数理解析研究所講究録 1374 巻 2004 年 40-51

40

AN INTRODUCTION TO ARC SPACES
1

東京工業大学大学院理工学研究科
石井志保子 (Shihoko Ishii)
ABSTRACT. This paper is an introduction to the structure of thearc space of an algebraic variety.
Keywords: arc space, valuation, toric variety, Nash problem

1. INTRODUCTION
The concept jet space and arc space over an algebraic variety or ananalytic space is introduced by Nash in his preprint in 1968 which islater published as [20]. The study of these spaces was further developed
by Kontsevich, Denef and Loeser as the theory of motivic integration,
see $[14, 5]$ . These spaces are considered as something to represent thenature of the singularities of the base space. In fact, papers [7], [18],[19] by Mustafa, Ein and Yasuda show that geometric properties of thejet schemes determine certain properties of the singularities of the basespace.
In this paper, we provide the beginners with the basic knowledgeof these spaces. One of the most powerful arms to work on thesespace is the motivic integration. But this paper does not step into
this theory, as there are already very good introduction papers on the
motivic integration by A. Craw [4] and W. Veys [24]. We devote into
the basic study of geometric structure of arc spaces. We also give theintroduction to the Nash problem which was posed in [20]Throughout this paper the base field A is algebraically close field ofarbitrary characteristic and a variety is an irreducible reduced schemeof finite type over .
$k$

2. CONSTRUCTION OF JET SPACES AND ARC SPACESaDefinition 2.1. Let $X$ be a scheme of finite type over andfield extension. A morphism
is called an ra-jet$X$$X$
is called an arc of . We denote the closedof andpoint ofby 0 and the generic point by .
$k$

$K\supset k$$8\mathrm{p}\mathrm{e}\mathrm{c}K[t]/(t^{m+1})arrow X$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[[t]]arrow X$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[[t]]$$\eta$

lpartially supported by Grand-In-Aid of Ministry of Science and Education in

Japan

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AN INTRODUCTION TO ARC SPACES

Proposition 2.2. Let $X$ be a scheme of finite type over A. Let $Sch/k$
be the category of -schemes and Set the category of sets. Define: $Sch/k$ $arrow Set$ bycontravariant functor
$k$

$a$

$F_{m}^{X}$$F_{m}^{X}(Y)=\mathrm{H}\mathrm{o}\mathrm{m}_{k}(Y\cross_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[t]/(t^{m+1}), X)$

Then,
is

$F_{m}^{X}$

is representable by a scheme$X_{m}$.

of finite type over , that$k$.

$\mathrm{H}\mathrm{o}\mathrm{m}_{k}(Y, X_{m})\simeq \mathrm{H}\mathrm{o}\mathrm{m}_{k}(Y\cross_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[t]/(t^{m+1}), X)$This

is called the space of $m$ -jets of $X$ .

$X_{m}$

Proof. This proposition is proved in [3, p. 276]. In this paper, we prove

this by a concrete construction for affine $X$ first and then patching them
together for a general, $R=k[x_{1}, . . , x_{n}]/(f_{1}, .., f_{r})$ . It is sufficient to
Let $X$ be. Then, we
prove the representability for an affine variety
obtain that
$X\mathrm{t}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R$$Z$(2.2.1)$\simeq${$=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A$$\mathrm{H}\mathrm{o}\mathrm{m}(Z\cross \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[t]f(t^{m+1}), X)\simeq \mathrm{H}\mathrm{o}\mathrm{m}(R, A[t]/(t^{m+1}))$$\varphi\in \mathrm{H}\mathrm{o}\mathrm{m}$($k[x_{1}$,,$x_{n}]$, $A[t]/(t^{m+1})$ ) $|\varphi(f_{i})=0$ for $i=1$ , ..,

$r$

}.

