{\mathscr{F}}^{\prime \prime} \rightarrow 0\) is an exact sequence, show that \(\mathscr{F}^{\prime}\) is isomorphic to a subsheaf of \(\overline{\mathscr{F}},\) and that \(\mathscr{F}^{\prime \prime}\) is isomorphic to the quotient of \(\mathscr{F}\) by this subsheaf.

Product Schemes. (a) Let \(X\) and \(Y\) be schemes over another scheme \(S\). Use (8.10) and (8.11) to show that \(\Omega_{X \times_{s} Y / S} \cong p_{1}^{*} \Omega_{X / S} \oplus p_{2}^{*} \Omega_{Y / S}\) (b) If \(X\) and \(Y\) are nonsingular varieties over a field \(k\), show that \(\omega_{X \times Y} \cong p_{1}^{*} \omega_{X} \otimes\) \(p_{2}^{*} \omega_{Y}\) (c) Let \(Y\) be a nonsingular plane cubic curve, and let \(X\) be the surface \(Y \times Y\) Show that \(p_{g}(X)=1\) but \(p_{a}(X)=-1\) (I, Ex. 7.2). This shows that the arithmetic genus and the geometric genus of a nonsingular projective variety may be different.

Vector Bundles. Let \(Y\) be a scheme. \(A\) (geometric) vector bundle of rank \(n\) over \(Y\) is a scheme \(X\) and a morphism \(f: X \rightarrow Y\), together with additional data consisting of an open covering \(\left\\{U_{i}\right\\}\) of \(Y\), and isomorphisms \(\psi_{i}: f^{-1}\left(U_{i}\right) \rightarrow \mathbf{A}_{U_{i}}^{n}\) such that for any \(i, j,\) and for any open affine subset \(V=\operatorname{Spec} A \subseteq U_{i} \cap U_{j}\) the automorphism \(\psi=\psi_{j} \circ \psi_{i}^{-1}\) of \(\mathbf{A}_{V}^{n}=\operatorname{Spec} A\left[x_{1}, \ldots, x_{n}\right]\) is given by a linear automorphism \(\theta\) of \(A\left[x_{1}, \ldots, x_{n}\right],\) i.e., \(\theta(a)=a\) for any \(a \in A,\) and \(\theta\left(x_{i}\right)=\) \(\sum a_{i j} x_{j}\) for suitable \(a_{i j} \in A\) An isomorphism \(g:\left(X, f,\left\\{U_{i}\right\\},\left\\{\psi_{i}\right\\}\right) \rightarrow\left(X^{\prime}, f^{\prime},\left\\{U_{i}^{\prime}\right\\},\left\\{\psi_{i}^{\prime}\right\\}\right)\) of one vector bundle of rank \(n\) to another one is an isomorphism \(g: X \rightarrow X^{\prime}\) of the underlying schemes, such that \(f=f^{\prime} \circ g,\) and such that \(X, f,\) together with the covering of \(Y\) consisting of all the \(U_{i}\) and \(U_{i}^{\prime},\) and the isomorphisms \(\psi_{i}\) and \(\psi_{i}^{\prime} \circ g,\) is also a vector bundle structure on \(X\) (a) Let \(\mathscr{E}\) be a locally free sheaf of rank \(n\) on a scheme \(Y\). Let \(S(\mathscr{E})\) be the symmetric algebra on \(\mathscr{E},\) and let \(X=\operatorname{Spec} S(\mathscr{E}),\) with projection morphism \(f: X \rightarrow Y\) For each open affine subset \(U \subseteq Y\) for which \(\left.\mathscr{E}\right|_{U}\) is free, choose a basis of \(\mathscr{E}\) and let \(\psi: f^{-1}(U) \rightarrow \mathbf{A}_{U}^{n}\) be the isomorphism resulting from the identification of \(S(\mathscr{E}(U))\) with \(\mathscr{O}(U)\left[x_{1}, \ldots, x_{n}\right] .\) Then \((X, f,\\{U\\},\\{\psi\\})\) is a vector bundle of rank \(n\) over \(Y\), which (up to isomorphism) does not depend on the bases of \(\mathscr{E}_{U}\) chosen. We call it the geometric vector bundle associated to \(\delta,\) and denote it by \(\mathbf{V}(\mathscr{E})\). (b) For any morphism \(f: X \rightarrow Y\), a section of \(f\) over an open set \(U \subseteq Y\) is a morphism \(s: U \rightarrow X\) such that \(f \circ s=\) id \(_{U} .\) It is clear how to restrict sections to smaller open sets, or how to glue them together, so we see that the presheaf \(U \mapsto\\{\text { set of sections of } f \text { over } U\\}\) is a sheaf of sets on \(Y\), which we denote by \(\mathscr{S}(X / Y) .\) Show that if \(f: X \rightarrow Y\) is a vector bundle of \(\operatorname{rank} n,\) then the sheaf of sections \(\mathscr{S}(X / Y)\) has a natural structure of \(\mathscr{O}_{Y}\) -module, which makes it a locally free \(\mathscr{O}_{Y}\) -module of rank \(n\). [Hint: It is enough to define the module structure locally, so we can assume \(Y=\operatorname{Spec} A\) is affine, and \(X=\mathbf{A}_{Y}^{n} .\) Then a section \(s: Y \rightarrow X\) comes from an \(A\) -algebra homomorphism \(\theta: A\left[x_{1}, \ldots, x_{n}\right] \rightarrow\) \(A,\) which in turn determines an ordered \(n\) -tuple \(\left\langle\theta\left(x_{1}\right), \ldots, \theta\left(x_{n}\right)\right\rangle\) of elements of \(A .\) Use this correspondence between sections \(s\) and ordered \(n\) -tuples of elements of \(A \text { to define the module structure. }]\) (c) Again let \(\delta\) be a locally free sheaf of rank \(n\) on \(Y\), let \(X=\mathbf{V}(\delta)\), and let \(\mathscr{S}=\) \(\mathscr{S}(X / Y)\) be the sheaf of sections of \(X\) over \(Y\). Show that \(\mathscr{S} \cong \mathscr{E}^{\curlyvee},\) as follows. Given a section \(s \in \Gamma\left(V, \delta^{\curlyvee}\right)\) over any open set \(V\), we think of \(s\) as an element of \(\operatorname{Hom}\left(\left.\mathscr{E}\right|_{V}, \mathcal{O}_{V}\right) .\) So \(s\) determines an \(\mathscr{O}_{V^{-} \text {algebra homomorphism }} S\left(\left.\mathscr{E}\right|_{V}\right) \rightarrow \mathcal{O}_{V}\) This determines a morphism of spectra \(V=\operatorname{Spec} O_{V} \rightarrow \operatorname{Spec} S\left(\left.\mathscr{E}\right|_{V}\right)=\) \(f^{-1}(V),\) which is a section of \(X / Y .\) Show that this construction gives an isomorphism of \(\mathscr{E}^{\curlyvee}\) to \(\mathscr{S}\) (d) Summing up, show that we have established a one-to-one correspondence between isomorphism classes of locally free sheaves of rank \(n\) on \(Y\), and isomorphism classes of vector bundles of rank \(n\) over \(Y\). Because of this, we sometimes use the words "locally free sheaf" and "vector bundle" interchangeably, if no confusion seems likely to result.
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