Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Example 20.2.2 (continued). Now, using the fact that \( {H}^{1}\left( {\mathcal{N}}_{X}\right) = \) \( {H}^{1}\left( {{\mathcal{O}}_{X}\left( d\right) }\right) = 0 \) for any surface in \( {\mathbb{P}}^{3} \), we can construct the following table for the dimensions of the groups of (20.2): | <table><thead><tr><th>\( d \)</th><th>\( {h}^{0}\left( {T}_{X}\right) \)</th><th>\( {h}^{0}\left( {{T}_{{\mathbb{P}}^{3}}{|}_{X}}\right) \)</th><th>\( {h}^{0}\left( {\mathcal{N}}_{X}\right) \)</th><th>\( {h}^{1}\left( {T}_{X}\right) \)</th><th>\( {h}^{1}\left( {T}_{{\mathbb{P}}^{3}}\middle| {}_{X}\right) \)</th></tr></... | Yes |
We examine more closely the case of a nonsingular surface of degree 4 in \( {\mathbb{P}}^{3} \), which is a \( {K3} \) surface. The functor of embedded deformations is unobstructed, since \( {h}^{1}\left( {\mathcal{N}}_{X}\right) = 0 \) as noted above. Also the functor \( {F}_{1} \) is pro-representable, so the univers... | Computing the image step by step, we see that the image, which corresponds to abstract deformations that lift to embedded deformations, is a smooth subspace of \( \operatorname{Spec}{R}_{2} \) of dimension 19. In particular, there are abstract deformations of \( {X}_{0} \) that cannot be realized as embedded deformatio... | Yes |
Using (20.3.1), we can give an example of an obstructed deformation of a line bundle. Let \( {X}_{0} \) be a nonsingular quartic surface in \( {\mathbb{P}}^{3} \). Let \( X \) be a deformation over the dual numbers \( D \) that does not lift to \( {\mathbb{P}}^{3} \). Let \( {\mathcal{L}}_{0} \) be the invertible sheaf... | For suppose it did lift to an invertible sheaf \( \mathcal{L} \) on \( X \). Then the exact sequence\n\n\[ 0 \rightarrow {\mathcal{L}}_{0} \rightarrow \mathcal{L} \rightarrow {\mathcal{L}}_{0} \rightarrow 0 \]\n\nand \( {H}^{1}\left( {{\mathcal{O}}_{{X}_{0}}\left( 1\right) }\right) = 0 \) would show that the sections \... | Yes |
Proposition 20.4. Let \( X \) be a nonsingular quartic surface in \( {\mathbb{P}}^{3} \) over a field \( k \) of characteristic 0 . Then for any nontrivial line bundle \( \mathcal{L} \) on \( X \), there is an abstract deformation \( {X}^{\prime } \) of \( X \) over the dual numbers to which \( \mathcal{L} \) does not ... | Proof. Recall (Ex. 10.6) that deformations of the pair \( \left( {X,\mathcal{L}}\right) \) are given by \( {H}^{1}\left( {X,{\mathcal{P}}_{\mathcal{L}}}\right) \) and that there is an exact sequence\n\n\[ \cdots \rightarrow {H}^{1}\left( {\mathcal{O}}_{X}\right) \rightarrow {H}^{1}\left( {\mathcal{P}}_{\mathcal{L}}\rig... | "No" |
Lemma 20.5. Let \( X \) be a nonsingular surface in \( {\mathbb{P}}^{3} \) over a field \( k \) of characteristic 0 . Let \( \mathcal{L} \) be a nontrivial line bundle. Then \( c\left( \mathcal{L}\right) \in {H}^{1}\left( {X,{\Omega }_{X}^{1}}\right) \) is not zero. | Proof. The formation of the cohomology class is compatible with intersection theory on the surface \( \left\lbrack {{57},\mathrm{\;V},\mathrm{{Ex}}.{1.8}}\right\rbrack \), in the sense that for any two divisor classes \( D, E \) on \( X \), the cup product of the cohomology classes \( c\left( {{\mathcal{O}}_{X}\left( D... | Yes |
Lemma 20.7. Let \( X \) be a nonsingular surface in \( {\mathbb{P}}^{3} \) (of any degree), and let \( C \) be an effective Cartier divisor on \( X \) having the same degree and genus as a complete intersection of \( X \) with another surface. Then \( C \) itself is a complete intersection of \( X \) with another surfa... | Proof. Let \( H \) be a hyperplane class, and let \( D \) be a complete intersection curve having the same degree and genus as \( C \) . Since \( C \) and \( D \) have the same degree, \( C.H = D.H \) . Furthermore, since \( D \sim {mH} \) for some \( H \), this implies \( C.D = {D}^{2} \) . Also the canonical class \(... | Yes |
Proposition 21.1. Let \( R,\mathfrak{m} \) be a complete local ring with residue field \( k \), and suppose we are given a formal family of deformations of \( {X}_{0} \) over \( R \), that is, for each \( n \), schemes \( {X}_{n} \) flat and of finite type over \( {R}_{n} = R/{\mathfrak{m}}^{n + 1} \) and maps \( {X}_{... | Proof. We define \( \mathcal{X} \) to be the locally ringed space formed by taking the topological space \( {X}_{0} \), together with the sheaf of rings \( {\mathcal{O}}_{\mathcal{X}} = \underline{\lim }{\mathcal{O}}_{{X}_{n}}\lbrack {57},\mathrm{{II}} , 9.2]. To show that \( \mathcal{X} \) is a noetherian formal schem... | Yes |
Theorem 21.2 (Grothendieck). Let \( \mathcal{X} \) be a formal scheme, proper over \( \operatorname{Spf}R \), where \( R,\mathfrak{m} \) is a complete local ring, and suppose there exists an invertible sheaf \( \mathcal{L} \) on \( \mathcal{X} \) such that \( {\mathcal{L}}_{0} = \mathcal{L}{ \otimes }_{R}k \) is ample ... | Proof. We refer to \( \left\lbrack {{48}\text{, III,}{5.4.5}}\right\rbrack \) for the proof. | No |
Example 21.2.1 (A noneffective formal deformation). This example shows that in the theorem, it is not enough to assume \( {X}_{0} \) projective. It must admit an ample invertible sheaf that lifts to the formal scheme \( \mathcal{X} \) . | Let \( {X}_{0} \) be a nonsingular quartic surface in \( {\mathbb{P}}^{3} \) over a field of characteristic zero. Then we have seen (20.4) that for any nonzero divisor \( D \) on \( {X}_{0} \), there is some deformation of \( X \), already over the dual numbers, to which \( L = {\mathcal{O}}_{X}\left( D\right) \) does ... | Yes |
In Situation A we start with a closed subscheme \( {X}_{0} \subseteq {\mathbb{P}}_{k}^{n} \). The Hilbert functor of deformations of \( {X}_{0} \) as a closed subscheme of \( {\mathbb{P}}^{n} \) is pro-representable (17.1). | Then by (21.1) we obtain a formal scheme \( \mathcal{X} \subseteq {\widehat{\mathbb{P}}}_{R}^{n} \). It is projective by construction, so (21.2) applies, and the formal family is effective. | No |
In Situation B we start with an invertible sheaf \( {\mathcal{L}}_{0} \) on a scheme \( {X}_{0} \) . Assuming \( {X}_{0} \) projective and \( {H}^{0}\left( {\mathcal{O}}_{{X}_{0}}\right) = k \), the local Picard functor of deformations of \( {\mathcal{L}}_{0} \) on \( {X}_{0} \) is pro-representable (17.2), so by (21.1... | We use the theorem of Grothendieck [48, III,5.1.4], which says that if \( X \) is a scheme, projective over a complete local ring \( R \), and if \( \widehat{X} \) is the completion along the closed fiber, then the functor \( \mathcal{F} \mapsto \widehat{\mathcal{F}} \) is an equivalence of the category of coherent she... | Yes |
In Situation D, suppose we start with an (abstract) projective scheme \( {X}_{0} \) . Then the local deformation functor has a miniversal family (18.1), which is universal if \( {H}^{0}\left( {{X}_{0},{\mathcal{T}}_{{X}_{0}}}\right) = 0 \) (18.3). Since \( {X}_{0} \) is projective, there is an ample invertible sheaf \(... | In this case the formal family of deformations of \( {X}_{0} \) is effective, as in the case of (21.2.1). This applies in particular to any projective curve, since then there is no \( {H}^{2} \) . If \( {X}_{0} \) is a nonsingular projective surface, then \( {H}^{2}\left( {\mathcal{O}}_{{X}_{0}}\right) \) is dual to \(... | Yes |
