论文标题
有限$ f $ - 代表类型的非fano品种的均匀坐标环
Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties
论文作者
论文摘要
有限的$ f $ - 代表类型是特征 - $ p $交换代数的重要概念，但是有或没有此属性的品种的明确示例很少。我们证明，具有积极特征的大量均质坐标环将无法具有有限的$ f $ - 代表类型。为此，我们证明了$ x $的均匀坐标环上的差分运算符与$（\ mathrm {sym}^Mω_x）^\ vee $的全局部分的存在之间的连接。通过高税和高桥的结果，这使我们能够将FFRT排除在与$（\ Mathrm {sym}^Mω_x）^\ vee $的坐标坐标。通过使用（CO）切线式带束带的结果阳性和可准性条件，我们表明几类品种无法具有有限的$ f $ apressentation类型，包括Abelian品种，大多数Calabi-calabi-Yau-Yau品种以及一般类型的完整交叉点。我们的工作还提供了非差异型二型葡萄酒的差分运算符结构的示例，到目前为止，这在很大程度上尚未探索。
Finite $F$-representation type is an important notion in characteristic-$p$ commutative algebra, but explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings in positive characteristic will fail to have finite $F$-representation type. To do so, we prove a connection between differential operators on the homogeneous coordinate ring of $X$ and the existence of global sections of a twist of $(\mathrm{Sym}^m Ω_X)^\vee$. By results of Takagi and Takahashi, this allows us to rule out FFRT for coordinate rings of varieties with $(\mathrm{Sym}^m Ω_X)^\vee$ not ``positive''. By using results positivity and semistability conditions for the (co)tangent sheaves, we show that several classes of varieties fail to have finite $F$-representation type, including abelian varieties, most Calabi--Yau varieties, and complete intersections of general type. Our work also provides examples of the structure of the ring of differential operators for non-$F$-pure varieties, which to this point have largely been unexplored.
