利用者:Angol Mois/sandbox2

因子の線形系とは、代数幾何学における幾何学的な「曲線の族」の一般化となる概念である。線形系の次元は、族のパラメータの個数である。

この概念は射影平面上の代数曲線たちのなす「線形系」として最初に現れた。より一般には、スキーム、あるいは環付き空間上のある与えられた因子と線形同値な有効因子からなる部分集合であると考えることができる^[1]。

線形因子によって定まる射は「小平写像」と呼ばれることがある。

定義

代数多様体 $X$ 上の二つの因子 $D,E$ が線形同値であるとは、ある有理型関数、すなわち関数体の元 $f\in K(X)^{\times }$ によって

D=E+(f)

と表せることを指す。ただし、ここで $(f)$ は、零点の極のなす因子である。ここで注意すべきは、 $X$ に特異点が存在してしまうと、Cartier 因子と Weil 因子が対応するとは限らないということである。この場合は考慮すべきことが増える(詳しくは後述)。

完備な線形系とは、 $X$ 上の因子 $D$ と線形同値な有効因子全体のなす集合のことで、 $|D|$ という記号で表される。 ${\mathcal {L}}$ を $D$ と対応する可逆層とするとき、 $X$ が非特異射影スキームであるとき、 $|D|$ と $(\Gamma (X,{\mathcal {L}})\smallsetminus \{0\})/k^{\ast },$ の間に一対一対応が存在し^[2]、したがって $|D|$ には射影空間の閉点の構造が入る。

線形系は、射影空間としての $|D|$ の部分空間のことをさす。すなわち、 $\Gamma (X,{\mathcal {L}})$ の部分ベクトル空間 $W$ と対応する有効因子のなすしゅうごうのことである。線形系 ${\mathfrak {d}}$ の次元は、空間の次元として定義される。すなわち、 $\dim {\mathfrak {d}}=\dim W-1$ である。

Cartier 因子類と直線バンドルの間に対応関係があるので、線形系は因子に触れずに、直線バンドル、つまり可逆層の言葉で表すことができる。このとき、因子は可逆層に、因子の同値は可逆層の同型に言い換えることができる。

Examples

Linear equivalence

Consider the line bundle ${\mathcal {O}}(2)$ on $\mathbb {P} ^{3}$ whose sections $s\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))$ define quadric surfaces. For the associated divisor $D_{s}=Z(s)$ , it is linearly equivalent to any other divisor defined by the vanishing locus of some $t\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))$ using the rational function $\left(t/s\right)$ ^[2] (Proposition 7.2). For example, the divisor $D$ associated to the vanishing locus of $x^{2}+y^{2}+z^{2}+w^{2}$ is linearly equivalent to the divisor $E$ associated to the vanishing locus of $xy$ . Then, there is the equivalence of divisors

$D=E+\left({\frac {x^{2}+y^{2}+z^{2}+w^{2}}{xy}}\right)$

Linear systems on curves

One of the important complete linear systems on an algebraic curve $C$ of genus $g$ is given by the complete linear system associated with the canonical divisor $K$ , denoted $|K|=\mathbb {P} (H^{0}(C,\omega _{C}))$ . This definition follows from proposition II.7.7 of Hartshorne^[2] since every effective divisor in the linear system comes from the zeros of some section of $\omega _{C}$ .

Hyperelliptic curves

One application of linear systems is used in the classification of algebraic curves. A hyperelliptic curve is a curve $C$ with a degree $2$ morphism $f:C\to \mathbb {P} ^{1}$ .^[2] For the case $g=2$ all curves are hyperelliptic: the Riemann–Roch theorem then gives the degree of $K_{C}$ is $2g-2=2$ and $h^{0}(K_{C})=2$ , hence there is a degree $2$ map to $\mathbb {P} ^{1}=\mathbb {P} (H^{0}(C,\omega _{C}))$ .

g_r^d

A $g_{d}^{r}$ is a linear system ${\mathfrak {d}}$ on a curve $C$ which is of degree $d$ and dimension $r$ . For example, hyperelliptic curves have a $g_{2}^{1}$ since $|K_{C}|$ defines one. In fact, hyperelliptic curves have a unique $g_{2}^{1}$ ^[2] from proposition 5.3. Another close set of examples are curves with a $g_{3}^{1}$ which are called trigonal curves. In fact, any curve has a $g_{d}^{1}$ for $d\geq (1/2)g+1$ .^[3]

