Flat deformations of ${\mathbb P}^n$

TL;DR

This paper investigates how projective spaces can be deformed flatly, revealing the nature of singular fibers and providing criteria for flatness, with implications for understanding the structure of such deformations.

Contribution

It characterizes the singular fibers in projective flat deformations and describes conditions for flatness when the total space is smooth.

Findings

Singular fibers appear in codimension 1 or over singular base points.

Provides flatness criteria for projective deformations.

Describes deformations with smooth total space.

Abstract

In this paper we study projective flat deformations of projective spaces. We prove that the singular fibers of projective flat deformations of projective spaces appear either in codimension 1 or over singular points of the base. We also describe projective flat deformations of projective spaces with smooth total space, and discuss flatness criteria.