F-regularity does not deform

Deformation of F-purity and F-regularity

Hochster and Huneke showed that the property of F-regularity deforms for Gorenstein rings, i.e., if (R,m) is a Gorenstein local ring such that R/tR is F-regular for some nonzerodivisor t in m, then R is F-regular. This result was later extended to the case of Q-Gorenstein rings by Smith (for rings of characteristic zero) and Aberbach, Katzman, and MacCrimmon (for rings of positive characteristic). We investigate the deformation of strong F-regularity using an anti-canonical cover of R, i.e., a symbolic Rees algebra S = R + It + I^(2)t^2 + ..., where I is an inverse for the canonical module in …

When does the F-signature exist?

We show that the F-signature of an F-finite local ring R of characteristic p >0 exists when R is either the localization of an $\mathbf{N}$-graded ring at its irrelevant ideal or $\mathbf{Q}$-Gorenstein on its punctured spectrum. This extends results by Huneke, Leuschke, Yao and Singh and proves the existence of the F-signature in the cases where weak F-regularity is known to be equivalent to strong F-regularity.

残差交集的变形

Deformation of Residual Intersections

研究显示在Cohen-Macaulay局部环中，理想I的通用链接是其任意链接的变形，无需I为Cohen-Macaulay理想。对于s-残差交集，当s不超过I的高度加一时，同样成立。在某些条件下，该原理可推广到任意s-残差交集。

F-内射性不发生形变

$F$-injectivity does not deform

研究特征为2的F-有限四维局部整环的F-内射性形变，构造出非F-内射但模非零因子后为F-内射的反例，证明F-内射性不发生形变。

关于纤维的半连续性问题与整体$F$-正则性

On the semicontinuity problem of fibers and global $F$-regularity

本文研究概形态射下纤维性质的半连续性问题，从局部层面通过解决局部环的提升与局部化问题得到半正规环、$F$-有理环的局部化定理，又从整体层面以整体$F$-正则性的形变为测试示例开展研究，推进了相关代数几何问题的探索。