Linear systems of hypersurfaces in $\mathbb {P} ^{n}$

Consider the line bundle ${\mathcal {O}}(d)$ over $\mathbb {P} ^{n}$ . If we take global sections $V=\Gamma ({\mathcal {O}}(d))$ , then we can take its projectivization $\mathbb {P} (V)$ . This is isomorphic to $\mathbb {P} ^{N}$ where

N={\binom {n+d}{n}}-1

Then, using any embedding $\mathbb {P} ^{k}\to \mathbb {P} ^{N}$ we can construct a linear system of dimension $k$ .

Linear system of conics

→詳細は「Linear system of conics」を参照

Other examples

The Cayley–Bacharach theorem is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th.

Linear systems in birational geometry

In general linear systems became a basic tool of birational geometry as practised by the Italian school of algebraic geometry. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions — the Riemann–Roch problem as it can be called — can be better phrased in terms of homological algebra. The effect of working on varieties with singular points is to show up a difference between Weil divisors (in the free abelian group generated by codimension-one subvarieties), and Cartier divisors coming from sections of invertible sheaves.

The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over Henri Poincaré's characteristic linear system of an algebraic family of curves on an algebraic surface.

Base locus

The base locus of a linear system of divisors on a variety refers to the subvariety of points 'common' to all divisors in the linear system. Geometrically, this corresponds to the common intersection of the varieties. Linear systems may or may not have a base locus – for example, the pencil of affine lines $x=a$ has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus.

More precisely, suppose that $|D|$ is a complete linear system of divisors on some variety $X$ . Consider the intersection

\operatorname {Bl} (|D|):=\bigcap _{D_{\text{eff}}\in |D|}\operatorname {Supp} D_{\text{eff}}\

where $\operatorname {Supp}$ denotes the support of a divisor, and the intersection is taken over all effective divisors $D_{\text{eff}}$ in the linear system. This is the base locus of $|D|$ (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of $\operatorname {Bl}$ should be).

One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system). Suppose $|D|$ is such a class on a variety $X$ , and $C$ an irreducible curve on $X$ . If $C$ is not contained in the base locus of $|D|$ , then there exists some divisor ${\tilde {D}}$ in the class which does not contain $C$ , and so intersects it properly. Basic facts from intersection theory then tell us that we must have $|D|\cdot C\geq 0$ . The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the 'smaller' the base locus, the 'more likely' it is that the class is nef.

In the modern formulation of algebraic geometry, a complete linear system $|D|$ of (Cartier) divisors on a variety $X$ is viewed as a line bundle ${\mathcal {O}}(D)$ on $X$ . From this viewpoint, the base locus $\operatorname {Bl} (|D|)$ is the set of common zeroes of all sections of ${\mathcal {O}}(D)$ . A simple consequence is that the bundle is globally generated if and only if the base locus is empty.

The notion of the base locus still makes sense for a non-complete linear system as well: the base locus of it is still the intersection of the supports of all the effective divisors in the system.

→「Theorem of Bertini」も参照

Example

Consider the Lefschetz pencil $p:{\mathfrak {X}}\to \mathbb {P} ^{1}$ given by two generic sections $f,g\in \Gamma (\mathbb {P} ^{n},{\mathcal {O}}(d))$ , so ${\mathfrak {X}}$ given by the scheme

${\mathfrak {X}}={\text{Proj}}\left({\frac {k[s,t][x_{0},\ldots ,x_{n}]}{(sf+tg)}}\right)$

This has an associated linear system of divisors since each polynomial, $s_{0}f+t_{0}g$ for a fixed $[s_{0}:t_{0}]\in \mathbb {P} ^{1}$ is a divisor in $\mathbb {P} ^{n}$ . Then, the base locus of this system of divisors is the scheme given by the vanishing locus of $f,g$ , so

${\text{Bl}}({\mathfrak {X}})={\text{Proj}}\left({\frac {k[s,t][x_{0},\ldots ,x_{n}]}{(f,g)}}\right)$

