The Han–Monsky lower-bound and rigidity conjecture for Hilbert–Kunz multiplicity

Let $d\geq1$ be an integer and let $p>2$ be prime. Define

$$A_{p,d}=\overline{\mathbb F_p}[[X_0,X_1,\ldots,X_d]]/(X_0^2+\cdots+X_d^2).$$

Let $(A,\mathfrak m,k)$ be a $d$-dimensional unmixed local ring with $k=\overline{\mathbb F_p}$. Han–Monsky lower-bound and rigidity conjecture. If $A$ is not regular, then

$$e_{\rm HK}(A)\geq e_{\rm HK}(A_{p,d})\geq1+\frac{c_d}{d!},$$

and in particular $s_{\rm HK}(p,d)=e_{\rm HK}(A_{p,d})$. Moreover, if $e_{\rm HK}(A)=e_{\rm HK}(A_{p,d})$, then $\widehat A\cong A_{p,d}$ as local rings. The conjecture generalizes the paper's low-dimensional results to arbitrary dimension; the supplied text notes that it is proved when $\dim A=4$ but does not establish it in general.