数理解析研究所講究録
1078 巻 1999 年 86-92

86

Effective base point freeness on normal surfaces

象工穴理工添 a 玄斜$\mathrm{t}^{\iota}|\hslash$$\mathrm{k}^{\mathrm{R}}$(Takeshi Kawachi)

1. INTRODUCTION
Let $M$ be a divisor on a normal variety $Y$ . Our main aim is to get criteria which

provide the base point freeness of the adjoint linear system$|K_{Y}+\lceil M\rceil|$

where$\lceil M\rceil[mathring]_{1}\mathrm{S}$theround-up of $M$ . For smooth manifolds, there are many good results in higher dimension.

On the other hand, since singularity has much information, we would conclude the same
result by a weaker condition. It is true in the two dimensional case, we introduce thatworse singularity causes better base point freeness.
2. THE INVARIANT
Let $Y$ be a projective normal two dimensional variety oversurface” for short), andif

$y$

$y$

$\mathrm{m}[mathring]_{\mathrm{l}}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l}$Definition 1. (MRLT) Let $Y,$$(\mathrm{Y}_{\gamma}B)$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}[mathring]_{\mathrm{l}}\mathrm{o}\mathrm{n}\mathrm{s}$are(we will call “normal

be a fixed point on $Y$ . Let $f:Xarrow \mathrm{Y}$ be the blowing up at

is a smooth point, or theon Y.

$\mathbb{C}$

$y$

and

resolution of
$f$

$y$

if

$y$

is singular.

be as above. Let $B$ be an effective

is called minimal resolutional $log$ terminal (MRLT) at$\mathrm{s}\mathrm{a}\mathrm{t}_{[mathring]_{1}\mathrm{S}}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d}$:

$y$

$y$

$\mathbb{Q}$-divisor

if the following

87(1) the round-down$\lfloor B\rfloor=0$,(2) if we write $K_{X}+f^{-1}B=f^{*}(K_{Y}+B)-\triangle_{B}$ and$\Delta_{B}=\sum e_{i}E_{i}$where $f^{-1}B$ means the strict transformation of $B$ by .
$f$then all $e_{i}<1$ ,$\square$

Definition 2. Let $Z$ be the fundamental cycle of . We define $\delta_{B,y}=-(Z-\Delta_{B})^{2}$ .

$\square$$y$

We setwe have$\triangle=\triangle 0$$\Delta_{B}>\triangle$, which is the case of $B=0$ ; and also

and then$0\leq\delta_{B,y}\leq\delta_{y}$$\delta_{y}=\triangle 0_{y},\cdot$

Since

$B$

is effective,(cf. [F]). We have the following bound of$\delta_{y}$.

Proposition 1. [KM, Theorem 1](1)$\delta_{y}=4$

if

$y$

is a smooth point, and(2) $0<\delta_{y}<2$ if$\mathrm{Y}$

if

$y$

is a rational double point.

is Kawamata $log$ terminal at .
$y$Note that if $(Y, B)$ is MSLT atis also bounded if$\delta_{y}=2$$(\mathrm{Y}, B)$then $Y$ is Kawamata$y$$\log$

terminal at . Hence$y$is MRLT. Now we will take the above invariant a little bit

smaller.

Definition 3.
$\delta_{\min}=\min${ $-(z-\triangle_{B}+x)^{2}|x$ is an effective$\delta=\{0\delta_{\min},$is of type

$A_{n}$-exceptional divisor.}$y$$\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}(\mathrm{Y},B\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e})\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}$$\delta’=$
$y$

$f$

MRLT at’Note that if$\delta_{B,y}$,$\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}y\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}y\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{y}\mathrm{P}^{\mathrm{e}}A\mathrm{r}\mathrm{w}\mathrm{i}_{\mathrm{S}\mathrm{e}}.D_{n}n\square ’$, the indices are taken in the standard way.

88

3. THE MAIN RESULT
Theorem 2. Let $M$ be a $nef$ and big

thatthrough$y_{f}$thenNote that if$M\cdot C>0$ by

Proof. If
thatis Cartier.