Theorem 22.1. Let \( {X}_{0} \) be a nonsingular projective variety over a perfect field \( k \) of characteristic \( p > 0 \) . Assume that \( {H}^{2}\left( {{X}_{0},{\mathcal{O}}_{{X}_{0}}}\right) = 0 \) and \( {H}^{2}\left( {{X}_{0},{\mathcal{I}}_{{X}_{0}}}\right) = 0 \) . Let \( \left( {R,\mathfrak{m}}\right) \) be... | Proof. Since \( {H}^{2}\left( {{X}_{0},{\mathcal{T}}_{{X}_{0}}}\right) = 0 \), the obstructions to infinitesimal lifting are zero, so we obtain a compatible sequence of liftings \( {X}_{n} \) flat over \( {R}_{n} \) . Their limit gives a noetherian formal scheme \( \mathcal{X} \) by (21.1). Since \( {X}_{0} \) is assum... | Yes |
Theorem 22.3. Let \( {X}_{0} \) be a closed subscheme of \( {\mathbb{P}}_{{k}_{0}}^{r} \), with \( {k}_{0} \) a perfect field of characteristic \( p \) . Assume that \( {X}_{0} \) is locally unobstructed (e.g., \( {X}_{0} \) is a locally complete intersection (9.2), or \( {X}_{0} \) is locally Cohen-Macaulay of codimen... | Proof. The method is already contained in the proof of the previous theorem. Because of \( {H}^{1}\left( {{X}_{0},{\mathcal{N}}_{0}}\right) = 0 \), one can lift \( {X}_{0} \) stepwise to a sequence of closed subschemes \( {X}_{n} \) of \( {\mathbb{P}}^{r} \), flat over \( {R}_{n} \) . The limit of these is a projective... | Yes |
Theorem 22.4 (Serre). Over an algebraically closed field \( k \) of characteristic \( p \geq 5 \), there is a nonsingular projective 3 -fold \( Z \) that cannot be lifted to characteristic 0 , even in the weak sense. | Proof. Let \( k \) be algebraically closed of characteristic \( p \geq 5 \) . Let \( r \geq 5 \), and let \( G = {\left( \mathbb{Z}/p\right) }^{r} \) . Then \( G \) is a finite abelian group, and by choosing elements \( {e}_{1},\ldots ,{e}_{r} \in k \) that are linearly independent over \( {\mathbb{F}}_{p} \), we can f... | No |
Proposition 22.5. Let \( Z \) be a scheme over a field \( k \), let \( Y \rightarrow Z \) be a finite étale cover, let \( R \) be a complete local ring with residue field \( k \), and suppose there exists a scheme \( {Z}^{\prime } \), flat over \( R \), with \( {Z}^{\prime }{ \times }_{R}k = Z \) . Then there is a fini... | Proof. By definition \( Y \rightarrow Z \) is a finite, affine, smooth morphism of relative dimension zero. As we showed in (4.11), for any smooth ring extension \( A \rightarrow B \) , the functors \( {T}^{i}\left( {B/A, M}\right) \) are 0 for \( i = 1,2 \) and for all \( M \) . If \( A \rightarrow B \) is étale, then... | Yes |
Let \( \mathcal{M} \) be the set of nonsingular projective curves of genus 0 over \( k \), up to isomorphism. Then \( \mathcal{M} \) has a single element, namely \( {\mathbb{P}}^{1} \). | We will see (25.1) that \( M = \operatorname{Spec}k \) is a coarse moduli space, and the trivial family \( {\mathbb{P}}^{1}/k \) becomes a tautological family. | No |
Proposition 23.1. If the functor \( \mathcal{F} \) associated to a moduli problem \( \mathcal{M} \) is representable by a scheme \( M \), then \( M \) is also a coarse moduli scheme for \( \mathcal{F} \) , and the universal family \( {X}_{u}/M \) is a tautological family. | Proof. Since \( \mathcal{F} \cong {h}_{M} \), we take the isomorphism as our morphism of functors \( \varphi : \mathcal{F} \rightarrow {h}_{M} \) . Property (a) of the definition, namely \( \mathcal{F}\left( k\right) \rightarrow {h}_{M}\left( k\right) \) bijective, follows from the isomorphism. We have only to check th... | No |
Proposition 23.2. Let \( M \) be a fine moduli scheme for the moduli problem \( \mathcal{M} \), and let \( {X}_{0} \in \mathcal{M} \) correspond to a point \( {x}_{0} \in M \) . Then the Zariski tangent space to \( M \) at \( {x}_{0} \) is in one-to-one correspondence with the set of families \( X \) over the dual numb... | Proof. Indeed, the Zariski tangent space to \( M \) at \( {x}_{0} \) can be identified with \( {\operatorname{Hom}}_{{x}_{0}}\left( {D, M}\right) ,\left\lbrack {{57},\mathrm{{II}},\mathrm{{Ex}}.{2.8}}\right\rbrack \), and this in turn corresponds to the subset of those elements of \( \mathcal{F}\left( D\right) \) restr... | No |
Proposition 23.3. Let \( \mathcal{F} \) be the functor associated to a moduli problem \( \mathcal{M} \), let \( {X}_{0} \in \mathcal{M} \), and consider the functor on Artin rings \( {\mathcal{F}}_{0} \) that to each local Artin ring \( A \) over \( k \) assigns the set of families of elements of \( \mathcal{M} \) over... | Proof. Let \( M \) be a fine moduli scheme for \( \mathcal{M} \), let \( {x}_{0} \in M \) correspond to \( {X}_{0} \in \mathcal{M} \), and let \( R \) be the completion of the local ring of \( {x}_{0} \) on \( M \) . Since \( M \) is a fine moduli space, each element of \( {\mathcal{F}}_{0}\left( A\right) \) correspond... | Yes |
Corollary 23.4. Let \( \mathcal{F} \) be a representable functor, represented by a scheme \( M \), and let \( {x}_{0} \in M \) be a point. If we have an obstruction theory for the local functor \( {\mathcal{F}}_{0} \), then knowing its tangent space \( {t}_{0} \) and its obstruction space \( {V}_{0} \) , the dimension ... | Proof. Just apply (11.2) to the local ring of \( {x}_{0} \) on \( M \) . | No |
Proposition 23.5. If the moduli problem \( \mathcal{M} \) has a fine moduli space, then the associated functor \( \mathcal{F} \) is a sheaf for the Zariski topology. | Proof. Indeed, if \( \mathcal{F} = {h}_{M} \), then for any scheme \( S,\mathcal{F}\left( S\right) = \operatorname{Hom}\left( {S, M}\right) \) , and one knows that morphisms from one scheme to another are determined locally, and can be glued together if they are given locally and are compatible on overlaps [57, II.3.3,... | No |
Theorem 24.1. The Hilbert functor, which to every scheme \( S/k \) associates the set of subschemes \( Y \subseteq {\mathbb{P}}_{S}^{N} \), flat over \( S \), whose fibers all have a given Hilbert polynomial \( P \), is representable by a scheme \( M \), projective over \( k \) . | We have already stated this theorem, in different words, as (1.1a). A complete proof can be found in the article of Nitsure [124]. | No |
Proposition 24.2. If the functor \( \mathcal{F} \) is represented by a scheme \( M \) of finite type over \( k \), then \( \mathcal{F} \) is bounded. In that case the scheme \( M \) is separated (resp., proper over \( k \) ) if and only if \( \mathcal{F} \) is separated (resp., separated and complete). | Proof. Left to reader as (Ex. 24.1). | No |
Proposition 24.3 (Mumford). If \( \mathcal{F} \) is \( m \) -regular, then \( \mathcal{F} \) is also \( {m}^{\prime } \) -regular for all \( {m}^{\prime } \geq m \) . Furthermore, \( \mathcal{F}\left( m\right) \) is generated by global sections. | Proof. \( \left\lbrack {115}\right\rbrack \) or \( \left\lbrack {{124},{5.1}}\right\rbrack \) . | No |