If we write $\varphi(x_{j})=a_{j}^{(0)}+a_{j}^{(1)}t+a_{j}^{(2)}t^{2}+..+a_{j}^{(m)}t^{m}$, it follows that$\varphi(f_{i})=F_{i}^{(0)}(a_{j}^{(l)})+F_{i}^{(1)}(a_{j}^{(l)}$for polynomials$F_{i}^{(s)}$in$a_{j}^{(l)}$$\mathrm{s}$)$t$$+..+F_{i}^{(m)}(a^{(l)}j)t^{m}$. Then the above set (2.2.1) is as follows:$=\{\varphi\in \mathrm{H}\mathrm{o}\mathrm{m}(k[x_{j}, x_{j}^{(1)}, ., x_{j}^{(m)}|7=1, ., n], A)|\varphi(x_{j}^{(l)})=a_{j}^{(l)}, F_{i}^{(s)}(a_{j}^{(l)})=0\}$$=\mathrm{H}\mathrm{o}\mathrm{m}(k[x_{j}, x_{j}^{(1)}, ., x_{j}^{(m)}]\mathit{1}(F_{i}^{(s)}(x_{j}^{(l)})), A)$

If we write(2.2.1) is$X_{m}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[x_{j}, x_{j}^{(1)}, ., x_{j}^{(m)}]/$$\mathrm{H}\mathrm{o}\mathrm{m}(Z, X_{m})$((s)$F_{i}$$(x_{j}^{(l)}$.

)), the above set.

For a general $X$ , cover it by affine open subsets $U_{i}(i\in I)$ . Then, we’s together by the following lemma.
can patch$\square$$(U_{i})_{m}$is representable byLemma 2.3. Assume the functorfor: $X_{m}arrow X$ be the morphism induced fromscheme X. Let
the canonical surjection $k[t]/(t^{m+1})arrow k.$ Then for an open subset$U\subset X,$
is representable by $X_{m}|_{U}:=\pi_{m}^{-1}(U)$ .
This lemma follows immediately from a more general statement PropOsition 3.3.
$F_{m}^{X}$$k$$\pi_{m}$$F_{m}^{U}$$X_{m}$$a$42
AN INTRODUCTION TO ARC SPACES

are obtained as’s ofRemark 2.4. The defining equations
defined by $D(x_{j}^{(l)})=$
follows: Let be a derivation of, where we define $x_{j}^{(l)}=0$ for $l>m.$ Then, it follows that
$X_{m}$$F_{i}^{s}(x_{j}^{(l)})$$D$$k[x_{j}, x_{j}^{(1)}, ., x_{j}^{(m)}]$$x_{j}^{(l+1)}$$F_{i}^{s}(x_{j}^{(l)})=D^{s}(f_{i})$.

from the proof, it follows
Example 2.5. For
of Proposition 2.2.
defined by $f=0.$ Then,
Example 2.6. Let $X$ be a hypersurface in$=D^{m}(f)=0.$ For
by $f=D(f)=$is defined inis defined byexample, if $f=xy+z^{2}$ and $m=2,$ we obtain that
$X_{m}=\mathrm{A}_{k}^{n(m+1)}$$X=\mathrm{A}_{k}^{n}$$\mathrm{A}_{k}^{3}$$\mathrm{A}_{k}^{3(m+1)}$$X_{m}$$X_{2}$$xy+z^{2}=x0)y+xy^{(1)}+2zz^{(1)}=x(2)\mathrm{t}7$ $+2x^{(1)}y^{(1)}+xy^{(2)}+2z^{(1)}z^{(1)}+$

is irreducible and not normal.
One can see that
2.7. The canonical surjection $k[t]$ 1 $(t^{m+1})arrow k[t]/(t^{m})$ induces a mor: $X_{m}arrow X.$: $X_{m}arrow X_{m-1}$ . Definephism
We denote thegives an $m$-jet a :point ofby the same
corresponding to :point of,symbol . Then,
is not ofand call it the space of arcs of X.
Let
finite type over but it is a scheme, see [5].
we obtain the following universal
Using the representability of:property of
Proposition 2.8. Let X be a scheme of finite type over k. Then
$2zz^{(2)}=0.$$X_{2}$$0\phi_{m}$$7\mathrm{r}_{m}=\phi_{1}0$$j_{m}$$\mathrm{A}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[t]/(t^{m+1})arrow X.$$X_{m}$$X_{m}$$\alpha$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[t]/(t^{m+1})arrow X$$\pi_{m}(\alpha)=\alpha(0)$$\alpha$$X_{\infty}=\varliminf_{m}X_{m}$$X_{\infty}$$k$$F_{m}$$X_{\infty}$$\mathrm{H}\mathrm{o}\mathrm{m}_{k}(Y, X_{\infty})\simeq \mathrm{H}\mathrm{o}\mathrm{m}_{k}$(c7$Y\hat{\cross}\mathrm{s}_{\mathrm{p}\mathrm{e}}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$$k[[t]]$ , $X$)$k[[t]]$ means the for, where }’’$Y$.
along the subscheme
Corollary 2.9. There is a universal family of arcs

