We will restrict now to the two-dimensional homogeneous case in order to work on the corresponding projective curve.
Let
be a two-dimensional standard-graded normal domain over an algebraically closed field
. Let
-
![{\displaystyle {}C=\operatorname {Proj} {\left(R\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/618e824cfea008c4803a06e3d1cd854ec5723ecf)
be the corresponding smooth projective curve and let
-
![{\displaystyle {}I={\left(f_{1},\ldots ,f_{n}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe0e095be425fc8946f1c9fbadd6cbf3ee152e6)
be an
-primary homogeneous ideal with generators of degrees
. Then we get on
the short exact sequence
-
Here
is a vector bundle, called the syzygy bundle, of rank
and of degree
-
Our approach to the computation of the Hilbert-Kunz multiplicity is by using the presenting sequence
-
and twists of its
-th Frobenius pull-backs, that is
-
(where
),
and to relate the asymptotic behavior of
-
![{\displaystyle {}\operatorname {length} \,(R/I^{[q]})=\operatorname {dim} _{K}\,(R/I^{[q]})=\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(R/I^{[q]})_{m}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba68ed97fd09208c4945b22575ee632bf1d087f)
to the asymptotic behavior of the global sections of the Frobenius pull-backs
-
![{\displaystyle {}(F^{e*}(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})(m)=\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a584c524721e2fc0ea2a3295c1e7a6c98b1585b)
What we want to compute is just the cokernel of the complex of global sections of the above sequence, namely
-
![{\displaystyle \operatorname {dim} _{K}\,(R/I^{[q]})_{m}=h^{0}(C,{\mathcal {O}}_{C}(m))-\sum _{i=1}^{n}h^{0}(C,{\mathcal {O}}_{C}(m-qd_{i}))+h^{0}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/469554273b6dc9915a41a925229d8e81ffe17a1b)
The summation over
is finite
(but the range depends on
),
and the terms
-
![{\displaystyle {}h^{0}(C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,\Gamma (C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,R_{m}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd27d985f87020532be1aaf14a5d7dd36022a8d7)
are easy to control, so we have to understand the behavior of the global syzygies
-
for all
and
,
at least asymptotically. This is a Frobenius-Riemann-Roch problem
(so far this works for all normal standard-graded domains).
The strategy for this is to use Riemann-Roch to get a formula for
and then use semistability properties to show that
or
are
in certain ranges.
We need the concept of
(strong)
semistability.
The rational number
is called the slope of a vector bundle.
Let
be a vector bundle on a smooth projective curve
over an algebraically closed field
. Then the
(uniquely determined)
filtration
-
![{\displaystyle {}0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}={\mathcal {S}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b561226bb10edb3f43f5e3adc61a868988ce3e)
of subbundles such that all quotient bundles
are semistable with decreasing slopes
,
is called the
Harder-Narasimhan filtration of
![{\displaystyle {}{\mathcal {S}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac87e311246f158781b37105eb040ded6029c15f)
.
An immediate consequence of this is that the Harder-Narasimhan filtration of all higher Frobenius pull-backs are just the pull-backs of this filtration. With these filtration we can at least Frobenius-asymptotically control the global sections of the pull-backs and hence also the Hilbert-Kunz multiplicity. This implies the following theorem.
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
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.