Proposition 24.4. A family \( \mathcal{M} \) of coherent sheaves on a projective scheme \( X/k \), all having the same Hilbert polynomial, is bounded (meaning the functor of flat families of sheaves in \( \mathcal{M} \) is bounded) if and only if there is a uniform \( {m}_{0} \) such that all members of \( \mathcal{M} ... | Proof (in outline). One direction is easy. If \( \mathcal{M} \) is a bounded family, then there is a scheme \( S \) of finite type over \( k \) together with a coherent sheaf \( \mathcal{F} \) on \( X \times S \), flat over \( S \), containing among its fibers at closed points of \( S \) all elements of \( \mathcal{M} ... | Yes |
Proposition 24.5. The set of subschemes \( Y \) of \( {\mathbb{P}}_{k}^{n} \) with Hilbert polynomial \( P \) forms a bounded family. | Proof. Using the previous proposition it is enough to show that there is a uniform \( {m}_{0} \) such that the ideal sheaves \( {\mathcal{I}}_{Y} \) of all such \( Y \) in \( {\mathbb{P}}_{k}^{n} \) are \( {m}_{0} \) -regular. This is accomplished by induction on \( \dim Y \), using a generic hyperplane section. See \(... | No |
Theorem 24.6. With the above hypotheses, assume furthermore that \( X \) is integral and projective. Then the functor \( {\operatorname{Pic}}_{X/k, P} \) is represented by a separated scheme, locally of finite type over \( k \), which we call the Picard scheme of \( X/k \) . | Proof. See [79, 9.4.8]. | No |
Theorem 24.7. The functor \( \mathcal{F} \) is represented by a scheme, projective over \( k \) , which we call the Hilbert-flag scheme. | Proof. One can deduce this from the existence of the relative Hilbert scheme: First let \( H \) be the Hilbert scheme associated to the Hilbert polynomial \( Q \), with universal family \( {Z}_{u}/H \) . Then take the Hilbert scheme of relative subschemes \( Y \subseteq {Z}_{u} \times S/H \times S \) with Hilbert polyn... | No |
Lemma 24.8. To give a deformation of a morphism \( f : X \rightarrow Y \) (keeping \( X \) and \( Y \) fixed) it is equivalent to give a deformation of the graph \( {\Gamma }_{f} \) as a closed subscheme of \( X \times Y \) . | Proof. To any deformation \( {f}^{\prime } \) of \( f \) we associate its graph \( {\Gamma }_{{f}^{\prime }} \), which will be a closed subscheme of \( X \times Y \times A \) . It is a deformation of \( {\Gamma }_{f} \) . Conversely, given a deformation \( Z \) of \( {\Gamma }_{f} \) over \( A \), we need only verify t... | Yes |
Proposition 24.9. Assume that \( Y \) is nonsingular. Then the tangent space to the deformation functor of \( f : X \rightarrow Y \) (keeping \( X \) and \( Y \) fixed) is \( {H}^{0}\left( {X,{f}^{ * }{T}_{Y}}\right) \), and the obstructions to deforming \( f \) lie in \( {H}^{1}\left( {X,{f}^{ * }{T}_{Y}}\right) \). I... | Proof. From (24.8) we must consider the deformations of \( {\Gamma }_{f} \) as a closed subscheme of \( X \times Y \). Note that \( {\Gamma }_{f} = {\left( f \times \mathrm{{id}}\right) }^{-1}{\Delta }_{Y} \), where \( {\Delta }_{Y} \subseteq Y \times Y \) is the diagonal. Since \( Y \) is nonsingular, \( {\Delta }_{Y}... | Yes |
Theorem 24.10. Given \( X, Y \) projective schemes over \( k \), the global functor of families of morphisms \( f : X \times S \rightarrow Y \times S \) over a scheme \( S \) is represented by a union of quasi-projective schemes over \( k \) . | Proof. This follows from the existence of the Hilbert scheme of closed sub-schemes of \( X \times Y \) (24.1), and the observation that the set of subschemes \( Z \) representing graphs of morphisms is an open subset of the Hilbert scheme. There will be different quasi-projective components depending on the Hilbert pol... | Yes |
Proposition 25.1. The one-point space \( M = \operatorname{Spec}k \) is a coarse moduli scheme for curves of genus 0 , and it has a tautological family. | Proof. The first condition (a) for a coarse moduli scheme (§23) is satisfied because the one point of \( M \) corresponds to the one curve \( {\mathbb{P}}_{k}^{1} \) . We can also see right away that there is a tautological family: just take \( {\mathbb{P}}_{k}^{1}/\operatorname{Spec}k \) . For any family \( X/S \) of ... | Yes |
Lemma 25.2. If \( C \) is an Artin ring with residue field \( k \) algebraically closed, then any family \( X/\operatorname{Spec}C \) of curves of genus 0 is trivial, namely isomorphic to \( {\mathbb{P}}_{\operatorname{Spec}C}^{1} \) . | Proof. Since \( k \) is algebraically closed, the special fiber \( {X}_{0} \) is just \( {\mathbb{P}}_{k}^{1} \) . Then by our infinitesimal study of deformations (10.3) the choices at each step are given by \( {H}^{1}\left( {{X}_{0},{\mathcal{T}}_{{X}_{0}}}\right) = 0 \) . Thus at each step there is a unique deformati... | No |
Here we show that the one-point space \( M \) is not a fine moduli space for curves of genus 0. | Just think of the theory of ruled surfaces. A ruled surface is a nonsingular projective surface \( X \) together with a morphism \( \pi \) to a nonsingular projective curve \( C \) whose fibers are copies of \( {\mathbb{P}}^{1} \) and that has a section, and therefore is isomorphic to \( \mathbb{P}\left( \mathcal{E}\ri... | Yes |
Here we show that families of curves of genus 0 need not even be locally trivial. Let \( A = k\left\lbrack {t, u}\right\rbrack \), and consider the curve in \( {\mathbb{P}}_{A}^{2} \) defined by \( t{x}^{2} + u{y}^{2} + {z}^{2} = 0 \). We take \( S = \operatorname{Spec}A - \{ {tu} = 0\} \), and take \( X \) to be this ... | A rational point would be given by taking \( x = f\left( {t, u}\right), y = g\left( {t, u}\right), z = h\left( {t, u}\right) \), where \( f, g, h \) are rational functions in \( t \) and \( u \), not all zero, satisfying the above equation. Clearing denominators, we may assume that \( f, g, h \) are polynomials. Then, ... | Yes |
Proposition 25.3. Any family \( X/S \) of pointed curves of genus 0 is locally trivial, that is, every point \( s \in S \) has an open neighborhood \( U \) such that \( {\pi }^{-1}\left( U\right) \cong {\mathbb{P}}_{U}^{1} \) . In particular, a pointed curve of genus 0 over any field \( k \) (not necessarily algebraica... | Proof. (Cf. [57, V,2.2] for a special case.) Given the family \( \pi : X \rightarrow S \) and the section \( \sigma : S \rightarrow X \), we let \( D \) be the scheme-theoretic image of \( \sigma \) . Then \( D \) is flat over \( S \), and its restriction to any fiber is one point, so \( D \) is a Cartier divisor on \(... | Yes |
Proposition 26.2. The functor \( F \) does not have a fine moduli space. | Proof. There are several reasons one can give for this. One is that the crude local functor \( {F}_{1} \) of local families \( C/A \) such that \( C{ \otimes }_{A}k \cong {C}_{0} \), but without specifying the inclusion \( {C}_{0} \subseteq C \), is not pro-representable (18.4.2). We have seen that this would be a nece... | Yes |