for an arbitrary -scheme
mal completion of
Spec
$k$

$Y$

$\hat{\mathrm{x}}_{\mathrm{S}\mathrm{p}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$$k\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[[t]]$$\cross$$Y_{\mathrm{X}\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{c}k}}}\{0\}$$X_{\infty^{\hat{\mathrm{X}}}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[[t]]arrow X.$

and the2.10. Denote the canonical projection $X_{\infty}arrow X_{m}$ byby . When we need to specify the base space $X$ ,composite:
gives an arc
we write it by . A point$K$
is the residue field at . As an $m$-jet weand $\pi(x)=\alpha_{x}(0)$ , whereby the same symbol .
anddenotewhich, thenExample 2.11. If
is. Here, we note that the set of closed points of
we denote bynecessarily
in the set
not$\eta_{m}$$\pi$$\pi_{m}\circ\eta_{m}$$x\in X_{\infty}$$\pi_{X}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[[t]]arrow X$$\alpha_{x}$$x$$x\in X_{\infty}$$\alpha_{x}$$X=\mathrm{A}_{k}^{n}$$\alpha$$X_{\infty}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[x_{j}, x_{j}^{(1)}, x_{j}^{(2)} .]$$\mathrm{A}_{k}^{\infty}$$\mathrm{A}_{k}^{\infty}$$k^{\infty}:=\{(a_{1}, a_{2}, \ldots)|a_{i}\in k\}$see the following theorem.

,

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AN INTRODUCTION TO ARC SPACES

Theorem 2.12 ([11]). Every closed point ofand only if $7; is not countable.

is a -valued point if$k$$\mathrm{A}_{k}^{\infty}$

3. MISCELLANEOUS PROPERTIES OF JET SPACES AND ARC SPACES
Proposition 3.1. Let : $Xarrow Y$ be a morphism of -schemes offinite type. Then the canonical $mor$ hism : $X_{m}arrow Y_{m}$ is induced.
for every$k$$f$$f_{m}$$m\in \mathrm{N}\cup\{\infty\}$

Proof. For an-jet (or arc) ofmorphism.

$m$

is an
of $X$ , the composite
is our required
This map $X_{m}arrow Y_{m}$ ,-jet (or arc )

$m$

$Y$

$\alpha\in X_{m}$$f\circ\alpha$$\alpha\mapsto f\circ\alpha$$\square$Proposition 3.2. Let : $Xarrow Y$ be a proper birational morphism
where $W\subset X$ and:of -schernes such that$V\subset Y$
are closed. Thengives a bisection
$f$

$k$

$X\backslash W\simeq Y\backslash V,$$f|_{X\backslash W}$$f_{\infty}$$X_{\infty}s$

Proof. Let a$\in Y_{\infty}\mathrm{Z}$$V_{\infty}$.

$W_{\infty}arrow Y_{\infty}\backslash V_{\infty}$, then$\alpha(\eta)\in X\backslash V$

As$X\backslash W\simeq Y\backslash V$We

obtain the following commutative diagram:$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K((t))$$X$$arrow$

1$\downarrow$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[[t]]$Then, as$f$$arrow\alpha$$Y$:
is a proper morphism, there is a unique morphism. This shows the bijectivity as
such that$\tilde{\alpha}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[[t]]arrow X$$f\circ\tilde{\alpha}=\alpha$required.

