A geometric realisation of 0-Schur and 0-Hecke algebras

We define a new product on orbits of pairs of flags in a vector space over a field $k$, using open orbits in certain varieties of pairs of flags.
This new product defines an associative $\mathbb{Z}$-algebra, denoted by $G(n,r)$. We show that $G(n,r)$ is a geometric realisation of the $0$-Schur algebra $S_0(n, r)$ over $\mathbb{Z}$, which is the $q$-Schur algebra $S_q(n,r)$ at $q=0$. A pair of flags naturally determines a pair of projective resolutions for a quiver of type $\mathbb{A}$ with linear orientation, and we study $q$-Schur algebras from this point of view. This allows us to understand
the relation between $q$-Schur algebras and Hall algebras and to construct bases of $q$-Schur algebras. Using the geometric realisation, we construct idempotents and multiplicative bases for $0$-Schur algebras.
We also give a geometric realisation of $0$-Hecke algebras and a presentation of the $q$-Schur algebra over a ground ring, where $q$ is not invertible.