Reflection principle for strange-duality sections

Determine whether, whenever a theta line bundle \(\Theta\) has negative intersection \((\Theta,\Delta)=-k<0\) with an effective divisor \(\Delta\) of square \(-2\), every theta section produced by the strange duality map lies in \(s_\Delta^k H^0(M,s_\Delta(\Theta))\), and whether the rank of the strange duality map equals the dimension of the section space obtained by successive reflections in such divisors.

Background

The three counterexamples share a mechanism involving an effective square-$-2$ divisor on a moduli space and a theta line bundle with negative degree on that divisor. The strange-duality sections vanish along the divisor to an order greater than the fixed multiplicity forced by the complete linear system.

Reflecting the theta class in the square-$-2$ divisor produces a better-behaved line bundle in the examples. The authors ask whether this reflection procedure universally controls both the image of the strange duality map and its rank.