Well formedness vs weak well formedness

TL;DR

This paper compares two definitions of well formedness in weighted projective spaces, showing they differ generally but coincide for certain quasi-smooth weighted complete intersections of dimension three or more.

Contribution

It clarifies the relationship between two existing definitions of well formedness and identifies conditions under which they agree.

Findings

The two definitions of well formedness differ in general.

They coincide for quasi-smooth weighted complete intersections of dimension at least 3.

The paper provides a precise comparison and conditions for equivalence.

Abstract

In the literature there are two definitions of well formed varieties in weighted projective spaces. According to the first one, well formed variety is the one whose intersection with the singular locus of the ambient weighted projective space has codimension at least two, while, according to the second one, well formed variety is the one who does not contain in codimension one a singular stratum of the ambient weighted projective space. We show that these two definitions indeed differ, and show that they coincide for quasi-smooth weighted complete intersections of dimension at least 3.