$K_{Y}+\lceil M\rceil$

$y$

$y$

$y$

$y$

If

$\mathbb{Q}$$M^{2}>\delta$-Weil divisor on $Y$ , and $B=\lceil M\rceil-M$ . Assume
and $M\cdot C\geq\delta’$ for any curve $C$ on $Y$ passing

is not a base point $of|K_{Y}+\lceil M\rceil|$ .

is of type

$D_{n}$

the definition ofthen the assumption$\delta’$$M\cdot C\geq\delta’$is equivalent to assume.

is not an MRLT, the proof is well known. (cf. [KM, (2.1)]). So we assume

is an MRLT point.

Since the assertion is local, we may assume $Y-\{y\}$ is smooth.
First we take a good effective$\mathbb{Q}$-divisor $D$ such that

Lemma 3. There exists an effective$\mathbb{Q}$$M^{2}>\delta_{\min}$$\triangle_{B}+x))>0$$\mathbb{Q}$-linearly equivalent to $M$ .

-divisor $D$ on $Y$ such that $D\equiv M$ (numericallyequivalent) and $f^{*}D>Z-\Delta_{B}+x$ where

Proof. Since$\mathbb{Q}$$x$

attains the minimum. Hence $f^{*}M-(Z-\triangle_{B}+x)$ is big, we can get an effective

Let $D$ be an$\mathbb{Q}$$\mathbb{Q}$-divisor$\square$-divisor satisfying the above lemma. We set $D= \sum d_{i}c_{i},$ $B= \sum b_{i}C_{i}$ ,$D_{i}=f^{-1}C_{i},$ $f^{*}D= \sum d_{i}D_{i}+\sum d_{j}’E_{j},$ $f^{*}B= \sum b_{i}D_{i}+\sum b_{j}’E_{j}$$c$.

, we have $(f^{*}M-(Z-\Delta_{B}+x))^{2}>0$ and $f^{*}M\cdot(f^{*}M-(Z-$-linearly equivalent to $f^{*}M-(Z-\Delta_{B}+x)$ .

number$\delta_{\min}$. We choose the rationalas the following.

$c= \min\{\frac{1-b_{i}}{d_{i}},$ $\frac{1-e_{j}}{d_{j}’}|d_{i}>0,$ $D_{i}\cap f^{-1}(y)\neq\emptyset$

and $f(E_{j})=\{y\}\}$ .

Since $(Y, B)$ is MRLT and the choice of $D$ , we have $0<c<1$ .

89Let $R=f^{*}M-cf^{*}D$ . Since $0<c<1$ and $D\equiv M$ is nef and big,

$R$is also nef andbig. By a simple calculation, we have$\lceil R\rceil=f^{*}(K_{Y}+\lceil M1)-K_{x}-\lfloor cf^{*}D+f*B+\triangle\rfloor=R+\{cf*D+f*B+\Delta\}$where$\{\cdot\}$means the fractional part. Hence we have
$K_{X}+ \mathrm{r}R\rceil=f^{*}(K_{Y}+M)-\sum \mathrm{L}Cd_{i}+b_{i}\rfloor Di+\sum\lfloor cd_{jj}’+e\rfloor E_{j}$We write $\sum\lfloor cd_{i}+b_{i}\rfloor D_{i}=A+N$ where all components ofis disjoint,$\mathrm{h}\mathrm{o}\mathrm{m}f^{-1}(y)$. Let$E= \sum\lfloor cd_{j}’+e_{j}\rfloor E_{j}$$A$.

meet with $f^{-1}(y)$ and $N$. By the choice of , both$c$$A$

and $E$ are

reduced or only one of them is zero. Let $A=D_{1}+\cdots+D_{t}$ .

Lemma 4. If $A\neq 0$ then$(\mathrm{Y}, f_{*}A)$is $log$ canonical at$y$and the dual graph is one of thefollowings.
(1)
(2)

(3)

$\wedge.- \mathrm{O}^{-}\bullet$

In the above lemma, we denote prime components of $E$ and $f_{*}A$ by
respectively. Note that only the case (1) is$\log$$\mathrm{O}$and$\bullet$terminal.

