Singularities of generic projection hypersurfaces

TL;DR

This paper investigates the singularities of hypersurfaces obtained by generic linear projections of smooth projective varieties, showing they are Du Bois and semi log canonical in low dimensions but not in high dimensions.

Contribution

It establishes the nature of singularities in low-dimensional cases and provides counterexamples in high dimensions, advancing understanding of projection hypersurfaces.

Findings

Low-dimensional projections have Du Bois singularities.

Such singularities are semi log canonical in low dimensions.

Counterexamples exist in high dimensions where singularities are worse.

Abstract

Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension -- the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois.