A $\mathrm{GL}$-variety $X$ is an (infinite-dimensional) affine variety with an action of the infinite general linear group $\mathrm{GL}$ such that the coordinate ring of $X$ is a polynomial  $\mathrm{GL}$-representation and generated by finitely many $\mathrm{GL}$-orbits of elements. In earlier work, we established that $\mathrm{GL}$-varieties are topologically Noetherian: they satisfy the descending chain condition on closed subvarieties. The proof of this fact led to a coarse understanding of the geometry of $\mathrm{GL}$-varieties, e.g. to a version of Chevalley's theorem on constructible sets and to the insight that $\mathrm{GL}$-varieties are ``unirational in the $\mathrm{GL}$-direction'': they admit a dominant $\mathrm{GL}$-equivariant morphism from a $\mathrm{GL}$-variety of the form $B \times A$ where $B$ is a finite-dimensional affine variety with trivial $\mathrm{GL}$-action and $A$ is an affine space with linear $\mathrm{GL}$-action. In this talk I will present recent work with Bik, Eggermont, and Snowden on finer aspects of the geometry of $\mathrm{GL}$-varieties. For instance, we show that any two points in an irreducible $\mathrm{GL}$-variety can be joined by an (ordinary) curve, and we use this to establish uniformity results for limits of tensor decompositions.