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A theorem stating that there is a strong restriction for the [[canonical divisor]] of an [[Algebraic variety|algebraic variety]] to be negative while the positivity is arbitrary. It is closely related to the structure of the cone of curves and the existence of rational curves.
 
A theorem stating that there is a strong restriction for the [[canonical divisor]] of an [[Algebraic variety|algebraic variety]] to be negative while the positivity is arbitrary. It is closely related to the structure of the cone of curves and the existence of rational curves.
  
 
==Definitions and terminology.==
 
==Definitions and terminology.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200501.png" /> be a normal algebraic variety (cf. [[Algebraic variety|Algebraic variety]]). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200503.png" />-divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200504.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200505.png" /> is a formal linear combination of a finite number of prime divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200506.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200507.png" /> with rational number coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200508.png" /> (cf. also [[Divisor|Divisor]]). The canonical divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k1200509.png" /> is a Weil divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005010.png" /> corresponding to a non-zero rational differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005011.png" />-form for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005012.png" /> (cf. also [[Differential form|Differential form]]). The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005013.png" /> is said to be weakly log-terminal if the following conditions are satisfied:
+
Let $X$ be a normal algebraic variety (cf. [[Algebraic variety|Algebraic variety]]). A $\mathbf{Q}$-divisor $B = \sum _ { j = 1 } ^ { t } b _ { j } B _ { j }$ on $X$ is a formal linear combination of a finite number of prime divisors $B _ { j }$ of $X$ with rational number coefficients $b _ { j }$ (cf. also [[Divisor|Divisor]]). The canonical divisor $K _ { X }$ is a Weil divisor on $X$ corresponding to a non-zero rational differential $n$-form for $n = \operatorname { dim } X$ (cf. also [[Differential form|Differential form]]). The pair $( X , B )$ is said to be weakly log-terminal if the following conditions are satisfied:
  
The coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005014.png" /> satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005015.png" />.
+
The coefficients of $B$ satisfy $0 \leq b _ { j } \leq 1$.
  
There exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005017.png" /> is a Cartier divisor (cf. [[Divisor|Divisor]]).
+
There exists a positive integer $r$ such that $r ( K _ { X } + B )$ is a Cartier divisor (cf. [[Divisor|Divisor]]).
  
There exists a projective [[Birational morphism|birational morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005018.png" /> from a smooth variety such that the union
+
There exists a projective [[Birational morphism|birational morphism]] $\mu : Y \rightarrow X$ from a smooth variety such that the union
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005019.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { j = 1 } ^ { t } \mu _ { * } ^ { - 1 } B _ { j } + \sum _ { k = 1 } ^ { s } D _ { k } \end{equation*}
  
is a normal crossing divisor (cf. [[Divisor|Divisor]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005020.png" /> is the strict transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005022.png" /> coincides with the smallest closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005025.png" /> is an isomorphism.
+
is a normal crossing divisor (cf. [[Divisor|Divisor]]), where $\mu_* ^ {- 1 } B _ { j }$ is the strict transform of $B _ { j }$ and $\cup _ { k = 1 } ^ { s } D _ { k }$ coincides with the smallest closed subset $E$ of $Y$ such that $\mu | _ { Y \backslash E } : Y \backslash E \rightarrow X \backslash \mu ( E )$ is an isomorphism.
  
 
One can write
 
One can write
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005026.png" /></td> </tr></table>
+
\begin{equation*} \mu ^ { * } ( K _ { X } + B ) = K _ { Y } + \sum _ { j = 1 } ^ { t } b _ { j } \mu _ { * } ^ { - 1 } B _ { j } + \sum _ { k = 1 } ^ { s } d _ { k } D _ { k } \end{equation*}
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005028.png" />.
+
such that $d _ { k } &lt; 1$ for all $k$.
  
There exist positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005029.png" /> such that the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005031.png" />-ample (cf. also [[Ample vector bundle|Ample vector bundle]]).
+
There exist positive integers $e_k$ such that the divisor $- \sum _ { k = 1 } ^ { s } e _ { k } D _ { k }$ is $\mu$-ample (cf. also [[Ample vector bundle|Ample vector bundle]]).
  