$\square$

The following is the generalization of Lemma 2.3.
Proposition 3.3. If : $Xarrow Y$ is an \’etale morphism, then, for every.
$f$$Y_{m}\mathrm{x}_{Y}X$

Proof. As$X_{m}\simeq$$m\in \mathrm{N}\cup\{\infty\}$, it is sufficient to prove$\underline{1\mathrm{i}\mathrm{m}}_{m}(\}_{m}\cross_{Y}X)=(\underline{1\mathrm{i}\mathrm{m}}_{m}Y_{m})\mathrm{x}_{Y}X$

the assertion for $m\in$ N. By the commutative diagram:$arrow f_{m}$$X_{m}\downarrow$$X$

we obtain a morphism
projection$\varphi$$Y_{m}\downarrow$$arrow f$,$Y$: $X_{m}arrow Y_{m}\cross_{Y}X$ . On the other hand, the$Y_{m}\mathrm{x}_{Y}Xarrow Y_{m}$corresponds to a morphism$Y_{m}\mathrm{x}_{Y}\mathrm{X}$$\cross \mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{c}k}}$

44
AN INTRODUCTION TO ARC SPACES

which completes the following commutative$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[t]/(t^{n+1})arrow Y$

diagram:$\mathrm{c}-arrow$$Y_{m}\mathrm{x}_{Y}X\mathrm{x}_{\mathrm{S}_{\mathrm{P}}k}\mathrm{S}\mathrm{e}\mathrm{c}_{l}\mathrm{p}\mathrm{e}\mathrm{c}k[t]/(t^{m+1})$$Y_{m}\mathrm{x}_{Y}X\downarrow$$arrow f$

$X$

$Y$

is formally \’etale, there is a morphism
which make the diagram comis the inverse
mutative. The corresponding morphism
morphism of .

As

$f$

$Y_{m}\cross_{Y}X\cross_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[t]/(t^{m+1})arrow X$$Y_{m}\mathrm{x}_{Y}Xarrow p$ $X_{m}$$\square$$\varphi$

Proposition 3.4. There is a canonical isomorphism:$(X\cross Y)_{m}\simeq X_{m}\cross Y_{m}$.
for every
Proof. For an arbitrary -scheme,$m\in \mathrm{N}\cup\{\infty\}$

$k$

$Z$

,

$\mathrm{H}\mathrm{o}\mathrm{m}_{k}(Z, X_{m}\cross Y_{m})\simeq \mathrm{H}\mathrm{o}\mathrm{m}_{k}(Z, X_{m})\cross \mathrm{H}\mathrm{o}\mathrm{m}_{k}(Z, Y_{m})$,and the right hand side is isomorphic to$\mathrm{H}\mathrm{o}\mathrm{m}_{k}(Z\cross_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} k[t]/(t^{m+1}), X)$$\cross \mathrm{H}\mathrm{o}\mathrm{m}_{k}$($Z\mathrm{x}_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$$\simeq \mathrm{H}\mathrm{o}\mathrm{m}_{k}(Z\mathrm{x}_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k[t]/(t^{m+1}), X\cross Y)$$\simeq \mathrm{H}\mathrm{o}\mathrm{m}_{k}(Z, (X\cross Y)_{m})$A $[t]f(t^{m+1})$ ,

$Y$

)

.

.

For $m=\infty$ , the proof is similar.

$\square$Proposition 3.5. Let : $Xarrow Y$ is an open immersion (resp. closed
immersion) of -schemes of finite type. Then the induced morphism: $X_{m}arrow Y_{m}$ is also an open immersion (resp. closed immersion)
$f$

$k$

$f_{m}$.
for every
Proof. The open case follows from Lemma 2.3. For the closed case, we$m\in \mathrm{N}\cup\{\infty\}$

may assume that is affine. If is defined by $(i=1, ., r)$ in an
affine space, then $X$ is defined by $(i=1, ., r, ., u)$ in the same affineisis defined by $D^{s}(f_{i})(i=1, ., r, s\leq m)$ andspace. Then,defined by $D^{s}(f_{i})(i=1, ., r, ., u, s\leq m)$ in the corresponding affine.
is a closed subscheme ofspace. This shows that
$Y$

$Y$

$f_{i}$$f_{i}$$X_{m}$$Y_{m}$$X_{m}$$Y_{m}$$\square$Remark 3.6. In the above proposition we see that the property open
or closed immersion of the base spaces is inherited by the morphism of
the space of jets and arcs. But some properties are not inherited. For
example, surjectivity and closedness are not inherited.

