Refinement of geometric vertex decomposition lifting to explain double determinantal splittings

Develop a refinement of the lifting of Frobenius splittings through geometric vertex decomposition that accounts for and explains the explicit Frobenius splitting of Li’s double determinantal ideals defined by maximal minors, namely ideals I = I_n(H) + I_n(V) formed from the horizontal and vertical concatenations of r generic m×n matrices X_1,…,X_r over a perfect field of characteristic p with m = n, where the current lifting theorem cannot be directly applied to this case.

Background

The paper proves a partial converse to a result of Knutson on Frobenius splittings and geometric vertex decompositions, showing that under additional hypotheses a splitting that compatibly splits both the link and deletion can be lifted to a splitting of the original ideal. An example demonstrates that the additional hypothesis cannot be removed.

Later, the authors construct Frobenius splittings for Li’s double determinantal varieties defined by maximal minors. Although the construction is guided by the main theorem’s framework, the main lifting theorem does not directly apply to these double determinantal ideals. The authors therefore highlight a gap in the current theory and call for a refinement that would explain why these examples are Frobenius split within a unified theoretical framework.