Let
be a two-dimensional standard-graded normal domain over an algebraically closed field of positive characteristic. Let
be a homogeneous
-primary ideal with homogeneous generators of degree
. Let
be the syzygy bundle on
and suppose that the Harder-Narasimhan filtration of
is strong, and let
,
,
be the corresponding slopes. We set
and
.
Then the Hilbert-Kunz multiplicity of
![{\displaystyle {}I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f0f26f3c8795b5a82e275342dcc3bd42e64d8e)
is
-
![{\displaystyle {}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)}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55eade08fbef02a2d3cbea2fd4f93d50c00ec3c5)
In particular, it is a rational number.