For example, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005032.png" /> is weak log-terminal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005033.png" /> is smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005034.png" /> is a normal crossing divisor, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005035.png" /> has only quotient singularities and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005036.png" />.
+
For example, the pair $( X , B )$ is weak log-terminal if $X$ is smooth and $B = \sum _ { j = 1 } ^ { t } B _ { j }$ is a normal crossing divisor, or if $X$ has only quotient singularities and $B = 0$.
  
 
==Rationality theorem.==
 
==Rationality theorem.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005037.png" /> be a normal algebraic variety defined over an algebraically closed field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005038.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005039.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005040.png" />-divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005041.png" /> such that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005042.png" /> is weakly log-terminal. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005043.png" /> be a projective morphism (cf. [[Projective scheme|Projective scheme]]) to another algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005045.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005046.png" />-ample Cartier divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005047.png" />. Then (the rationality theorem, [[#References|[a1]]])
+
Let $X$ be a normal algebraic variety defined over an algebraically closed field of characteristic $0$, and let $B$ be a $\mathbf{Q}$-divisor on $X$ such that the pair $( X , B )$ is weakly log-terminal. Let $f : X \rightarrow S$ be a projective morphism (cf. [[Projective scheme|Projective scheme]]) to another algebraic variety $S$, and let $H$ be an $f$-ample Cartier divisor on $X$. Then (the rationality theorem, [[#References|[a1]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005048.png" /></td> </tr></table>
+
\begin{equation*} \lambda = \operatorname { sup } \{ t \in \mathbf{Q} : H + t ( K _ { X } + B ) \text { is } f\square \text{ ample} \} \end{equation*}
  
is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005049.png" /> or a rational number. In the latter case, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005050.png" /> be the smallest positive integer such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005051.png" /> is a Cartier divisor, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005052.png" /> be the maximum of the dimensions of geometric fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005053.png" />. Express <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005054.png" /> for relatively prime positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005056.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005057.png" />.
+
is either $+ \infty$ or a rational number. In the latter case, let $r$ be the smallest positive integer such that $r ( K _ { X } + B )$ is a Cartier divisor, and let $d$ be the maximum of the dimensions of geometric fibres of $f$. Express $\lambda / r = p / q$ for relatively prime positive integers $p$ and $q$. Then $q \leq r ( d + 1 )$.
  
For example, equality is attained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005060.png" /> is a point, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005061.png" /> is a hyperplane section.
+
For example, equality is attained when $X = {\bf P} ^ { d }$, $B = 0$, $S$ is a point, and $H$ is a hyperplane section.
  
The following theorem asserts the existence of a rational curve, a birational image of the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005062.png" />, and provides a more geometric picture. However, the estimate of the denominator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005063.png" /> obtained is weaker: In the situation of the above rationality theorem, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005064.png" />, then there exists a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005065.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005066.png" /> is a point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005067.png" /> [[#References|[a2]]].
+
The following theorem asserts the existence of a rational curve, a birational image of the projective line $\mathbf{P}^{1}$, and provides a more geometric picture. However, the estimate of the denominator $q \leq 2 d r$ obtained is weaker: In the situation of the above rationality theorem, if $\lambda \neq + \infty$, then there exists a morphism $g : \mathbf{P} ^ { 1 } \rightarrow X$ such that $f \circ g ( \mathbf{P} ^ { 1 } )$ is a point and $0 &lt; - ( K _ { X } + B ) , g ( \mathbf{P} ^ { 1 } ) \leq 2 d$ [[#References|[a2]]].
  
