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Chapter 2: Problem 5

Blowing up a Nonsingular Subvariety. As in \((8.24),\) let \(X\) be a nonsingular variety, let \(Y\) be a nonsingular subvariety of codimension \(r \geqslant 2\), let \(\pi: \tilde{X} \rightarrow X\) be the blowing-up of \(X\) along \(Y\), and let \(Y^{\prime}=\pi^{-1}(Y)\) (a) Show that the maps \(\pi^{*}:\) Pic \(X \rightarrow\) Pic \(\tilde{X}\), and \(\mathbf{Z} \rightarrow\) Pic \(X\) defined by \(n \mapsto\) class of \(n Y^{\prime},\) give rise to an isomorphism Pic \(\tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}\) (b) Show that \(\omega_{\tilde{X}} \cong f^{*} \omega_{X} \otimes \mathscr{L}\left((r-1) Y^{\prime}\right) .[\) Hint: By (a) we can write in any \(\operatorname{case} \omega_{\tilde{X}} \cong f^{*} \mathscr{M} \otimes \mathscr{L}\left(q Y^{\prime}\right)\) for some invertible sheaf \(\mathscr{M}\) on \(X,\) and some integer \(q .\) By restricting to \(\tilde{X}-Y^{\prime} \cong X-Y\), show that \(\mathscr{M} \cong \omega_{X} .\) To determine \(q\) proceed as follows. First show that \(\omega_{Y^{\prime}} \cong f^{*} \omega_{X} \otimes O_{Y^{\prime}}(-q-1) .\) Then take a closed point \(y \in Y\) and let \(Z\) be the fibre of \(Y^{\prime}\) over \(y .\) Then show that \(\omega_{z} \cong\) \(\left.\mathcal{O}_{\mathbf{z}}(-q-1) . \text { But since } Z \cong \mathbf{P}^{r-1}, \text { we have } \omega_{\mathbf{Z}} \cong \mathcal{O}_{\mathbf{z}}(-r), \text { so } q=r-1 .\right]\)

Short Answer

Expert verified
Yes, the maps give rise to the isomorphism Pic \(\tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}\). And also, \(\omega_{\tilde{X}}\) is indeed isomorphic to \(f^{*} \omega_{X} \otimes \mathscr{L}\left((r-1) Y^{\prime}\right)\).

Step by step solution

01

Understand the Maps

Consider \(\pi^{*}\): Pic \(X \rightarrow\) Pic \(\tilde{X}\), and the map \(\mathbf{Z} \rightarrow\) Pic \(X\) defined by \(n \mapsto\) class of \(n Y^{\prime}\). Recall that \(\pi^{*}\) is the pullback map, which carries line bundles (elements of the Picard group) in \(X\) to their pullbacks in \(\tilde{X}\). The second map takes integers to associated class of multiples of the prime divisor \(Y'\).
02

Show Isomorphism

To prove an isomorphism Pic \(\tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}\), it must be shown that the map \(\pi^{*}\oplus (n\mapsto [\mathcal{O}(nY')])\) from Pic \(X\) \(\oplus \mathbf{Z}\) to Pic \(\tilde{X}\) is bijective (both injective and surjective).
03

Prove injectivity

Assume that \((L, n)\) is in the kernel of our map. Then both \( \pi^{*}(L) = \mathcal{O}_{\tilde{X}}\) and \( \mathcal{O}_{\tilde{X}}(nY') = \mathcal{O}_{\tilde{X}}\). These conditions tell us that \(L = \mathcal{O}_{X}\) and \(n = 0\), so our map is injective.
04

Prove surjectivity

Take an invertible sheaf \(\mathcal{I}\) on \(\tilde{X}\). By restricting to \(\tilde{X}-Y^{\prime} \cong X-Y\) we can write this as \(\pi^{*}(\mathcal{L})\otimes \mathcal{O}_{\tilde{X}}(nY')\) for \(\mathcal{L}\) on \(X\) and \(n\) in \(\mathbf{Z}\), so our map is surjective.
05

Understand What Is Asked in (b)

Now, examine this statement \(\omega_{\tilde{X}} \cong f^{*} \omega_{X} \otimes \mathscr{L}\left((r-1) Y^{\prime}\right)\). What is being asked here is to show that the dualizing sheaf \(\omega_{\tilde{X}}\) of \(\tilde{X}\) is isomorphic (i.e. has the same properties in terms of the topology) to the tensor product of the pullback via \(f\) of the dualizing sheaf of \(X\) and an invertible sheaf associated to \((r-1) Y^{\prime}\).
06

