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.