Hilbert-Kunz multiplicity/Two-dimensional graded/Brenner-Trivedi-formula/Fact

Let ${}R$ be a two-dimensional standard-graded normal domain over an algebraically closed field of positive characteristic. Let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be a homogeneous ${}R_{+}$-primary ideal with homogeneous generators of degree ${}d_{i}$. Let ${}{\mathcal {S}}=\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}$ be the syzygy bundle on ${}C=\operatorname {Proj} {\left(R\right)}$ and suppose that the Harder-Narasimhan filtration of ${}F^{e*}({\mathcal {S}})$ is strong, and let ${}\mu _{k}$, ${}k=1,\ldots ,t$, be the corresponding slopes. We set ${}\nu _{k}={\frac {-\mu _{k}}{\operatorname {deg} \,(C)p^{e}}}$ and ${}r_{k}=\operatorname {rk} \,({\mathcal {S}}_{k}/{\mathcal {S}}_{k-1})$.

Then the Hilbert-Kunz multiplicity of ${}I$ is

${}e_{HK}(I)={\frac {\operatorname {deg} \,(C)}{2}}{\left(\sum _{k=1}^{t}r_{k}\nu _{k}^{2}-\sum _{i=1}^{n}d_{i}^{2}\right)}\,.$

In particular, it is a rational number.

Write a proof