Reduced complements and linear-type maps on D-type klt singularities

Determine whether a non-canonical D-type klt surface singularity admitting a curve smooth at the singular point necessarily admits a reduced 2-complement.

Background

The paper proves that a reduced 2-complement on a non-canonical D-type klt surface singularity yields a curve smooth at the singular point. Since a smooth curve is equivalent in this setting to the existence of a singular linear-type map, the converse would identify linear-type-map existence with reduced-complement existence.

The authors explain that reduced 2-complements exist formally but need not exist on the original algebraic germ, making the converse implication a genuinely unresolved algebraic question.

References

Let $x\in X$ be a non-canonical $D$-type klt surface singularity. If there is a curve $$ C\subset X $$ which is smooth at $x$, does $x\in X$ admit a reduced $2$-complement?

— Singularities of rational maps: foundations and surfaces  (2608.16218 - Birkar, 17 Aug 2026) in Question labeled (\ref{q-D-type-smooth-curve-reduced-complement}), Section 10.5, “D-type klt singularities”