# Hilbert-Kunz theory/Graded situation/Introduction/Section

We will restrict now to the two-dimensional homogeneous case in order to work on the corresponding projective curve.
Let $R$ be a two-dimensional standard-graded normal domain over an algebraically closed field $K$. Let
    $C=\operatorname {Proj} {\left(R\right)}\,$
be the corresponding smooth projective curve and let
    $I={\left(f_{1},\ldots ,f_{n}\right)}\,$
be an $R_{+}$-primary homogeneous ideal with generators of degrees $d_{1},\ldots ,d_{n}$. Then we get on $C$ the short exact sequence
    $0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0.$
Here $\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)$ is a vector bundle, called the syzygy bundle, of rank $n-1$ and of degree
    $((n-1)m-\sum _{i=1}^{n}d_{i})\operatorname {deg} \,(C).$
Our approach to the computation of the Hilbert-Kunz multiplicity is by using the presenting sequence
    $0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}\longrightarrow 0$
and twists of its $e$-th Frobenius pull-backs, that is
    $0\longrightarrow \operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-qd_{i}){\stackrel {f_{1}^{q},\ldots ,f_{n}^{q}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0$
(where $q=p^{e}$), and to relate the asymptotic behavior of
    $\operatorname {length} \,(R/I^{[q]})=\operatorname {dim} _{K}\,(R/I^{[q]})=\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(R/I^{[q]})_{m}\,$
to the asymptotic behavior of the global sections of the Frobenius pull-backs
    $(F^{e*}(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})(m)=\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\,.$
What we want to compute is just the cokernel of the complex of global sections of the above sequence, namely
    $\operatorname {dim} _{K}\,(R/I^{[q]})_{m}=h^{0}(C,{\mathcal {O}}_{C}(m))-\sum _{i=1}^{n}h^{0}(C,{\mathcal {O}}_{C}(m-qd_{i}))+h^{0}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))\,.$
The summation over $m$ is finite (but the range depends on $q$), and the terms
    $h^{0}(C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,\Gamma (C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,R_{m}\,$
are easy to control, so we have to understand the behavior of the global syzygies
    $H^{0}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))$
for all $q$ and $m$, 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 $H^{0}-H^{1}$ and then use semistability properties to show that $H^{0}$ or $H^{1}$ are $0$ in certain ranges.
We need the concept of (strong) semistability.

## Definition

Let ${\mathcal {S}}$ be a vector bundle on a smooth projective curve $C$. It is called semistable, if $\mu ({\mathcal {T}})={\frac {\deg({\mathcal {T}})}{\operatorname {rk} ({\mathcal {T}})}}\leq {\frac {\deg({\mathcal {S}})}{\operatorname {rk} ({\mathcal {S}})}}=\mu ({\mathcal {S}})$ for all subbundles ${\mathcal {T}}\subseteq {\mathcal {S}}$.
Suppose that the base field has positive characteristic $p>0$. Then ${\mathcal {S}}$ is called strongly semistable, if all (absolute)
$F^{e*}({\mathcal {S}})$
The rational number $\mu ({\mathcal {S}})={\frac {\operatorname {deg} ({\mathcal {S}})}{\operatorname {rk} ({\mathcal {S}})}}$ is called the slope of a vector bundle.

## Definition

Let ${\mathcal {S}}$ be a vector bundle on a smooth projective curve $C$ over an algebraically closed field $K$. Then the (uniquely determined) filtration
    $0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}={\mathcal {S}}\,$
of subbundles such that all quotient bundles ${\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}$ are semistable with decreasing slopes $\mu _{k}=\mu ({\mathcal {S}}_{k}/{\mathcal {S}}_{k-1})$,
${\mathcal {S}}$

## Theorem

Let $C$ denote a smooth projective curve over an algebraically closed field of positive characteristic $p$, and let ${\mathcal {S}}$ be a vector bundle on $C$. Then there exists a natural number $e\in \mathbb {N}$ such that the Harder-Narasimhan filtration of the $e$th Frobenius pull-back $F^{e*}({\mathcal {S}})$, say
    $0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}=F^{e*}({\mathcal {S}})\,$
${\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}$
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.

## Theorem

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

## Corollary

Let $R=K[x,y,z]/(H)$ be a normal homogeneous hypersurface domain of dimension two and degree $\delta$ over an algebraically closed field of positive characteristic. Then there exists a rational number $\nu _{2}$, ${\frac {3}{2}}\leq \nu _{2}\leq 2$, such that the Hilbert-Kunz multiplicity of $R$ is
    $e_{HK}(R)=\delta {\left(\nu _{2}^{2}-3\nu _{2}+3\right)}\,.$