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AN INTRODUCTION TO ARC SPACES

Example 3.7. There is an example that
: $Xarrow Y$ is surjective(resp. closed) but:
is not surjective (resp. closed). Let$f$$X_{\infty}arrow Y_{\infty}$$f_{\infty}$$X=\mathrm{A}_{\mathbb{C}}^{2}$and$G=\langle(\begin{array}{ll}\epsilon 00 \epsilon^{n-1}\end{array})\rangle$

be a finite cyclic subgroup in$\mathrm{G}\mathrm{L}(2, \mathbb{C})$acting on $X$ , where $n\geq 2$ and is a primitive -th root of unity. Let$Y=X/G$ be the quotient of $X$ by the action of . Then, it is wellknown that the singularity appeared insingularity Thenisthe canonical projection : $Xarrow Y$ is closed and surjective. We will:.
see that these two properties are not inherited by
Let be the image 7
Then, by the commutativity$n$$\epsilon$

$G$

$Y$

$A_{n-1}$$f$$f_{\infty}$$X_{\infty}arrow Y_{\infty}$$(\mathrm{O})\in Y$$p$$arrow f_{\infty}$$X_{\infty}$$Y_{\infty}$$\downarrow\pi_{X}$,$arrow f$$X$

we obtain$\downarrow\pi_{Y}$$)’$. Here,is irreducible, since$X$ is non-singular.
On the other hand
has $(n-1)$ irreducible
components by [20], [12]. Therefore the morphism
is not surjective.
As
is \’etale, Proposition 3.3 yields the surjectivityof the morphism. Sinceis irreducible,
is dominant. Therefore,
is not closed.
Next we discuss about the irreducibility of the arc space or jet spaces.
$\pi_{X}^{-1}(0)=f_{\infty}^{-1}\circ\pi_{Y}^{-1}(p)$$rr_{X}^{-1}(0)$$\pi_{Y}^{-1}(p)$$f_{\infty}$$X\mathrm{s}$$\{\mathrm{O}\}arrow Y\mathrm{s}$$\{p\}$$(X\mathrm{z}\{0\})_{\infty}arrow(Y\backslash \{p\})_{\infty}$$f_{\infty}$$Y_{\infty}$$f_{\infty}$

Theorem 3.8. If characteristic ofa variety $X$ is irreducible.

$k$is zero,then the space of arcs ofProof This is proved in [13]. The lemma [12, 2.12] also gives a proof ofthis statement. It proves that every arc
Sing
is in the closureSingof an arcSing
. We can see thatis irreducible, since this is the image ofby a, where : $Yarrow X$ is a resolution of the singularities ofmorphism
X. ( , Proposition 3.2). As(Sing $X$ )is an open subset, it follows the irreducibility ofof an irreducible.
$\alpha\in$$\beta\in X_{\infty}\backslash$$($$X)_{\infty}$$($$X)_{\infty}$$X_{\infty}\mathrm{S}$$($$X)_{\infty}$$Y_{\infty}\backslash (\varphi^{-1}(\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}X))_{\infty}$$\varphi_{\infty}$$\varphi$$\mathrm{c}.\mathrm{f}.$$Y_{\infty}\backslash (\varphi^{-1}$$)_{\infty}$$\square$$X_{\infty}$$Y_{\infty}$

Example 3.9 ([12]). If the characteristic of is $p>0$ ,is not$X$
necessarily irreducible. For example, the hypersurface
defined by
has an irredicible component in Sing
which is notin the closure of.
Example 3.10 ([11]). Let $X$ be a toric variety over an algebraicallyclosed field of arbitrary chaiacterictic. Then,is irreducible.
A space of $m$-jets is not necessarily irreducible even if the characteristic of is zero.
$k$$x^{\mathrm{p}}-y^{\mathrm{p}}z=0$$($$X_{\infty}s$$(\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}X)_{\infty}$$X_{\infty}$$k$$X_{\infty}$$X)_{\infty}$

