论文标题
F-Pure阈值和极端奇异性的下限
Lower Bounds on the F-pure Threshold and Extremal Singularities
论文作者
论文摘要
我们证明，如果$ f $是$ d $的同质多项式，那么在唯一同质最大理想下的$ f $ pure-pure阈值至少为$ \ frac {1} {d-1} $。此外，我们还表明其$ f $ pure阈值等于$ \ frac {1} {d-1} $，并且仅当$ f \ in \ mathfrak m^{[q]} $和$ d = q+1 $，其中$ q $是$ p $的功率。直到坐标的线性变化（在固定的代数封闭场上），我们对这种“极端奇点”进行了分类，并表明最多有孤立的奇异性。最后，我们指出了几种方式，例如，这种形式定义的投影性高空是“极端”，例如，就它们可以包含的线条配置而言。
We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities," and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain.
