Algebra & Number Theory ( IF 0.9 ) Pub Date : 2021-11-10 , DOI: 10.2140/ant.2021.15.2019 Devlin Mallory
We consider the question of simplicity of a -algebra under the action of its ring of differential operators . We give examples to show that even when is Gorenstein and has rational singularities, need not be a simple -module; for example, this is the case when is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano varieties, and our proof proceeds by showing that the tangent bundle of such a variety need not be big. We also give a partial converse showing that when is the homogeneous coordinate ring of a smooth projective variety , embedded by some multiple of its canonical divisor, then simplicity of as a -module implies that is Fano and thus has rational singularities.
中文翻译:
del Pezzo 曲面的切丛的大小和 D-简单性
我们考虑一个简单的问题 -代数 在其微分算子环的作用下 . 我们举例说明,即使当 是 Gorenstein 并且具有有理奇点, 不必是一个简单的 -模块; 例如,当是光滑三次曲面的齐次坐标环。我们的例子是光滑 Fano 簇的齐次坐标环,我们的证明是通过证明这种簇的切丛不需要很大来进行的。我们还给出了部分逆向证明,当 是光滑射影簇的齐次坐标环 ,嵌入其规范除数的一些倍数,然后简单 作为一个 -module 意味着 是法诺,因此 有理性的奇点。