Buchweitz–Greuel–Schreyer conjecture on minimal MCM rank

Let $P$ be a regular local ring with algebraically closed residue field and let $f\in P$. Define

$$e:=\left\lfloor \frac{1}{2}\left(\operatorname{codim}_{P/f}\operatorname{Sing}(P/f)-2\right)\right\rfloor,$$

where $\operatorname{Sing}(P/f)$ denotes the singular locus of $P/f$. Buchweitz–Greuel–Schreyer conjecture. The minimal rank of a nonfree maximal Cohen–Macaulay module on $P/f$ is at least $2^e$. This conjecture gives a lower bound for the ranks of nonfree maximal Cohen–Macaulay modules over hypersurface singularities; the paper proves it for generic homogeneous polynomials of prescribed degree and strength, with sharp examples.