Differential operators, $n$-branch curve singularities and the $n$-subspace problem

Abstract:

Let $R$ be the coordinate ring of a smooth affine curve over an algebraically closed field of characteristic zero $k$. For $S$ a subalgebra of $R$ with integral closure $R$ denote by $\mathcal {D}(S)$ the ring of differential operators on $S$ and by $H(S)$ the finite-dimensional factor of $\mathcal {D}(S)$ by its unique minimal ideal. The theory of diagonal $n$-subspace systems is introduced. This is used to show that if $A$ is a finite-dimensional $k$-algebra and $t \geqslant 1$ is any integer there exists such an $S$ with \[ H(S) \cong \left ( {\begin {array}{*{20}{c}} A & {\ast } \\ 0 & {{M_t}(k)} \\ \end {array} } \right ).\] Further, the Morita classes of $H(S)$ are classified for curves with few branches, and it is shown how to lift Morita equivalences from $H(S)$ to $\mathcal {D}(S)$.