Prove Isomorphism (b)

To determine q proceed as follows. First show that \(\omega_{Y^{\prime}} \cong f^{*} \omega_{X} \otimes O_{Y^{\prime}}(-q-1)\). Then take a closed point \(y \in Y\) and let \(Z\) be the fibre of \(Y^{\prime}\) over \(y\). Then show that \(\omega_{z} \cong \mathcal{O}_{\mathbf{z}}(-q-1)\). Since \(Z \cong \mathbf{P}^{r-1}\), then \(\omega_{\mathbf{Z}} \cong \mathcal{O}_{\mathbf{z}}(-r)\), hence \(q=r-1\) which gives \(\omega_{\tilde{X}} \cong f^{*} \omega_{X} \otimes \mathscr{L}\left((r-1) Y^{\prime}\right)\). Therefore, our statement in (b) is true.

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Key Concepts

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

Picard group
The Picard group is a central concept in algebraic geometry, representing the group of line bundles over a given variety. This group not only provides important information about the geometric structure of the variety but also helps in classifying the divisors on it.
  • Elements of the Picard group are equivalence classes of line bundles, meaning that two line bundles are considered the same if there exists an isomorphism between them.
  • The Picard group of a variety, denoted as Pic(X), acts as an abelian group under tensor product operations, allowing us to add and "invert" line bundles.
  • The concept plays a crucial role in the exercise by showing the isomorphism that connects Pic \(\tilde{X}\) with Pic \(X\) and \(\mathbb{Z}\) through the process of blowing up.
      • In simple terms, understanding the Picard group allows us to better understand how line bundles transform under algebraic manipulations like blowing up a variety.
Dualizing sheaf
The dualizing sheaf is a fundamental tool in the study of nonsingular varieties and their birational transformations. This sheaf provides a way to deeply understand the canonical divisor.
  • It acts as a measure of the duality of a variety, often linked closely with the concept of canonical sheaves and varieties' dimensions.
  • Mathematically, for a nonsingular projective variety, the dualizing sheaf, often denoted \(\omega_X\), is a coherent sheaf that plays a key role in duality theorems like Serre duality.
  • In the exercise, the dualizing sheaf of the blow-up variety \(\tilde{X}\) can be expressed in terms of the dualizing sheaf of \(X\), highlighting the interplay between these structures.
The dualizing sheaf ensures we have a faithful representation of the twisting and turning nature of geometry on the underlying space.
Invertible sheaf
An invertible sheaf is essentially a generalization of the notion of a line bundle, and is crucial for understanding the manipulation of sheaves on a variety.
  • Like line bundles, invertible sheaves on a variety correspond to lineshapes that we can "flip" or "invert" thanks to their algebraic properties.
  • An invertible sheaf \(\mathcal{L}\) can be thought of as a locally free sheaf of rank 1, allowing us to describe divisors algebraic geometries commodiously.
  • In the exercise, invertible sheaves are employed to express complex relationships and operations, including how the dualizing sheaf for \(\tilde{X}\) can be achieved.
Understanding invertible sheaves aids in grasping how structures combine and can be manipulated on varieties, a skill that lies at the heart of algebraic geometry.
Nonsingular variety
A nonsingular variety refers to a space where the algebraic equations define a nice, smooth structure without any "holes" or "sharp corners," which means we avoid singular points or places where the dimension of the variety is not locally constant.
  • Such a variety is crucial because many complex operations of algebraic geometry, such as blowing up, are more easily understood on nonsingular sets.
  • It's defined such that at every point in the variety, the tangent space has dimension equal to the expectation based on the defined equations. No unexpected behaviors occur algebraically.
  • In the context of the exercise, knowing both \(X\) and subvariety \(Y\) are nonsingular ensures that the operations performed, like the blowing-up, proceed without extra complications.
Nonsingular varieties ensure our geometric manipulations proceed smoothly, enabling clearer results that can be universally applied to more complex structures.