Corollary 26.5. If \( Y/T \) is a fiberwise trivial family of elliptic curves, then there exists a finite étale map \( {T}^{\prime } \rightarrow T \) such that the base extension \( {Y}^{\prime }/{T}^{\prime } \) is isomorphic to the trivial family. In other words, the family \( Y/T \) is isotrivial (Ex. 25.1). | Proof. Indeed, let \( X/S \) be a modular family. Then there is a surjective étale morphism \( {T}^{\prime } \rightarrow T \) together with a morphism \( {T}^{\prime } \rightarrow S \) such that the extended families over \( {T}^{\prime } \) are isomorphic. But \( Y/T \) is fiberwise trivial, so the image of \( {T}^{\p... | Yes |
Proposition 26.6. If \( X/S \) is a modular family of elliptic curves, the corresponding map of \( S \) to the coarse moduli space \( {\mathbb{A}}_{j} \) is étale over points where \( j \neq 0,{12}^{3} \); ramified of order 2 over \( j = {12}^{3} \) and ramified of order 3 over \( j = 0 \) . | Proof. Writing\n\n\[ j = {256}\frac{{\left( \lambda + \omega \right) }^{3}{\left( \lambda + {\omega }^{2}\right) }^{3}}{{\lambda }^{2}{\left( \lambda - 1\right) }^{2}} \]\n\nwhere \( {\omega }^{3} = 1 \), shows that at \( \lambda = - \omega \), corresponding to \( j = 0 \), the map from the \( \lambda \) -line to the \... | Yes |
Theorem 27.1. The moduli functor \( \mathcal{F} \) of curves of genus \( g \geq 2 \) over \( k \) algebraically closed has a coarse moduli space \( {M}_{g} \), which is a normal quasi-projective variety of dimension \( {3g} - 3 \) having at most quotient singularities. | The existence and the fact that it is quasi-projective are proved in Mumford's book [119]; the irreducibility is proved in the article of Deligne and Mumford [21]. Fulton [35] improved the proof of Deligne and Mumford, making it purely algebraic. | Yes |
Theorem 27.2. For any \( g \geq 2 \), the class of nonsingular projective curves of genus \( g \) over \( k \) has a modular family. | Proof. On a curve of genus \( g \), any divisor of degree \( \geq {2g} + 1 \) is nonspecial and very ample. In particular, if we take the tricanonical divisor \( {3K} \), where \( K \) is the canonical divisor, then for any \( g \geq 2 \), its degree \( d = {6g} - 6 \) is \( > {2g} + 1 \) , so we can use it to embed th... | No |
Theorem 27.3. A general curve of genus \( g > {2d} - 2 \) does not have a \( {g}_{d}^{1} \) . | Proof. First of all, to explain the word \ | No |
Lemma 27.5. Let \( X \) be a reduced curve with at most nodes as singularities, and let \( Z \) denote the set of nodes (with the reduced induced scheme structure). Then there are natural exact sequences\n\n\[ 0 \rightarrow {\mathcal{O}}_{Z} \rightarrow {\Omega }_{X}^{1} \rightarrow {\omega }_{X} \rightarrow {\mathcal{... | Proof. The second sequence follows by dualizing the first, so we have only to prove the first. To construct the natural map \( {\Omega }_{X}^{1} \rightarrow {\omega }_{X} \), we embed \( X \) in a nonsingular variety \( P \) (such as projective space), with ideal sheaf \( \mathcal{I} \) . Then there is an exact sequenc... | Yes |
Lemma 27.6. If \( X \) is a reduced curve that is a union \( X = C \cup D \) of two curves \( C \) and \( D \) meeting transversally at a finite set of nodes \( S \), then \( {T}_{X} \cong \) \( \left( {{\mathcal{I}}_{S, C} \otimes {T}_{C}}\right) \oplus \left( {{\mathcal{I}}_{S, D} \otimes {T}_{D}}\right) | Proof. As before, the question is local around each node, so we take \( X \) to be the curve \( {xy} = 0 \) in \( {\mathbb{A}}^{2} \), with \( C = \operatorname{Spec}k\left\lbrack x\right\rbrack \) and \( D = \operatorname{Spec}k\left\lbrack y\right\rbrack \) . Now \( {T}_{X} = \operatorname{Hom}\left( {{\Omega }_{X}^{... | Yes |
Proposition 27.7. Let \( X \) be a stable curve of (arithmetic) genus \( g \geq 2 \). Then the functor of local infinitesimal deformations of \( X \) is pro-representable by a regular complete local ring of dimension \( {3g} - 3 \). | Proof. The functor is pro-representable because \( X \) is projective and satisfies the critical condition \( {H}^{0}\left( {X,{T}_{X}}\right) = 0\left( {18.3}\right) \). The local ring representing it is regular because there are no obstructions to deforming \( X \) (Ex. 10.4). Since the ring is regular, to find its d... | No |
Lemma 28.1. For a bundle \( \mathcal{E} \) on a curve \( X \) , (a) \( \mathcal{E} \) stable \( \Rightarrow \mathcal{E} \) semistable. | Proof. (a) is trivial. | No |
Corollary 28.3. The families of stable bundles or simple bundles of given rank and degree are bounded. | Proof. Follows from (28.1) and (28.2). | No |
Lemma 28.5. Let \( S \) be a scheme of finite type over \( k \), let \( {\mathcal{E}}_{1} \) and \( {\mathcal{E}}_{2} \) be two families of vector bundles of rank \( r \) and degree \( d \) on \( X \times S \), together with isomorphisms \( {\theta }_{i} : {\sigma }^{ * }\left( {\mathop{\bigwedge }\limits^{r}{\mathcal{... | Proof. Since \( S \) is of finite type, we can choose an \( {m}_{0} \) such that the fibers of \( {\mathcal{E}}_{1} \) and \( {\mathcal{E}}_{2} \) are all \( {m}_{0} \) -regular. Then, letting \( \pi : X \times S \rightarrow S \) be the projection, \( {\pi }_{ * }\left( {{\mathcal{E}}_{1}\left( {m}_{0}\right) }\right) ... | Yes |
Proposition 28.6. Let \( T \) be a nonsingular curve, \( 0 \in T \) a point, and let \( \mathcal{E} \) and \( {\mathcal{E}}^{\prime } \) be two families of vector bundles of rank \( r \) and degree \( d \) on \( X \), parametrized by \( T \), such that for each point \( t \neq 0 \) in \( T \), the fibers \( {\mathcal{E... | Proof. Since \( {\mathcal{E}}_{t} \) and \( {\mathcal{E}}_{t}^{\prime } \) are isomorphic for each \( t \neq 0 \), the function \( {h}^{0}\left( {\mathcal{H}{om}\left( {{\mathcal{E}}_{t},{\mathcal{E}}_{t}^{\prime }}\right) }\right) \) is nonzero at all points of \( T \) different from 0 . This function is upper semicon... | Yes |
We show that there exist simple unstable bundles in \( M \) . Let \( \mathcal{L},\mathcal{M} \) be line bundles of degrees \( 1,0 \), respectively on \( X \), with the property that \( {h}^{0}\left( {{\mathcal{M}}^{ \vee } \otimes \mathcal{L}}\right) = 0 \) . This is possible because \( {\mathcal{M}}^{ \vee } \otimes \... | \[ 0 \rightarrow \mathcal{L} \rightarrow \mathcal{E} \rightarrow \mathcal{M} \rightarrow 0 \] (12) Its stability degree is -1 ; hence \( \mathcal{E} \) is unstable. Now suppose \( \mathcal{E} \) were not simple. Then there would be a map \( \varphi : \mathcal{E} \rightarrow \mathcal{E} \) of rank 1, as in the proof of ... | Yes |
A hypersurface in \( {\mathbb{P}}^{n} \), or more generally, a complete intersection scheme in \( {\mathbb{P}}^{n} \), is smoothable. | This follows from repeated applications of Bertini’s theorem [57, II, 8.18], since the space of complete intersections of given degrees is irreducible. | Yes |
Proposition 29.1. Let \( {X}_{0} \) be an affine scheme of finite type over the algebraically closed field \( k \) . Let \( X/T \) be a flat family over a nonsingular curve \( T \) of finite type over \( k \) with \( X, T \) both affine, and let \( 0 \in T \) be a point such that the fiber of \( X \) over 0 is \( {X}_{... | Proof. We know from (4.11) that a morphism is smooth if and only if the relative \( {T}^{1} \) functor vanishes for all modules. Thus condition (i) says that the functor \( F \) is zero over \( T \smallsetminus \{ 0\} \) . In other words, for every \( B \) -module \( M \) of finite type, \( F\left( M\right) \) has supp... | No |