The two theorems are related in the following way: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005068.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005069.png" /> is no longer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005070.png" />-ample. However, there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005071.png" /> such that the natural homomorphism
+
The two theorems are related in the following way: If $\lambda \neq + \infty$, then $H + \lambda ( K _ { X } + B )$ is no longer $f$-ample. However, there exists a positive integer $m_0$ such that the natural homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005072.png" /></td> </tr></table>
+
\begin{equation*} f ^ { * } f_{*} \mathcal{O} _ { X } ( m q ( H + \lambda ( K _ { X } + B ) ) ) \rightarrow \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005073.png" /></td> </tr></table>
+
\begin{equation*} \rightarrow \mathcal{O} _ { X } ( m q ( H + \lambda ( K _ { X } + B ) ) ) \end{equation*}
  
is surjective for any positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005074.png" /> (the base-point-free theorem, [[#References|[a1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005075.png" /> be the associated morphism over the base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005076.png" />. Then any positive dimensional fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120050/k12005077.png" /> is covered by a family of rational curves as given in the second theorem [[#References|[a2]]].
+
is surjective for any positive integer $m \geq m _ { 0 }$ (the base-point-free theorem, [[#References|[a1]]]). Let $\phi : X \rightarrow Y$ be the associated morphism over the base space $S$. Then any positive dimensional fibre of $\phi$ is covered by a family of rational curves as given in the second theorem [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , ''Algebraic Geometry (Sendai 1985)'' , ''Adv. Stud. Pure Math.'' , '''10''' , Kinokuniya&amp; North-Holland (1987) pp. 283–360 {{MR|0946243}} {{ZBL|0672.14006}} </TD></TR>
+
<tr><td valign="top">[a1]</td> <td valign="top"> Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , ''Algebraic Geometry (Sendai 1985)'' , ''Adv. Stud. Pure Math.'' , '''10''' , Kinokuniya&amp; North-Holland (1987) pp. 283–360 {{MR|0946243}} {{ZBL|0672.14006}} </td></tr>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Kawamata, "On the length of an extremal rational curve" ''Invent. Math.'' , '''105''' (1991) pp. 609–611 {{MR|1117153}} {{ZBL|0751.14007}} </TD></TR>
+
<tr><td valign="top">[a2]</td> <td valign="top"> Y. Kawamata, "On the length of an extremal rational curve" ''Invent. Math.'' , '''105''' (1991) pp. 609–611 {{MR|1117153}} {{ZBL|0751.14007}} </td></tr>
 
</table>
 
</table>
  
 
[[Category:Algebraic geometry]]
 
[[Category:Algebraic geometry]]

Revision as of 15:31, 1 July 2020

A theorem stating that there is a strong restriction for the canonical divisor of an algebraic variety to be negative while the positivity is arbitrary. It is closely related to the structure of the cone of curves and the existence of rational curves.

Definitions and terminology.

Let $X$ be a normal algebraic variety (cf. Algebraic variety). A $\mathbf{Q}$-divisor $B = \sum _ { j = 1 } ^ { t } b _ { j } B _ { j }$ on $X$ is a formal linear combination of a finite number of prime divisors $B _ { j }$ of $X$ with rational number coefficients $b _ { j }$ (cf. also Divisor). The canonical divisor $K _ { X }$ is a Weil divisor on $X$ corresponding to a non-zero rational differential $n$-form for $n = \operatorname { dim } X$ (cf. also Differential form). The pair $( X , B )$ is said to be weakly log-terminal if the following conditions are satisfied:

The coefficients of $B$ satisfy $0 \leq b _ { j } \leq 1$.

There exists a positive integer $r$ such that $r ( K _ { X } + B )$ is a Cartier divisor (cf. Divisor).

There exists a projective birational morphism $\mu : Y \rightarrow X$ from a smooth variety such that the union

\begin{equation*} \sum _ { j = 1 } ^ { t } \mu _ { * } ^ { - 1 } B _ { j } + \sum _ { k = 1 } ^ { s } D _ { k } \end{equation*}

is a normal crossing divisor (cf. Divisor), where $\mu_* ^ {- 1 } B _ { j }$ is the strict transform of $B _ { j }$ and $\cup _ { k = 1 } ^ { s } D _ { k }$ coincides with the smallest closed subset $E$ of $Y$ such that $\mu | _ { Y \backslash E } : Y \backslash E \rightarrow X \backslash \mu ( E )$ is an isomorphism.