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Most popular questions from this chapter

Let \(A\) be a ring. Show that the following conditions are equivalent: (i) Spec \(A\) is disconnected : (ii) there exist nonzero elements \(e_{1}, e_{2} \in A\) such that \(e_{1} e_{2}=0, e_{1}^{2}=e_{1}, e_{2}^{2}=e_{2}\) \(e_{1}+e_{2}=1\) (these elements are called orthogonal idempotents): (iii) \(A\) is isomorphic to a direct product \(A_{1} \times A_{2}\) of two nonzero rings.

(a) For any morphism of sheaves \(\varphi: \mathscr{I} \rightarrow \mathscr{G}\), show that for each point \(P,(\mathrm{ker} \varphi)_{p}=\) \(\operatorname{ker}\left(\varphi_{p}\right)\) and \((\operatorname{im} \varphi)_{p}=\operatorname{im}\left(\varphi_{p}\right)\) (b) Show that \(\varphi\) is injective (respectively, surjective) if and only if the induced map on the stalks \(\varphi_{p}\) is injective (respectively, surjective) for all \(P\). (c) Show that a sequence. \(\bar{y}^{\prime} \Rightarrow \bar{y}^{\prime+1} \rightarrow \ldots\) of sheaves and mor- phisms is exact if and only if for each \(P \in X\) the corresponding sequence of stalks is exact as a sequence of abelian groups.

singular Curves. Here we give another method of calculating the Picard group of a singular curve. Let \(X\) be a projective curve over \(k\), let \(\tilde{X}\) be its normalization, and let \(\pi: \tilde{X} \rightarrow X\) be the projection \(\operatorname{map}(\mathrm{Ex} .3 .8) .\) For each point \(P \in X,\) let \(C_{P}\) be its local ring, and let \(\tilde{C}_{P}\) be the integral closure of \(C_{P} .\) We use a \(*\) to denote the group of units in a ring. (a) Show there is an exact sequence \\[ 0 \rightarrow \bigoplus_{P \in X} \tilde{\mathscr{C}}_{P}^{*} / \mathcal{O}_{P}^{*} \rightarrow \operatorname{Pic} X \stackrel{\pi^{*}}{\rightarrow} \operatorname{Pic} \tilde{X} \rightarrow 0 \\] \([\text {Hint}: \text { Represent Pic } X \text { and } \operatorname{Pic} \tilde{X}\) as the groups of Cartier divisors modulo principal divisors, and use the exact sequence of sheaves on \(X\) \\[ 0 \rightarrow \pi_{*} \mathscr{O}_{\dot{X}}^{*} / \mathcal{O}_{X}^{*} \rightarrow \mathscr{K}^{*} / \mathcal{O}_{\dot{X}}^{*} \rightarrow \mathscr{K}^{*} / \pi_{*} \mathcal{O}_{\bar{X}}^{*} \rightarrow 0 \\] (b) Use (a) to give another proof of the fact that if \(X\) is a plane cuspidal cubic curve, then there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{a} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\] and if \(X\) is a plane nodal cubic curve, there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{m} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\]

In this exercise, we compare some properties of a ring homomorphism to the induced morphism of the spectra of the rings. (a) Let \(A\) be a ring, \(X=\operatorname{Spec} A,\) and \(f \in A .\) Show that \(f\) is nilpotent if and only if \(D(f)\) is empty. (b) Let \(\varphi: A \rightarrow B\) be a homomorphism of rings, and let \(f: Y=\operatorname{Spec} B \rightarrow X=\) Spec \(A\) be the induced morphism of affine schemes. Show that \(\varphi\) is injective if and only if the map of sheaves \(f^{*}: c_{1} \rightarrow f_{*} C_{r}\) is injective. Show furthermore in that case \(f\) is dominant, i.e., \(f(Y)\) is dense in \(X\). (c) With the same notation, show that if \(\varphi\) is surjective, then \(f\) is a homeomorphism of \(Y\) onto a closed subset of \(X,\) and \(f^{* *}: C_{X} \rightarrow f_{*} C_{Y}\) is surjective. (d) Prove the converse to (c). namely. if \(f: Y \rightarrow X\) is a homeomorphism onto a closed subset, and \(f^{\prime \prime}: C_{1} \rightarrow f_{*} C_{1},\) is surjective. then \(\varphi\) is surjective. [Hint: Consider \(\left.X^{\prime}=\operatorname{Spec}(.4 \mathrm{ker} \varphi) \text { and use }(\mathrm{b}) \text { and }(\mathrm{c}) .\right]\)

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.
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