Proof. Because of $f^{*}(K_{Y}+f_{*}A)-K_{X}-A\leq E,$ $(Y, f_{*}A)$ is$\log$are classified as in [A] and [K], they are only above 3 cases.

$\square$canonical at . These$y$

We divide the proof of the main theorem in two cases according to $E$ .

Case 1: $E\neq 0$ .
If $t>0$ then$y$is of type

$A_{n}$

or$D_{n}$. Note that if$y$is of type

$E_{n}$

then$A$

must be

$0$.

90

Since

$R$

is nef and big, each

$D_{i}$

is integral in$R$

and $R\cdot D_{i}\geq\delta’>0$ , we have the following

vanishing due to Kawamata-Viehweg.
$H^{1}(X, K_{X}+\lceil R\rceil+A)=H^{1}(X, f^{*}(KY+\lceil M\rceil)-N-E)=0$.

Hence the morphism$H^{0}(X, f^{*}(KY+\lceil M\rceil)-N)arrow H^{0}(E, (f^{*}(KY+\lceil M\rceil)-N)|_{E})$is surjective.

Case 2: $E=0$ .
In this case, $(Y, f_{*}A)$ is$\log$

terminal of type

$A_{n}$

at$y$

and $t=1$ . So we let $A=D_{1}$ .

Hence the morphism$H^{0}(x, f*(K_{Y}+\lceil M1)-N)arrow H^{0}(D_{1}, (f^{*}(K_{Y}+\lceil M\rceil)-N)|_{D}1)$is surjective. Since $(f^{*}(K_{Y}+\lceil M\rceil)-N)|_{D_{1}}=K_{D_{1}}+\lceil R\rceil|_{D_{1}}$ , ifexists a section in$H^{0}(D_{1}, K_{D_{1}}+\lceil R\rceil|_{D_{1}})$Hence it is enough to show$\lceil R\rceil\cdot D_{1}>1$

If$D_{1}$meets.

. By changing the indices we may assume$E_{n}$

then there

which does not vanish at $D_{1}\cap f^{-}1(y)$ by [H].

Note that $\lceil R\rceil\cdot D_{1}\geq R\cdot D_{1}+\sum(Cd_{j}’+e_{j})E_{j}\cdot D_{1}$ and$R\cdot D_{1}\geq(1-c)\delta’$$\lceil R\rceil\cdot D_{1}>1$$y\in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{P}f_{*}D_{1}$$e_{1}\leq e_{n}$. Hence, we have$\delta’=1-e_{n}$.

then the inequalities $f^{*}D>Z-\Delta_{B}$ and$\lceil R\rceil\cdot D_{1}\geq(1-c)(1-e_{n})+cd_{n}’+e_{n}=1+c(d_{n}’+e_{n}-1)$

imply$\lceil R\rceil\cdot D_{1}>1$.

So we assume that

$D_{1}$meets$E_{1}$.

Let $A=A(w_{1}, \ldots, w_{n})=(-E_{i}\cdot E_{j})ij$ be the intersection matrix of the exceptional
divisors of type

$A_{n}$. Let $a(w_{1}, \ldots, w_{n})=\det A(w_{1}, \ldots, w_{n})$ be the determinant. We set91$a()=1$ for convenience.

Let$L_{i}$

becurve. on such that$f^{-1}L_{i}\cdot E_{i}=1$$Y$$\mathrm{a}.\mathrm{n}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{C},\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}$and $f^{-1}L_{i}\cdot E_{j}=0$ for all $j\neq i$ . We set $f^{*}L_{i}=f^{-1}L_{i}+ \sum c_{ij}E_{j}$ .
By simple calculation of matrices, we have the following proposition.

Proposition 5. Let$\triangle=\sum a_{j}E_{j}$.