Lemma 29.2. Let \( B \) be a noetherian ring, let \( F \) be a coherent functor on \( \operatorname{Mod}\left( B\right) \), and let \( t \in B \) be a non-zero-divisor. Suppose that for every \( M \in \) \( \operatorname{Mod}\left( B\right) \) there exists some \( n > 0 \) such that \( {t}^{n} \cdot F\left( M\right) = ... | Proof. Our hypothesis implies that the extension of \( F \) to the category of modules over the localized ring \( {B}_{t} \) is identically zero. Hence for any \( {B}_{t} \) -module \( N \), the map\n\n\[ \n{\operatorname{Hom}}_{{B}_{t}}\left( {{Q}_{t}, N}\right) \rightarrow {\operatorname{Hom}}_{{B}_{t}}\left( {{P}_{t... | Yes |
Proposition 29.3. Let \( {X}_{0} \) be an affine scheme with isolated singularities. Then \( {X}_{0} \) is formally smoothable if and only if each singular point has a formally smoothable affine neighborhood. | Proof. One direction is obvious. For the converse, let \( {U}_{0} \) be the disjoint union of formally smoothable open affine subsets containing all the singular points (each one once only). Then by hypothesis there is a formal family \( \left\{ {U}_{n}\right\} \) of deformations of \( {U}_{0} \) and an integer \( {n}_... | Yes |
Proposition 29.4. Let \( {X}_{0} \) and \( {X}_{0}^{\prime } \) be affine schemes each having a single isolated singularity at points \( P,{P}^{\prime } \), and assume that these singular points are analytically isomorphic. Then \( {X}_{0} \) is formally smoothable if and only if \( {X}_{0}^{\prime } \) is formally smo... | Proof. There is an isomorphism of the deformation functors \( \operatorname{Def}\left( {X}_{0}\right) \) and \( \operatorname{Def}\left( {X}_{0}^{\prime }\right) \), so that a formal family \( {X}_{n} \) of deformations of \( {X}_{0} \) corresponds to a formal family \( {X}_{n}^{\prime } \) of deformations of \( {X}_{0... | No |
Proposition 29.5. Let \( {X}_{0} \) be a formally smoothable closed subscheme of a nonsingular projective scheme \( Z \) over \( k \) . Then \( {X}_{0} \) is smoothable as a subscheme of \( Z \) . | Proof. We make use of the fact that deformations of closed subschemes of a projective scheme are represented by the Hilbert scheme \( H \) . Let \( {X}_{0} \) correspond to a point \( {x}_{0} \in H \) . Then a formal family \( {X}_{n} \) of deformations of \( {X}_{0} \) in \( Z \) corresponds to a series of compatible ... | Yes |
Proposition 29.6. A singular rigid scheme is not formally smoothable, and hence not smoothable. | Proof. If \( {X}_{0} \) is singular, then there is some \( {X}_{0} \) -module \( {M}_{0} \) for which \( {T}^{1}\left( {{X}_{0}/k,{M}_{0}}\right) \neq 0 \) . If \( {X}_{0} \) is rigid, and \( {X}_{n} \) is a deformation of \( {X}_{0} \) over \( {A}_{n} = k\left\lbrack t\right\rbrack /{t}^{n + 1} \), then \( {X}_{n} \co... | Yes |
Theorem 29.7. Let \( {X}_{0} \) be a closed subscheme of the nonsingular projective scheme \( Z \), and assume that \( {X}_{0} \) has isolated singularities. Let \( {U}_{0} \) be the disjoint union of open affine subsets of \( {X}_{0} \) containing all the singular points of \( {X}_{0} \) , each one once. Suppose that\... | Proof. Hypothesis (b) tells us that the map of functors is surjective on tangent spaces. To show strong surjectivity, suppose we are given a small extension \( {C}^{\prime } \rightarrow C \) of Artin rings and a deformation \( X \) of \( {X}_{0} \) in \( Z \) over \( C \) , restricting to a deformation \( U \) of \( {U... | Yes |
Corollary 29.8. Suppose that \( {X}_{0} \) is a closed subscheme of the nonsingular projective scheme \( Z \), having isolated singularities and satisfying conditions (a),(b) of (29.7). Suppose furthermore that each singular point \( {P}_{i} \) is locally formally smoothable. Then \( {X}_{0} \) is smoothable in \( Z \)... | Proof. By hypothesis we can find a formally smoothable open affine subset \( {U}_{i} \) of \( {X}_{0} \) containing \( {P}_{i} \) for each \( i \) . Let \( {U}_{0} \) be the disjoint union of the \( {U}_{i} \) . Then by (29.7) the restriction map of functors \( \operatorname{Hilb}\left( {{X}_{0}, Z}\right) \rightarrow ... | Yes |
Proposition 29.9. A reduced curve \( Y \) in \( {\mathbb{P}}^{n} \) with locally smoothable singularities and \( {H}^{1}\left( {Y,{\mathcal{O}}_{Y}\left( 1\right) }\right) = 0 \) is smoothable. In particular, this applies if \( Y \) has locally complete intersection singularities. | Proof. Since \( Y \) is reduced, its singularities are isolated. Since the singularities are locally smoothable, the obstructions to deforming \( Y \) in \( {\mathbb{P}}^{n} \) will lie in \( {H}^{1}\left( {Y,{\mathcal{N}}_{Y/{\mathbb{P}}^{n}}}\right) \). The defining sequence for the module \( {T}_{Y}^{1} \), supporte... | Yes |
The hypothesis \( {H}^{1}\left( {Y,{\mathcal{O}}_{Y}\left( 1\right) }\right) = 0 \) is necessary in (29.9). Let \( Y \) be the union of a plane quartic curve with a line meeting it at one point and not lying in the same plane. This curve is not smoothable in \( {\mathbb{P}}^{3} \) for the simple reason that there are n... | Even though this curve is not smoothable in \( {\mathbb{P}}^{3} \), it is smoothable as an abstract curve because of (29.10) below. | No |
Corollary 29.10. An abstract reduced curve \( Y \) with locally formally smoothable singularities is smoothable. | Proof. First embed \( Y \) in a complete curve \( \bar{Y} \) and normalize at points of \( \bar{Y} \smallsetminus Y \) so as to introduce no new singularities. Thus we reduce to the case \( Y \) complete. Taking a Cartier divisor of sufficiently high degree on each irreducible component of \( Y \), we can embed \( Y \)... | Yes |
A connected, reduced curve \( Y \) in \( {\mathbb{P}}^{n} \) with \( {p}_{a}\left( Y\right) = 0 \) is smoothable. | If \( Y \) is irreducible, being reduced with \( {p}_{a} = 0 \), it is already isomorphic to \( {\mathbb{P}}^{1} \), hence smooth. If it is reducible and connected, then we can find an irreducible component \( C \) such that the union of the remaining irreducible components, \( D \), is still connected. Let \( s \) be ... | Yes |
Changing notation, let \( X/S \) be an irreducible flat family of singular integral projective curves of arithmetic genus \( {p}_{a} \). Assume that \( \dim S \geq 3{p}_{a} - 3 \) and for each \( s \in S \) the set of \( {s}^{\prime } \in S \) for which \( {X}_{s} \cong {X}_{{s}^{\prime }} \) is finite. Then the genera... | We proceed by contradiction. If a fiber \( {X}_{s} \) is smoothable, then there exists a flat family \( {X}^{\prime }/T \) whose general fiber \( {X}_{t}^{\prime } \) is a smooth curve of genus \( g = \) \( {p}_{a}\left( {X}_{s}\right) \) and whose special fiber \( {X}_{0}^{\prime } \) is \( {X}_{s} \). Choose a projec... | Yes |
