论文标题
关于合理的乘法小组动作
On rational multiplicative group actions
论文作者
论文摘要
我们在代数品种$ x $上的理性乘法小组操作与衍生$ \ partial \ colon k_x \ to k_x $ of a $ k_x $ $ x $满足的$ k_x $ of ASSEATION $ k_x $ a i \ x $ \ x $ k_x $ \ x $ \ partial（a_i）=λ_ia_i $ at \ mathbb {z} $ in i $中的所有$ i \。我们称这种派生为理性的半圣事。此外，我们还证明了每个有理半衍生的有理切片，即k_x $中的元素$ s \，使得$ \ partial（s）= s $。通过类似于加法组动作案例的情况，我们证明$ k_x \ simeq k_x^{\ mathbb {g} _m} $} $，并且在此同构中，派生$ \ partial $由$ \ partial = s partial = s \ frac = s \ frac {d} {ds ds ds} $。在这里，$ k_x^{\ mathbb {g} _m} $是$ \ mathbb {g} _m $ -action的不变的字段。
We establish a one-to-one correspondence between rational multiplicative group actions on an algebraic variety $X$ and derivations $\partial\colon K_X\to K_X$ of the field of fractions $K_X$ of $X$ satisfying that there exists a generating set $\{a_i\}_{i\in I}$ of $K_X$ as a field such that $\partial(a_i)=λ_i a_i$ with $λ_i \in \mathbb{Z}$ for all $i\in I$. We call such derivations rational semisimple. Furthermore, we also prove the existence of a rational slice for every rational semisimple derivation, i.e., an element $s\in K_X$ such that $\partial(s)=s$. By analogy with the case of additive group actions case, we prove that $K_X\simeq K_X^{\mathbb{G}_m}(s)$ and that under this isomorphism the derivation $\partial$ is given by $\partial=s\frac{d}{ds}$. Here, $K_X^{\mathbb{G}_m}$ is the field of invariant of the $\mathbb{G}_m$-action.