A map determined by a linear system

Each linear system on an algebraic variety determines a morphism from the complement of the base locus to a projective space of dimension of the system, as follows. (In a sense, the converse is also true; see the section below)

Let L be a line bundle on an algebraic variety X and $V\subset \Gamma (X,L)$ a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when V is base-point-free; in other words, the natural map $V\otimes _{k}{\mathcal {O}}_{X}\to L$ is surjective (here, k = the base field). Or equivalently, $\operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {O}}_{X}$ is surjective. Hence, writing $V_{X}=V\times X$ for the trivial vector bundle and passing the surjection to the relative Proj, there is a closed immersion:

i:X\hookrightarrow \mathbb {P} (V_{X}^{*}\otimes L)\simeq \mathbb {P} (V_{X}^{*})=\mathbb {P} (V^{*})\times X

where $\simeq$ on the right is the invariance of the projective bundle under a twist by a line bundle. Following i by a projection, there results in the map:^[4]

f:X\to \mathbb {P} (V^{*}).

When the base locus of V is not empty, the above discussion still goes through with ${\mathcal {O}}_{X}$ in the direct sum replaced by an ideal sheaf defining the base locus and X replaced by the blow-up ${\widetilde {X}}$ of it along the (scheme-theoretic) base locus B. Precisely, as above, there is a surjection $\operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {I}}^{n}$ where ${\mathcal {I}}$ is the ideal sheaf of B and that gives rise to

i:{\widetilde {X}}\hookrightarrow \mathbb {P} (V^{*})\times X.

Since $X-B\simeq$ an open subset of ${\widetilde {X}}$ , there results in the map:

f:X-B\to \mathbb {P} (V^{*}).

Finally, when a basis of V is chosen, the above discussion becomes more down-to-earth (and that is the style used in Hartshorne, Algebraic Geometry).

Linear system determined by a map to a projective space

Each morphism from an algebraic variety to a projective space determines a base-point-free linear system on the variety; because of this, a base-point-free linear system and a map to a projective space are often used interchangeably.

For a closed immersion $f:Y\hookrightarrow X$ of algebraic varieties there is a pullback of a linear system ${\mathfrak {d}}$ on $X$ to $Y$ , defined as $f^{-1}({\mathfrak {d}})=\{f^{-1}(D)|D\in {\mathfrak {d}}\}$ ^[2] (page 158).

O(1) on a projective variety

A projective variety $X$ embedded in $\mathbb {P} ^{r}$ has a canonical linear system determining a map to projective space from ${\mathcal {O}}_{X}(1)={\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{\mathbb {P} ^{r}}}{\mathcal {O}}_{\mathbb {P} ^{r}}(1)$ . This sends a point $x\in X$ to its corresponding point $[x_{0}:\cdots :x_{r}]\in \mathbb {P} ^{r}$ .

References

^ Grothendieck, Alexandre; Dieudonné, Jean. EGA IV, 21.3.
^ ^a ^b ^c ^d ^e ^f Hartshorne, R. 'Algebraic Geometry', proposition II.7.2, page 151, proposition II.7.7, page 157, page 158, exercise IV.1.7, page 298, proposition IV.5.3, page 342
^ Kleiman, Steven L.; Laksov, Dan (1974). “Another proof of the existence of special divisors” (英語). Acta Mathematica 132: 163–176. doi:10.1007/BF02392112. ISSN 0001-5962.
^ Fulton, § 4.4.

P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 137. ISBN 0-471-05059-8
Hartshorne, R. Algebraic Geometry, Springer-Verlag, 1977; corrected 6th printing, 1993. ISBN 0-387-90244-9.
Lazarsfeld, R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. ISBN 3-540-22533-1.

[1] Grothendieck, Alexandre; Dieudonné, Jean. EGA IV, 21.3.

[:0-2] ^ ^a ^b ^c ^d ^e ^f Hartshorne, R. 'Algebraic Geometry', proposition II.7.2, page 151, proposition II.7.7, page 157, page 158, exercise IV.1.7, page 298, proposition IV.5.3, page 342

[3] Kleiman, Steven L.; Laksov, Dan (1974). “Another proof of the existence of special divisors” (英語). Acta Mathematica 132: 163–176. doi:10.1007/BF02392112. ISSN 0001-5962.

[4] Fulton, § 4.4.

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定義