48
AN INTRODUCTION TO ARC SPACES

Theorem 3.11 ([18]). If $X$ is locally a complete intersection varr.ety
is irthenover an algebraically closed field of characteristic
reducible for all $m\geq 1$ if and only if $X$ has rational singularities.
Another story in which a geometric property of space of jets determines the singularities on the base space is as follows:
Theorem 3.12 ([7]). Let $X$ be a reduced divisor on a nonsingular
is no rmal
variety over C. $X$ has terminal singularities if and only if
for every $m\in$ N.
$X_{m}$$zero_{f}$$\mathrm{x}_{m}$4. INTRODUCTION TO THE NASH PROBLEM
The Nash problem is a question about the Nash components and theessential divisors. First we introduce the concept of essential divisors.
: $X_{1}arrow X$ a proper biraDefinition 4.1. Let $X$ be a variety,an irreducible
andtional morphism from a normal variety: $X_{2}arrow X$ be another proper biexceptional divisor of . Let. The birational map
rational morphism from a normal varietyofis defined on a (nonempty) open subset: $X_{1}--*$$E$ . The closure of $(f^{-1}\circ g)(E^{0})$ is well defined. It is called the center.
of onis also
We say that $E$ appears in (or in ), if the center of ona divisor. In this case the birational map $f^{-1}\circ g:X_{1}--+X_{2}$ is a local
and we denote the birational
isomorphism at the generic point of
is identified
again by . For our purposestransform of on
about thetalking. (Strictly speaking, we should bewithcorresponding divisorial valuation instead.) Such an equivalence classis called an exceptional divisor over $X$ .
Definition 4.2. Let $X$ be a variety over . In this paper, by a resolution of the singularities of $X$ we mean a proper, birational morphism
non-singular such thatSing $X$ ): $Yarrow$? $X$ with
$Xs$ Sing $X$ is an isomorphism.
Definition 4.3. An exceptional divisor over $X$ is called an essential
divisor over $X$ if for every resolution : $Yarrow X$ the center of $E$ on
Sing $X$ ).
is an irreducible component of
For a given resolution : $Yarrow X,$ the set
Sing X) $X\}$
irreducible components ofwhich are centers of essential divisors overcorresponds bijectively to the set of all essential divisors over $X$ .
Therefor we call an element of an essential component on
C. Bourvier and G. Gonzalez-Sprinberg also work on “essential divisors” and “essential components” in [1] and [2], but we should note that$g$$E\subset X_{1}$$X_{1}$$f$$g$$X_{2}$$E^{0}$$X_{2}$$f^{-1}\mathrm{o}g$$E$$X_{2}$$E$$X_{2}$$f$$X_{2}$$E$$E$$E\subset X_{1}$$E$$X_{2}$$E\subset X_{2}$$k$$arrow$$Y\backslash f^{-1}$

$Y$

$f$

$E$

$f$

$f^{-1}$$Y$$f$$f^{-1}$$\mathcal{E}=\mathcal{E}_{Y/X}=\{$$\mathcal{E}$$\mathrm{e}$

$Y$

47
AN INTRODUCTION TO ARC SPACES

the definitions are different from ours. In order to distinguish them wegive different names to their “essential divisors” and “essential comp0nents”

Definition 4.4 ([1], [2]). An exceptional divisor $E$ over $X$ is called
a $BGS$-essential divisor over $X$ if $E$ appears in every resolution. An
exceptional divisor $E$ over $X$ is called a $BGS$-essential component over
$X$ if the center of $E$ on every resolutionof the singularity of $X$ is anirreducible component of $f^{-1}(E’)$ , where $E’$ is the center of $E$ on $X$ .
$f$

Proposition 4.5. If $X$ is a surface, then each set of “essential divi-essential components” aresors”, $BGS$-essential divisors” and(Sing $X$ ), where
bijective to the set of the components of the fiber$‘ {}^{t}BGS$$f^{-1}$$f$: $Yarrow X$ is the minimal resolution.

Remark 4.6. Four concepts “essential divisor” , “essential component”,“BGS-essential divisor” and “BGS-essential component” are mutuallydifferent in general.
First, our essential componet is different from the others, because
it is a closed subset on a specific resolution and the others are all
equivalence class of divisors.
Next, a BGS-essential divisor is different from a BGS-essential component or a essential divisor. Indeed, for $X=(xy-zw=0)\subset \mathrm{A}_{k}^{4}$ , theexceptional divisor obtained by a blow-up at the origin is the uniqueessential divisor and also the unique BGS-essential component, while
there is no BGS-essential divisor since $X$ has a small resolution whose
exceptional set is .
Finally a BGS-essential component and an essential divisor are different. Indeed, consider a cone generated by (0, 0, 1), (2, 0, 1), (1, 1, 1),. Let $X$ be the affine toric variety defined by this(0, 1, 1) incone. Then the canonical subdivision adding a one dimensional cone
is a resolution of $X$ . As the singular locus of $X$ is of dimension one, there is no small resolution. Therefore, the divisor $D(1,0,1)$ is
the unique essential divisor, while -0(1,1,2) and $D(2,1,2)$ are BGS-essential
components by the criterion [1, Theorem 2.3].
$\mathrm{P}_{k}^{1}$$N_{\mathrm{R}}=\mathbb{Z}^{3}$$\mathbb{R}_{\geq 0}(1,0,1)$

