On Nash Blowing-Up
On Nash Blowing-Up
复制标题
论纳什的爆发
DOI：
10.1007/978-1-4757-9286-7_6
复制
发表时间：
1983
影响因子：
1.8
通讯作者：
H. Hironaka
中科院分区：
文献类型：
--
作者：
H. Hironaka
Let X be an algebraic variety, reduced and equidimensional, over the base field k of characteristic zero. Let us consider a sequence of transformations
$$ {X_0} = X\xleftarrow{{{\sigma _1}}}{X_1}\xleftarrow{{{\sigma _2}}}{X_2} \leftarrow \cdots $$
where \( {\sigma _i}:{X_i} \to {X_{i - 1}} \) for each \( i \geqslant 1 \) is
(1)
birational, i.e., proper and almost everywhere isomorphic, while X i is reduced and equidimensional, and
(2)
\( \sigma _i^*\left( {{\Omega _{{X_{i - 1}}}}} \right) \) /(its torsion) is locally free as \( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} _{Xi}} \) -module. Here Ω denotes the sheaf of Kahler differentials on the variety and the torsion means the subsheaf consisting of those local sections whose supports are nowhere dense.
Let X be an algebraic variety, reduced and equidimensional, over the base field k of characteristic zero. Let us consider a sequence of transformations
$$ {X_0} = X\xleftarrow{{{\sigma _1}}}{X_1}\xleftarrow{{{\sigma _2}}}{X_2} \leftarrow \cdots $$
where \( {\sigma _i}:{X_i} \to {X_{i - 1}} \) for each \( i \geqslant 1 \) is
(1)
birational, i.e., proper and almost everywhere isomorphic, while X i is reduced and equidimensional, and
(2)
\( \sigma _i^*\left( {{\Omega _{{X_{i - 1}}}}} \right) \) /(its torsion) is locally free as \( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} _{Xi}} \) -module. Here Ω denotes the sheaf of Kahler differentials on the variety and the torsion means the subsheaf consisting of those local sections whose supports are nowhere dense.