So let \( Y \subseteq {\mathbb{P}}_{k}^{n} \) be a nonsingular projectively normal variety with \( {H}^{1}\left( {{\mathcal{O}}_{Y}\left( \nu \right) }\right) = {H}^{1}\left( {{\mathcal{T}}_{Y}\left( \nu \right) }\right) = 0 \) for all \( \nu \in \mathbb{Z} \) . Let \( {\bar{X}}_{0} \) be the projective cone over \( Y ... | \[ 0 \rightarrow {H}^{1}\left( {\mathcal{T}}_{{\bar{X}}_{0}}\right) \rightarrow \operatorname{Def}\left( {\bar{X}}_{0}\right) \rightarrow {H}^{0}\left( {\mathcal{T}}_{{\bar{X}}_{0}}^{1}\right) \] Now \( {\bar{X}}_{0} \) is nonsingular except at the vertex, so \( {\mathcal{T}}_{{\bar{X}}_{0}}^{1} \) is concentrated ther... | Yes |
Theorem 29.12 (Pinkham). Let \( Y \subseteq {\mathbb{P}}^{n} \) be a nonsingular projectively normal curve of genus \( g \geq 1 \) and degree \( d > {4g} + 5 \) if \( g = 1 \) or \( d > {4g} + 4 \) if \( g \geq 2 \) . Then the affine cone \( {X}_{0} \subseteq {\mathbb{A}}^{n + 1} \) is not smoothable. In fact, it is no... | Proof. By (29.11), if \( {\bar{X}}_{0} \) is the closure of \( {X}_{0} \) in \( {\mathbb{P}}^{n + 1} \), then the morphism of functors \( \varphi : \operatorname{Hilb}\left( {{\bar{X}}_{0},{\mathbb{P}}^{n + 1}}\right) \rightarrow \operatorname{Def}\left( {X}_{0}\right) \) is strongly surjective. Therefore, as in the pr... | Yes |
Corollary 29.14. There exist \( d \) -gonal curves of genus \( g \) for all \( d \geq 2 \) . | Proof. Indeed, since there exist hyperelliptic curves of every genus (for example curves of bidegree \( \left( {2, g + 1}\right) \) on nonsingular quadric surface in \( {\mathbb{P}}^{3} \) ), the theorem provides us with \( d \) -gonal curves for every \( d \geq 2 \) . | No |
Theorem 1.1. Let \( f : I = \\left( {c, d}\\right) \\rightarrow \\mathbb{R} \) be a n-times differentiable function. If \( a, b \) are distinct points in \( I \), then there exists a point \( \\bar{x} \) strictly between \( a \) and b such that\n\n\[ f\\left( b\\right) = f\\left( a\\right) + {f}^{\\prime }\\left( a\\ri... | Proof. The idea of the proof is similar to that in the case \( n = 1 \) : create a function \( g\\left( t\\right) \) such that \( {g}^{\\left( k\\right) }\\left( a\\right) = 0, k = 0,\\ldots, n - 1, g\\left( b\\right) = 0 \), and apply Rolle's theorem repeatedly.\n\nThe \( \\left( {n - 1}\\right) \) th-degree Taylor ap... | Yes |
Theorem 1.3. Let \( f \) satisfy the conditions of Theorem 1.1. We have\n\n\[ f\left( b\right) = f\left( a\right) + {f}^{\prime }\left( a\right) \left( {b - a}\right) + \frac{{f}^{\prime \prime }\left( a\right) }{2!}{\left( b - a\right) }^{2} + \cdots \]\n\n\[ + \frac{{f}^{\left( n - 1\right) }\left( a\right) }{\left( ... | Proof. The idea of the proof is to use the fundamental theorem of calculus repeatedly. First, we have \( f\left( b\right) - f\left( a\right) = {\int }_{a}^{b}{f}^{\prime }\left( x\right) {dx} \), or\n\n\[ f\left( b\right) = f\left( a\right) + {\int }_{a}^{b}{f}^{\prime }\left( {s}_{1}\right) d{s}_{1} \]\n\nSimilarly, \... | Yes |
Corollary 1.4. Theorem 1.1 follows from Theorem 1.3. | Proof. By the mean value theorem there exists \( \bar{x} \in \left( {a,{s}_{n - 1}}\right) \) such that\n\n\[{\int }_{a}^{{s}_{n - 1}}{f}^{\left( n\right) }\left( {s}_{n}\right) d{s}_{n} = {f}^{\left( n\right) }\left( \bar{x}\right) \left( {{s}_{n - 1} - a}\right) = {\int }_{a}^{{s}_{n - 1}}{f}^{\left( n\right) }\left(... | Yes |
Theorem 1.5. Let \( f \) satisfy the conditions of Theorem 1.1. We have\n\n\[ f\left( b\right) = f\left( a\right) + {f}^{\prime }\left( a\right) \left( {b - a}\right) + \frac{{f}^{\prime \prime }\left( a\right) }{2}{\left( b - a\right) }^{2} + \cdots + \frac{{f}^{\left( n - 1\right) }\left( a\right) }{\left( {n - 1}\ri... | Proof. The domain of the iterated integral in the statement of Theorem 1.3 is \( \left\{ {\left( {{s}_{1},\ldots ,{s}_{n}}\right) : a \leq {s}_{n} \leq {s}_{n - 1} \leq \cdots \leq {s}_{1} \leq b}\right\} \) . By Fubini’s theorem, this integral can be written as\n\n\[ {\int }_{a}^{b}{f}^{\left( n\right) }\left( {s}_{n}... | Yes |
Theorem 1.10. If \( U \subseteq {\mathbb{R}}^{n} \) is open and \( f : U \rightarrow \mathbb{R} \) is Fréchet differentiable at \( x \), then \( f \) is Gâteaux differentiable at \( x \) . | Thus, Fréchet differentiability implies Gâteaux differentiability, but the converse is not true; see the exercises at the end of the chapter. Consequently, Fréchet differentiability is a stronger concept than Gâteaux differentiability. In fact, the former concept is a uniform version of the latter: it is not hard to se... | No |
Lemma 1.12. (Mean value theorem) Let \( f : U \rightarrow \mathbb{R} \) be a Gâteaux differentiable function on an open set \( U \) in \( {\mathbb{R}}^{n} \). If \( x, y \) are distinct points in \( U \) such that the line segment \( \left\lbrack {x, y}\right\rbrack \) lies in \( U \), then there exists a point \( z \)... | Proof. Define the function \( h\left( t\right) = f\left( {x + t\left( {y - x}\right) }\right) \). Since \( f \) is Gâteaux differentiable, \( h\left( t\right) \) is differentiable and\n\n\[ {h}^{\prime }\left( t\right) = \langle \nabla f\left( {x + t\left( {y - x}\right) }\right), y - x\rangle . \]\n\nIt follows from t... | Yes |
Theorem 1.13. Let \( f : U \rightarrow \mathbb{R} \) be a function on an open set \( U \subseteq {\mathbb{R}}^{n} \) . If \( f\left( x\right) \) is Gâteaux differentiable at \( {x}_{0} \in U \) and the partial derivatives \( \partial f/\partial {x}_{j} \) \( \left( {j = 1,\ldots, n}\right) \) are continuous at \( {x}_{... | Proof. The mean value theorem (Lemma 1.12) implies that there exists a point \( \bar{x} \) strictly between \( {x}_{0} \) and \( {x}_{0} + h \) such that\n\n\[ f\left( {{x}_{0} + h}\right) - f\left( {x}_{0}\right) - \left\langle {\nabla f\left( {x}_{0}\right), h}\right\rangle = \left\langle {\nabla f\left( \bar{x}\righ... | Yes |
Theorem 1.16. Let \( F : U \rightarrow V, G : V \rightarrow {\mathbb{R}}^{k} \), where \( U \subseteq {\mathbb{R}}^{n} \) and \( V \subseteq {\mathbb{R}}^{m} \) are open sets, and let \( H = G \circ F : U \rightarrow {\mathbb{R}}^{k}, H\left( x\right) = G\left( {F\left( x\right) }\right) = \left( {G \circ F}\right) \le... | Proof. Set \( A = {DF}\left( x\right) \) and \( B = {DG}\left( y\right) \) . We have\n\n\[ \nF\left( {x + {td}}\right) = F\left( x\right) + {tDF}\left( x\right) d + o\left( t\right) = F\left( x\right) + {tAd} + o\left( t\right) \n\]\n\nand\n\n\[ \nH\left( {x + {td}}\right) = G\left( {F\left( {x + {td}}\right) }\right) ... | Yes |
Lemma 1.17. Let \( f : I \rightarrow {\mathbb{R}}^{m} \) be a map on an interval \( I = \left( {a, b}\right) \) . If \( f \) is differentiable at every point in \( I \), then\n\n\[ \parallel f\left( y\right) - f\left( x\right) \parallel \leq \left| {y - x}\right| \cdot \mathop{\sup }\limits_{{0 \leq t \leq 1}}\parallel... | Proof. Let \( M > \mathop{\sup }\limits_{{0 \leq t \leq 1}}\parallel {Df}\left( {x + t\left( {y - x}\right) }\right) \parallel \) and set\n\n\[ K \mathrel{\text{:=}} \{ t : 0 \leq t \leq 1,\parallel f\left( {x + t\left( {y - x}\right) }\right) - f\left( x\right) \parallel \leq {Mt}\left| {y - x}\right| \} . \]\n\nThe s... | Yes |