Definition 4.7. Let $X$ be a variety. An irreducible component of(Sing $X$ ) is called a Nash component if it contains an arc such.
Sing $X$ . This is equivalent to that
that
$C$

$\pi^{-1}$$\alpha$$C\not\subset(\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}X)_{\infty}$$\alpha(\eta)\not\in$

The following lemma is already quoted for the irreducibility of thespace of axes (Theorem 3.8).

Lemma 4.8 ([12]). If the characteris itic of the base field k is zero,(Sing X) is a Nash component
then every irreducible component of$\pi^{-1}$48
AN INTRODUCTION TO ARC SPACES

Example 4.9 ([12]). Let the characteristic of the base field be $p>0.$
Let $X$ be a hypersurface defined by $x^{\mathrm{p}}-y^{p}z=0$ . Then, $X$ has an.
(Sing X) contained in Singirredicible component in$k$$\pi^{-1}$$X)_{\infty}$$($: $Yarrow X$ be a resolution of the singularities of $X$ and(Sing $X$ ). Now we are$(l=1, .., r)$ the irreducible components ofLet$E_{l}$$f$$f^{-1}$going to introduce a map $N$ which is called the Nash map$\{\begin{array}{l}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{o}\mathrm{n}Y\end{array}\}\simeq\{\begin{array}{l}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}X\end{array}\}$$\{\begin{array}{ll}\mathrm{N}\mathrm{a}s\mathrm{h} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h} \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}X\end{array}\}arrow N$

4.10 (construction of the Nash map). The resolution : $Yarrow X$:
of schemes. Let:
induces a morphismisis non-singular,
be the canonical projection. As
the open subset of
irreducible for every 1. Denote byconsisting of the points corresponding to arcs
. Let $C_{i}(i\in I)$ be the Nash components ofsuch that$X$ . Denote byconsisting of the points correthe open subset ofSing $X$ . Bysuch thatsponding to arcs :. By Proposition:
gives
restriction$i\inI$
such that
there is3.2,
is surjective. Hence, for each
is mapped to the generic pointofthe generic point
we obtain a mapof . By this correspondence$f$$Y_{\infty}arrow X_{\infty}$$f_{\infty}$$\pi_{Y}$

$Y$

$Y$

$(\pi_{Y})^{-1}(E_{l})^{o}$a:$(\pi_{Y})^{-1}(E_{l})$$Y$$\mathrm{d}(\mathrm{y}\mathrm{y})$$\not\in$$Y_{\infty}arrow$$(\pi_{Y})^{-1}(E_{l})$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[[t]]arrow$$\mathrm{U}_{l}E_{l}$$C_{i}$$C_{i}^{o}$$\alpha$$\bigcup_{l=1}^{r}(\pi_{Y})^{-1}(E_{l})^{o}arrow\bigcup_{i\in I}C_{i}^{o}$$f_{\infty}’$$f_{\infty}$$\alpha(\eta)\not\in$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K[[t]]arrow X$$1\leq l_{i}\leq r$$f_{\infty}’$$\beta\iota_{:}$$(\pi_{Y})^{-1}(E_{l_{i}})^{o}$$\alpha_{\mathrm{i}}$$C_{i}\mapsto E_{l_{i}}$$C_{i}^{o}$$N$:$\{\begin{array}{ll}\mathrm{N}\mathrm{a}\mathrm{s}\mathrm{h} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h} \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}X\end{array}\}arrow\{\begin{array}{l}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{o}\mathrm{f}f^{-1}(\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}X)\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\end{array}\}$Lemma 4.11. The map N is an injective map to the subset { essential
components on Y}.
byand. Denote the generic point offor ) , thenby . If
the generic point of, a contradiction.
To prove that the $\{E_{l}. : i\in I\}$ are essential components on , leta divisorial resolution
be another resolution andbeandwhich factors through both and $Y’$ . Let
the irreducible components of the exceptional sets corresponding to .
are the image of . This shows
andThen, we can see thatis an essential divisor over $X$ and therefore : is an essential
thatcomponent on

