关于三项超曲面和阶乘三项簇的刚性
On rigidity of trinomial hypersurfaces and factorial trinomial varieties
摘要：
三项簇是由三项多项式组成的特殊方程组给出的仿射簇。这类簇是正规有理簇的全坐标空间，其环面作用为复杂作用。对于仿射簇X，我们考虑由同构于基域的加群的所有代数子群生成的自同构群的子群SAut（X）。一个仿射簇X是刚性的，如果SAut（X）是平凡的。相反，如果SAut（X）传递地作用在正则轨迹上，则仿射簇是柔性的。Arzhantsev证明了阶乘三项式超曲面是刚性的一个判据。本文给出了Arzhantsev结果的两个推广：任意三项超曲面为刚性的一个判别准则和阶乘三项簇为刚性的一个判别准则。并得到了三项式超曲面为柔性的一个充分条件。
英文摘要：
Trinomial varieties are affine varieties given by some special system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal rational varieties with torus action of complexity one. For an affine variety X we consider the subgroup SAut(X) of the automorphism group generated by all algebraic subgroups isomorphic to the additive group of the ground field. An affine variety X is rigid if SAut(X) is trivial. In opposite an affine variety is flexible if SAut(X) acts transitively on the regular locus. Arzhantsev proved a criterium for a factorial trinomial hypersurface to be rigid. We give two generalizations of Arzhantsev's result: a criterium for an arbitrary trinomial hypersurface to be rigid and a criterium for a factorial trinomial variety to be rigid. Also a sufficient condition for a trinomial hypersurface to be flexible is obtained.
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