One can write

\begin{equation*} \mu ^ { * } ( K _ { X } + B ) = K _ { Y } + \sum _ { j = 1 } ^ { t } b _ { j } \mu _ { * } ^ { - 1 } B _ { j } + \sum _ { k = 1 } ^ { s } d _ { k } D _ { k } \end{equation*}

such that $d _ { k } < 1$ for all $k$.

There exist positive integers $e_k$ such that the divisor $- \sum _ { k = 1 } ^ { s } e _ { k } D _ { k }$ is $\mu$-ample (cf. also Ample vector bundle).

For example, the pair $( X , B )$ is weak log-terminal if $X$ is smooth and $B = \sum _ { j = 1 } ^ { t } B _ { j }$ is a normal crossing divisor, or if $X$ has only quotient singularities and $B = 0$.

Rationality theorem.

Let $X$ be a normal algebraic variety defined over an algebraically closed field of characteristic $0$, and let $B$ be a $\mathbf{Q}$-divisor on $X$ such that the pair $( X , B )$ is weakly log-terminal. Let $f : X \rightarrow S$ be a projective morphism (cf. Projective scheme) to another algebraic variety $S$, and let $H$ be an $f$-ample Cartier divisor on $X$. Then (the rationality theorem, [a1])

\begin{equation*} \lambda = \operatorname { sup } \{ t \in \mathbf{Q} : H + t ( K _ { X } + B ) \text { is } f\square \text{ ample} \} \end{equation*}

is either $+ \infty$ or a rational number. In the latter case, let $r$ be the smallest positive integer such that $r ( K _ { X } + B )$ is a Cartier divisor, and let $d$ be the maximum of the dimensions of geometric fibres of $f$. Express $\lambda / r = p / q$ for relatively prime positive integers $p$ and $q$. Then $q \leq r ( d + 1 )$.

For example, equality is attained when $X = {\bf P} ^ { d }$, $B = 0$, $S$ is a point, and $H$ is a hyperplane section.

The following theorem asserts the existence of a rational curve, a birational image of the projective line $\mathbf{P}^{1}$, and provides a more geometric picture. However, the estimate of the denominator $q \leq 2 d r$ obtained is weaker: In the situation of the above rationality theorem, if $\lambda \neq + \infty$, then there exists a morphism $g : \mathbf{P} ^ { 1 } \rightarrow X$ such that $f \circ g ( \mathbf{P} ^ { 1 } )$ is a point and $0 < - ( K _ { X } + B ) , g ( \mathbf{P} ^ { 1 } ) \leq 2 d$ [a2].

The two theorems are related in the following way: If $\lambda \neq + \infty$, then $H + \lambda ( K _ { X } + B )$ is no longer $f$-ample. However, there exists a positive integer $m_0$ such that the natural homomorphism

\begin{equation*} f ^ { * } f_{*} \mathcal{O} _ { X } ( m q ( H + \lambda ( K _ { X } + B ) ) ) \rightarrow \end{equation*}

\begin{equation*} \rightarrow \mathcal{O} _ { X } ( m q ( H + \lambda ( K _ { X } + B ) ) ) \end{equation*}

is surjective for any positive integer $m \geq m _ { 0 }$ (the base-point-free theorem, [a1]). Let $\phi : X \rightarrow Y$ be the associated morphism over the base space $S$. Then any positive dimensional fibre of $\phi$ is covered by a family of rational curves as given in the second theorem [a2].

References

[a1] Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , Algebraic Geometry (Sendai 1985) , Adv. Stud. Pure Math. , 10 , Kinokuniya& North-Holland (1987) pp. 283–360 MR0946243 Zbl 0672.14006
[a2] Y. Kawamata, "On the length of an extremal rational curve" Invent. Math. , 105 (1991) pp. 609–611 MR1117153 Zbl 0751.14007
How to Cite This Entry:
Kawamata rationality theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kawamata_rationality_theorem&oldid=49934
This article was adapted from an original article by Yujiro Kawamata (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article