View source for Kawamata rationality theorem

A theorem stating that there is a strong restriction for the canonical [[Divisor|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.==
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:

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" />.

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 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

<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>

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.

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>

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" />.

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]]).

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" />.

==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]]])

<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>

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" />.

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.

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 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

<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>

<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>

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]]].

====References====
<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">[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>