Theorem 1.18. Let \( f : U \rightarrow {\mathbb{R}}^{m} \) be Gâteaux differentiable on an open set \( U \) in \( {\mathbb{R}}^{n} \) . If \( x, y \) are points in \( U \) such that the line segment \( \left\lbrack {x, y}\right\rbrack \) lies in \( U \), and \( T : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) is a ... | Proof. The map \( g\left( t\right) = f\left( {x + t\left( {y - x}\right) }\right) - {tT}\left( {y - x}\right) \) is differentiable with the derivative\n\n\[ {Dg}\left( t\right) = \left\lbrack {{Df}\left( {x + t\left( {y - x}\right) }\right) - T}\right\rbrack \left( {y - x}\right) . \]\n\nLemma 1.17 gives \( \parallel g... | Yes |
Theorem 1.19. Let \( F : U \rightarrow {\mathbb{R}}^{m} \) be a map on an open set \( U \subseteq {\mathbb{R}}^{n} \) . If \( F\left( x\right) = {\left( {f}_{1}\left( x\right) ,\ldots ,{f}_{m}\left( x\right) \right) }^{T} \) is Gâteaux differentiable at \( {x}_{0} \in U \) and the partial derivatives \( \partial {f}_{i... | Proof. By Theorem 1.18, we have\n\n\[ \parallel f\left( {x + h}\right) - f\left( x\right) - {Df}\left( x\right) h\parallel \leq \mathop{\sup }\limits_{{0 \leq t \leq 1}}\parallel {Df}\left( {x + {th}}\right) - {Df}\left( x\right) \parallel \cdot \parallel h\parallel = o\left( h\right) \]\n\nas \( h \rightarrow 0 \), be... | Yes |
Corollary 1.22. If \( f : U \rightarrow \mathbb{R} \) is \( k \) -times Fréchet differentiable on an open set \( U \) in \( {\mathbb{R}}^{n} \) and \( a \in U \), then \( {D}^{k}f\left( a\right) \) is a symmetric \( k \) -linear form, that is,\n\n\[ \n{D}^{k}f\left( a\right) \left\lbrack {{u}_{\sigma \left( 1\right) },... | The proof is obtained from Theorem 1.21 by induction. | No |
Theorem 1.23. (Multivariate Taylor’s formula) Let \( U \) be an open subset of \( {\mathbb{R}}^{n} \), and let \( x, y \) be distinct points in \( U \) such that the line segment \( \left\lbrack {x, y}\right\rbrack \) lies in \( U \) . If \( f : U \rightarrow \mathbb{R} \) has continuous kth-order partial derivatives o... | Proof. It follows from Taylor’s formula for \( h \) that there exists \( 0 < \bar{t} < 1 \) such\n\nthat\n\[ h\left( 1\right) = h\left( 0\right) + {h}^{\prime }\left( 0\right) + \frac{{h}^{\prime \prime }\left( 0\right) }{2!} + \cdots + \frac{{h}^{\left( k - 1\right) }\left( 0\right) }{\left( {k - 1}\right) !} + \frac{... | Yes |
Example 1.26. (Quadratic functions) Let\n\n\\[ f\\left( x\\right) = \\frac{1}{2}\\langle {Ax}, x\\rangle + \\langle c, x\\rangle + \\alpha = \\frac{1}{2}\\mathop{\\sum }\\limits_{{j = 1}}^{n}\\mathop{\\sum }\\limits_{{i = 1}}^{n}{a}_{ij}{x}_{i}{x}_{j} + \\mathop{\\sum }\\limits_{{j = 1}}^{n}{c}_{j}{x}_{j} + \\alpha \\]... | Alternatively,\n\n\\[ f\\left( {x + {td}}\\right) = \\frac{1}{2}\\langle A\\left( {x + {td}}\\right), x + {td}\\rangle + \\langle c, x + {td}\\rangle + \\alpha \\]\n\n\\[ = \\frac{1}{2}\\langle {Ax}, x\\rangle + t\\langle {Ax}, d\\rangle + \\frac{{t}^{2}}{2}\\langle {Ad}, d\\rangle + \\langle c, x\\rangle + t\\langle c... | Yes |
Theorem 1.29. (Danskin) Suppose \( f : X \times Y \rightarrow \mathbb{R} \) is a continuous function, where \( X \subseteq {\mathbb{R}}^{n} \) is an open set, \( Y \) is a compact set of a topological space \( F \), and \( {\nabla }_{x}f\left( {x, y}\right) \) exists and is continuous. Then the marginal function\n\n\[ ... | Proof. We first prove that \( \varphi \left( x\right) \) is continuous. Let \( {x}_{0} \in X \) and let \( {\left\{ {x}_{k}\right\} }_{1}^{\infty } \) be a sequence converging to \( {x}_{0} \) . Pick \( {y}_{k} \in Y \) such that \( \varphi \left( {x}_{k}\right) = f\left( {{x}_{k},{y}_{k}}\right) \) . Since \( Y \) is ... | Yes |
Corollary 1.30. Let \( {\left\{ {f}_{i}\right\} }_{1}^{k} \) be functions defined on a set \( X \) in \( {\mathbb{R}}^{n} \), and let \( \varphi \left( x\right) \mathrel{\text{:=}} \max \left\{ {{f}_{i}\left( x\right) : i = 1,\ldots, k}\right\} \) be their pointwise maximum. If all \( {f}_{i} \) are directionally diffe... | \[ {\varphi }^{\prime }\left( {x;h}\right) = \mathop{\max }\limits_{{y \in I}}{f}_{i}^{\prime }\left( {x;h}\right) \] where \( I = \left\{ {i : \varphi \left( x\right) = {f}_{i}\left( x\right) }\right\} \). | Yes |
Theorem 2.2. (Weierstrass) Let \( f : K \rightarrow \mathbb{R} \) be a continuous function defined on a compact metric space \( K \) . Then there exists a global minimizer \( {x}^{ * } \in K \) of \( f \) on \( K \), that is,\n\n\[ f\left( {x}^{ * }\right) \leq f\left( x\right) \text{ for all }x \in K. \] | Proof. Let \( \left\{ {x}_{k}\right\} \) in \( K \) be a minimizing sequence for \( f \), that is, \( f\left( {x}_{k}\right) \rightarrow \) \( \inf \{ f\left( x\right) : x \in K\} = : {f}^{ * } \), where we may have \( {f}^{ * } = - \infty \) . Since \( K \) is compact, there exists a subsequence \( \left\{ {x}_{{k}_{i... | Yes |
Theorem 2.3. Let \( f : E \rightarrow \mathbb{R} \) be a continuous function defined on a metric space \( E \) . If \( f \) has a nonempty, compact sublevel set \( \{ x \in E : f\left( x\right) \leq \alpha \} \), then \( f \) achieves a global minimizer on \( E \) . | Proof. Let \( \left\{ {x}_{n}\right\} \) be a minimizing sequence for \( f \), that is,\n\n\[ f\left( {x}_{n}\right) \rightarrow \inf \{ f\left( x\right) : x \in E\} = \mathop{\inf }\limits_{E}f = : {f}^{ * }.\]\n\nDenote by \( D \) the sublevel set above, that is, \( D = \{ x \in E : f\left( x\right) \leq \alpha \} \)... | Yes |
Corollary 2.5. If \( f : D \rightarrow \mathbb{R} \) is a continuous coercive function defined on a closed set \( D \subseteq {\mathbb{R}}^{n} \), then \( f \) achieves a global minimum on \( D \) . | Proof. The sublevel sets \( {l}_{\alpha }\left( f\right) = \{ x \in D : f\left( x\right) \leq \alpha \} \) are closed, since \( f \) is continuous, and bounded since \( f \) is coercive. Thus, \( f \) achieves its minimum on \( L \) at a point \( {x}^{ * } \), which is also a global minimizer of \( f \) on \( D \) . | Yes |
Example 2.6. (The fundamental theorem of algebra)\n\nThis famous theorem states that every polynomial\n\n\\[ \np\\left( z\\right) \\mathrel{\\text{:=}} {a}_{n}{z}^{n} + {a}_{n - 1}{z}^{-1} + \\cdots + {a}_{1}z + {a}_{0}, \n\\]\n\nwith leading coefficient \\( {a}_{n} \\neq 0 \\) and where the coefficients \\( {a}_{i} \\... | Consider minimizing the function\n\n\\[ \nf\\left( z\\right) = \\left| {p\\left( z\\right) }\\right| \n\\]\n\nover the complex numbers. We have\n\n\\[ \n\\left| {p\\left( z\\right) }\\right| = {\\left| z\\right| }^{n} \\cdot \\left| {{a}_{n} + \\frac{{a}_{n - 1}}{z} + \\frac{{a}_{n - 2}}{{z}^{2}} + \\cdots + \\frac{{a}... | Yes |
