A GL-variety is a reduced affine scheme whose coordinate ring is a polynomial representation that is finitely generated up to the action of the infinite general linear group GL.

Equivalently, GL-varieties are inverse limits of Vec-varieties, functors from the category of finite dimensional vector spaces to reduced affine varieties.

In a joint work with Jan Draisma and Christopher Chiu, taking the perspective of Vec-varieties, we show that the (inverse) limit of the singular loci of a Vec-variety evaluated in dimension n defines a closed Vec-subvariety of “singular points”.

In a forthcoming paper with Andrew Snowden, we show that the GL-variety obtained from these "singular points" is actually the locus of singular points of the initial GL-variety, namely, the following loci coincide:

1. The closed GL-variety arising from taking the limit of the singular points in the corresponding Vec-variety.
2. The locus of points where the module of Kähler differential is not flat.
3. The locus of points where the tangent space and tangent cone are not isomorphic.
4. The locus of points that are not formally smooth.
5. The locus of points where there is not an n_0 such that for all n> n_0 an n-jet lift to an n+1-jet.

In this talk, I will present the necessary background and more details about the aforementioned results.