Grassmannians and Singularities

TL;DR

This paper proves that near any singular point of an integral scheme over a perfect field, there exists an open subset that admits a resolution, ensuring local smooth models for singularities.

Contribution

It establishes the existence of resolutions of singularities in a local setting for integral schemes over perfect fields, extending the understanding of singularity resolution.

Findings

Existence of open subsets with resolutions around singular points

Construction of smooth schemes birational to neighborhoods of singularities

Advancement in local resolution techniques for algebraic schemes

Abstract

Let $X$ be an integral scheme of finite presentation over a perfect field. Let $q$ be a singular closed point of $X$. We prove that there exists an open subset $V$ of $X$ containing $q$ such that $V$ admits a resolution, that is, there exists a smooth scheme $\widetilde V$ and a proper birational morphism from $\widetilde V$ onto $V$.