Theorem 2.7. (First-order necessary condition for a local optimizer) Let \( f : U \rightarrow \mathbb{R} \) be a Gâteaux differentiable function on an open set \( U \subseteq {\mathbb{R}}^{n} \) . A local optimizer is a critical point, that is,\n\n\[ x\\text{ a local optimizer }\\; \\Rightarrow \\;\\nabla f\\left( x\\r... | Proof. We first assume that \( x \) is a local minimizer of \( f \) . If \( d \in {\mathbb{R}}^{n} \), then\n\n\[ {f}^{\\prime }\\left( {x;d}\\right) = \\mathop{\\lim }\\limits_{{t \\rightarrow 0}}\\frac{f\\left( {x + {td}}\\right) - f\\left( x\\right) }{t} = \\langle \\nabla f\\left( x\\right), d\\rangle .\n\]\n\nIf \... | Yes |
We determine the minimizers and the minimum value of the function\n\n\\[ \nf\\left( {{x}_{1},\\ldots ,{x}_{n}}\\right) = \\frac{1}{2}\\mathop{\\sum }\\limits_{1}^{n}{x}_{j}^{2} - \\mathop{\\sum }\\limits_{{1 \\leq i < j \\leq n}}\\ln \\left| {{x}_{i} - {x}_{j}}\\right| .\n\\] | Differentiate \\( f \\) with respect to each variable \\( {x}_{j} \\) and set to zero to obtain\n\n\\[ \n\\frac{\\partial f}{\\partial {x}_{j}} = {x}_{j} - \\mathop{\\sum }\\limits_{{i \\neq j}}\\frac{1}{{x}_{j} - {x}_{i}} = 0.\n\\]\n\nTo solve for \\( x \\), consider the polynomial\n\n\\[ \ng\\left( x\\right) = \\math... | Yes |
Theorem 2.12. (Second-order necessary condition for a local minimizer) Let \( f : U \rightarrow \mathbb{R} \) be twice Gâteaux differentiable on an open set \( U \subseteq {\mathbb{R}}^{n} \) in the sense that there exist a vector \( \nabla f\left( x\right) \) and a symmetric matrix \( {Hf}\left( x\right) \) such that ... | Proof. The first-order necessary condition implies \( \nabla f\left( x\right) = 0 \) . Since \( x \) is a local minimizer, we have \( f\left( {x + {th}}\right) \geq f\left( x\right) \) if \( \left| t\right| \) is small enough. Then,(2.1) gives\n\n\[ \frac{{t}^{2}}{2}\langle {Hf}\left( x\right) h, h\rangle + o\left( {t}... | Yes |
Theorem 2.13. (Second-order sufficient condition for a local minimizer) Let \( f : U \rightarrow \mathbb{R} \) be \( {C}^{2} \) on an open set \( U \subseteq {\mathbb{R}}^{n} \) . If \( x \in U \) is a critical point and \( {Hf}\left( x\right) \) is positive definite, then \( x \) is a strict local minimizer of \( f \)... | Proof. Define \( A \mathrel{\text{:=}} {Hf}\left( x\right) \) . Since \( g\left( d\right) \mathrel{\text{:=}} \langle {Ad}, d\rangle > 0 \) for all \( d \) on the unit sphere \( S \mathrel{\text{:=}} \left\{ {d \in {\mathbb{R}}^{n} : \parallel d\parallel = 1}\right\} \) and \( S \) is compact, it follows that there exi... | Yes |
Theorem 2.14. (Second-order sufficient condition for a global minimizer) Let \( f : U \rightarrow \mathbb{R} \) be a function with positive semidefinite Hessian on an open convex set \( U \subseteq {\mathbb{R}}^{n} \) . If \( x \in U \) is a critical point, then \( x \) is a global minimizer of \( f \) on \( U \) . | Proof. Let \( y \in U \) . It follows from the multivariate Taylor’s formula (Theorem 1.23) that there exists a point \( z \in \left( {x, y}\right) \) such that\n\n\[ f\left( y\right) = f\left( x\right) + \langle \nabla f\left( x\right), y - x\rangle + \frac{1}{2}{\left( y - x\right) }^{T}{Hf}\left( z\right) \left( {y ... | Yes |
Theorem 2.16. (Second-order sufficient condition for a saddle point)\n\nLet \( f : U \rightarrow \mathbb{R} \) be twice Gâteaux differentiable on an open set \( U \subseteq {\mathbb{R}}^{n} \) in the sense of (2.1). If \( x \in U \) is a critical point and \( {Hf}\left( x\right) \) is indefinite, that is, it has at lea... | Proof. Define \( A \mathrel{\text{:=}} {Hf}\left( x\right) \) . If \( \lambda > 0 \) is an eigenvalue of \( A \) with a corresponding eigenvector \( d \in {\mathbb{R}}^{n},\parallel d\parallel = 1 \), then \( \langle {Ad}, d\rangle = \langle {\lambda d}, d\rangle = \lambda \), and it follows from Corollary 1.24 that fo... | Yes |
Consider the family of problems\n\n\\[ \n\\min f\\left( {x, y}\\right) \\mathrel{\\text{:=}} {x}^{2} + {y}^{2} + {\\beta xy} + x + {2y}.\n\\]\n | We have\n\\[ \n\\nabla f\\left( {x, y}\\right) = \\left( \\begin{array}{l} {2x} + {\\beta y} + 1 \\\\ {2y} + {\\beta x} + 2 \\end{array}\\right) ,\\;{Hf}\\left( {x, y}\\right) = \\left\\lbrack \\begin{array}{ll} 2 & \\beta \\\\ \\beta & 2 \\end{array}\\right\\rbrack .\n\\]\n\nWe have \\( \\nabla f\\left( {x, y}\\right)... | Yes |
Theorem 2.19. (Spectral decomposition of a symmetric matrix) Let A be an \( n \times n \) real symmetric matrix. There exist a real diagonal matrix \( \Lambda = \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \) and a real orthogonal matrix \( U = \left\lbrack {{u}_{1},\ldots ,{u}_{n}}\right\rb... | Proof. It is well known from linear algebra that \( A \) has \( n \) real eigenvalues \( {\left\{ {\lambda }_{i}\right\} }_{1}^{n} \) with corresponding eigenvectors \( {\left\{ {u}_{i}\right\} }_{1}^{n},\begin{Vmatrix}{u}_{i}\end{Vmatrix} = 1 \), which are mutually orthogonal, that is, \( \left\langle {{u}_{i},{u}_{j}... | Yes |
Corollary 2.20. Let \( A \) be an \( n \times n \) symmetric matrix. Then \( A \) is positive semidefinite if and only if all eigenvalues of \( A \) are nonnegative. Moreover, \( A \) is positive definite if and only if all eigenvalues of \( A \) are positive. | Proof. We have\n\n\[ \n{d}^{T}{Ad} = {d}^{T}{U\Lambda }{U}^{T}d = {\left( {U}^{T}d\right) }^{T}\Lambda \left( {{U}^{T}d}\right) .\n\]\n\nSince \( U \) is nonsingular, we see that \( {d}^{T}{Ad} \geq 0 \) for all \( d \in {\mathbb{R}}^{n} \) if and only if \( {d}^{T}{\Lambda d} \geq 0 \) for all \( d \in {\mathbb{R}}^{n... | Yes |
Theorem 2.21. Let \( A \) and \( B \) be symmetric \( n \times n \) matrices such that at least one of the matrices is positive definite. The matrices can be simultaneously diagonalized in the sense that there exists a nonsingular matrix \( X \in {\mathbb{R}}^{n \times n} \) such that\n\n\[ \n{X}^{T}{AX} = \operatornam... | Proof. Suppose that \( B \) is positive definite. Then \( B \) has the spectral decomposition \( {U}^{T}{BU} = D \), where \( U \in {\mathbb{R}}^{n \times n} \) is orthogonal and \( D = \operatorname{diag}\left\{ {{d}_{1},\ldots ,{d}_{n}}\right\} \) is a diagonal matrix with all \( {d}_{i} > 0 \) . Define the square ro... | Yes |
Theorem 2.23. (Descartes’s rule of sign) Let \( p\left( x\right) = {a}_{0} + {a}_{1}x + {a}_{2}{x}^{2} + \cdots + {a}_{n}{x}^{n} \) be a polynomial of degree \( n \) with real coefficients. Then the number of positive roots \( {N}_{p}\left( {0,\infty }\right) \) of \( p \) is given by \[ {N}_{p}\left( {0,\infty }\right... | A simple proof of the theorem is given in Appendix B. | Yes |
Corollary 2.24. Let \( {A}_{n \times n} \) be a symmetric matrix and let \( p\left( \lambda \right) = \det ({\lambda I} - \) \( A) = {a}_{0} + {a}_{1}\lambda + \cdots + {a}_{n}{\lambda }^{n} \) be the characteristic polynomial of \( A \) . The number of positive eigenvalues of \( A \) is given by\n\n\[ \n{N}_{p}\left( ... | Proof. The characteristic polynomial has only real roots, these being the eigenvalues of \( A \) . This proves the first equality. The second equality follows by considering the polynomial \( q\left( \lambda \right) = - p\left( \lambda \right) \) and noting that the \( k \) coefficient of \( q \) is \( {\left( -1\right... | No |
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