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

Abstract

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