CANONICAL FACTORIZATION OF THE QUOTIENT MORPHISM FOR AN AFFINE G a -VARIETY
CANONICAL FACTORIZATION OF THE QUOTIENT MORPHISM FOR AN AFFINE G a -VARIETY
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仿射G a 簇的商态射的正则因式分解
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通讯作者：
A. H. Awad
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作者：
Ahmed M. Abdelbaky;W. G. Elmasry;A. H. Awad
.本文研究了特征为零的基域上的商态射X！Y对于a（cid：14）ne G a -簇X具有（cid：14）ne商Y .证明了与Ga-作用相关的度模给出了一个唯一确定的显性Ga-等变态射序列，其中Xi是a（cid：14）neG-簇，Xi +1！对于每个i（cid：21）1，X i是双有理的。这是（cid：25）的典型因式分解。我们给出了一个算法（cid：12）nding与G a -作用相关联的度模，这产生了（cid：25）的典范因子分解.该算法适用于计算的典范因式分解的几个例子，包括齐次（2 ; 5）行动的A3。在本文的最后一节中引入的自由猜想断言，对X = A3上的任何Ga-作用，多项式环k [ X ]是k [ X ] Ga上的自由模。
. Working over a ground (cid:12)eld of characteristic zero, this paper studies the quotient morphism (cid:25) : X ! Y for an a(cid:14)ne G a -variety X with a(cid:14)ne quotient Y . It is shown that the degree modules associated to the G a -action give a uniquely determined sequence of dominant G a -equivariant morphisms, ; where X i is an a(cid:14)ne G a -variety and X i +1 ! X i is birational for each i (cid:21) 1. This is the canonical factorization of (cid:25) . We give an algorithm for (cid:12)nding the degree modules associated to the G a -action, and this yields the canonical factorization of (cid:25) . The algorithm is applied to compute the canonical factorization for several examples, including the homogeneous (2 ; 5) action on A 3 . The Freeness Conjecture , introduced in the paper’s last section, asserts that, for any G a -action on X = A 3 , the polynomial ring k [ X ] is a free module over k [ X ] G a .