$1-a_{i}= \frac{a(w1,\ldots,wi-1)+a(wi+1,\ldots,wn)}{a(w_{1},\ldots,w_{n})}$$c_{ij}= \frac{a(w_{1},\ldots,w_{i}-1)a(wj+1\cdots,wn)}{a(w_{1,..\circ},w_{n})},$, if $i\leq j$ ,,$c_{ij}=c_{ji}$Let $f^{*}C_{1}=D_{1}+ \sum c_{j}E_{j}$ . Let $y_{D,j}=d_{j}’-d_{1}c_{j}$ , the coefficients of$D_{i}’ \mathrm{s}$except$D_{1}$. We also let$y_{B,j}=b_{j}’-b_{1}C_{j}$.
$E_{j}$

arising$\mathrm{h}\mathrm{o}\mathrm{m}$and $y_{j}=cy_{D,j}+y_{B,j}$ . Since the minimalityof , we have $cd_{1}+b_{1}=1$ . Hence we have $cd_{1}’+b_{1}’=c_{1}+y_{1}$ . Therefore we have$c$$\lceil R\rceil\cdot D_{1}\geq(1-C)\delta’+cd_{1}’+e_{1}=(1-c)(1-e_{n})+a_{1}+c_{1}+y_{1}$.

By Proposition 5, we have $a_{1}+c_{1}=1/\alpha$ , where $\alpha=\det A(w1, \ldots, wn)$ . Since $E=0$ ,we also have$y_{1}\leq 1/\alpha$.

Claim 6.
$(1-c)(1-e_{n})> \frac{a(w_{1},\ldots,w_{n-1})}{\alpha}$and$y_{n}\leq a(w_{1}, \ldots , W_{narrow 1})y_{1}$.

By this claim, we have $\lceil R\rceil\cdot D_{1}>1+(a(w_{1,\ldots,-1}w_{n})-1)(1/\alpha-y_{1})$ .
$a(w_{1}, \ldots, w_{n-1})\geq 1$

and $y_{1}<1/\alpha$ , we have

Proof of Claim 6. By the choice of$D$$\lceil R\rceil\cdot D_{1}>1$.

, we have $d_{n}’>1-a_{n}-b’n$ . Hence$(d_{n}’-1+an+b_{n}’) \frac{c}{1-a_{n}}>0=\frac{cd_{1}+b_{1}.-1}{1+a(w_{1},..,wn-1)}$,since $cd_{1}+b_{1}=1$ . We set $\alpha’=a(w_{1}, \ldots, w_{n-1})$ for convenience. Then we have$((d_{n}’-1+a_{n}+b_{n}’) \frac{1}{1-a_{n}}-\frac{d_{1}}{1+\alpha’})c>\frac{b_{1}-1}{1+\alpha},$.

Since $(1-a_{n})\alpha=1+\alpha’$ and $d_{n}’=d_{1}/\alpha+y_{D,n}$ , the left-hand-side equals toSince

92$( \frac{d_{n}’}{1-a_{n}}-1+\frac{b_{n}’}{1-a_{n}}-\frac{d_{1}}{1+\alpha’})c=(\frac{y_{D,n}}{1-a_{n}}+\frac{b_{n}’}{1-a_{n}}-1)c$.

On the other hand, the right-hand-side equals to$\frac{b_{1}-1}{1+\alpha},$$= \frac{b_{1}+\alpha yB,n}{1+\alpha},-\frac{1+\alpha y_{B,n}}{1+\alpha},=\frac{b_{n}’}{1-a_{n}}-1+\frac{\alpha’-\alpha yB,n}{1+\alpha},$.

Thus we have(1–c) $(1- \frac{b_{n}’}{1-a_{n}})>\frac{\alpha’/\alpha-y_{B},n-CyD,n}{1-a_{n}}$ .

The second assertion follows from Proposition 5 and the inequalities $c_{11}>c_{12}>\cdots>$$c_{1n}$and $c_{n1}<c_{n2}<\cdots<c_{nn}$ .

$\square$

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[F]

T. Fujita, An appendix to Kawachi-Ma9ek’s paper on global generation of adjoint bundles on

nomal surfaces, J. Alg. Geom. 7 (1998), 251-252.
[H]

R. Hartshone, Generalized divisor on Gorenstein curves and a theorem of Noether, J. Math.
Kyoto Univ. 26 (1986), 375-386.

[K]

Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its applications

to degeneration of surfaces, Ann. Math. 127 (1988), 93-163.
[KM]

T. Kawachi and V. Masek, Reider-type theorems on nomal surfaces, J. Alg. Geom. 7 (1998),

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