Proof. Let$N(C_{i})=E_{l_{i}}$$C_{i}$$(\pi_{Y})^{-1}(E_{l})$$\alpha_{i}=7\infty’(\mathrm{d}\mathrm{x}_{i})$$\beta_{l}$$\alpha_{i}$$i$$E_{l_{i}}=E_{l_{j}}$$j$$=f_{\infty}’(\beta_{l_{j}})=\alpha_{j}$$Y$$\tilde{Y}arrow X$$\mathit{1}’arrow X$$Y$$\tilde{E}_{l}.\cdot\subset\tilde{Y}$$E_{l_{j}}’\subset Y’$$C\dot{.}$$\tilde{E}_{l_{i}}$$E\iota_{:}$$E_{l}’\dot{.}$$E_{l}$$\tilde{E}\iota_{:}$$Y$$\square$43
AN INTRODUCTION TO ARC SPACES

Problem 4.12. Is the Nash mapi;$\{\begin{array}{ll}Nash componentsofthespace ofarcsthrough \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}X\end{array}\}arrow N\mathfrak{i}^{(}comQ^{e}CSo\mathit{8}n_{ts}\}\simeq\{e\mathrm{S}Sdivove_{r}^{nt}.so$bijective

$l$

$\}$

$($

After Nash’s preprint which posed this problem was circulated in
1968, Bouvier, Gonzalez-Sprinberg, Hickel, Lejeune-Jalabert, Nobile,
Reguera-Lopez and others (see, [1], [9], [10], [15], [16], [17], [21], [22])worked on the arc space of a singular variety related to this problem.
Recently for a toric variety of arbitrary dimension the Nash problem

is affirmatively answered but is negatively answered in general in [12].
Here, we introduce a brief history of this problem.
singularity
Theorem 4.13 ([20]). The Nash problem holds true for an-singularity is the hypersurface singularity definedwhere aby $xy-z^{n+1}=0$ in Ag.
Theorem 4.14 ([22]). The Nash problem holds true for a rninirnal surface singularity. Here, a minimal surface singularity means a rationalsurface singularity with the reduced fundamental cycle.
Theorem 4.15 ([17], [23]). The Nash problem holds true for a sandwiched surface singularity. Here, a sanclwichted surface singularity
means the formal neighborhood of a singular point on a surface oblained by blowing up a complete ideal in the local ring of a closed point
on a non-singular algebraic surface.
Theorem 4.16 ([12]). The Nash problem holds true for a toric singularity of arbitrary dimension.
So far we have seen the affirmative answers. The last year, negative
examples are given in [12].
Example 4.17. Let $X$ be a hypersurface defined by. Then the number of the Nash components is one,inwhile the number of the essential divisors is two. Therefore the Nash
map is not bijective.
By the above example we can construct counter examples to the
Nash problem for dimension greater than 3. At this moment the Nashproblem is still open for two and three dimensional variety.
$\mathrm{A}_{n}$$(n\in \mathrm{N})_{f}$$A_{n}$$x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+$$x_{4}^{3}+x_{5}^{6}=0$$\mathrm{A}_{\mathbb{C}}^{5}$

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singuli\‘eres, Duke Math. J. 91 (1998) 609-620

50
AN INTRODUCTION TO ARC SPACES

2. C. Bouvier and G. Gonzalez-Sprinberg, Syst\‘eme g\’en\’erateur $minimal_{J}$ diviseurs
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$G$$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{A}\mathrm{G}/9911179$$\mathrm{A}\mathrm{G}/0312324$$d^{f}$$\mathrm{M}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\check{\mathrm{a}}$51
AN INTRODUCTION TO ARC SPACES

DEPARTMENT OF MATHEIVIATICS, TOkyO INSTITUTE OF TECHNOLOGY, OHOKAYAMA, MEGURO, Tokyo, JAPANE-MAIL :SHIHOKO@MATH.TITECH.AC.-.TP