Quasi-Gorenstein (Normal) Finite Covers in Arbitrary Characteristic
Ehsan Tavanfar
TL;DR
The paper proves that every normal complete local domain admits a finite quasi-Gorenstein extension and that every normal projective variety over a field admits a finite quasi-Gorenstein cover, covering residual characteristic two as well. It develops two parallel constructions: a complete local Bertini-based method and a projective-geometry approach using integral closures in function-field extensions, both yielding normal, quasi-Gorenstein finite covers and leveraging graded structures to establish $Q$-Gorenstein-ness. The work extends Kawamata’s and Griffith’s frameworks to arbitrary characteristic, provides descent and base-change results to handle finite and infinite base fields, and yields applications such as a converse to Roberts’ module-freeness result and a Patankar–Vanishing type Corollary in dimension two. Overall, it offers robust finite-cover techniques to improve singularities while respecting canonical divisors, with potential implications for factoriality and rational singularity theory in mixed characteristics. Key innovations include carefully orchestrated Bertini steps in both local and global settings and detailed control of the canonical/anticanonical algebras to guarantee quasi-Gorenstein outcomes.
Abstract
We show that any complete local (normal) domain admits a module-finite quasi-Gorenstein normal (complete local) domain extension. In the geometric vein, we show that any normal projective variety $X$ over a field admits a finite surjective morphism $Y\rightarrow X$ from a normal quasi-Gorenstein projective variety $Y$. Notably, our results resolve the previously open case for residual characteristic two.