# 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 ${\displaystyle {}R}$ be a two-dimensional standard-graded normal domain over an algebraically closed field ${\displaystyle {}K}$. Let

${\displaystyle {}C=\operatorname {Proj} {\left(R\right)}\,}$

be the corresponding smooth projective curve and let

${\displaystyle {}I={\left(f_{1},\ldots ,f_{n}\right)}\,}$

be an ${\displaystyle {}R_{+}}$-primary homogeneous ideal with generators of degrees ${\displaystyle {}d_{1},\ldots ,d_{n}}$. Then we get on ${\displaystyle {}C}$ the short exact sequence

${\displaystyle 0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0.}$

Here ${\displaystyle {}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)}$ is a vector bundle, called the syzygy bundle, of rank ${\displaystyle {}n-1}$ and of degree

${\displaystyle ((n-1)m-\sum _{i=1}^{n}d_{i})\operatorname {deg} \,(C).}$

Our approach to the computation of the Hilbert-Kunz multiplicity is by using the presenting sequence

${\displaystyle 0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}\longrightarrow 0}$

and twists of its ${\displaystyle {}e}$-th Frobenius pull-backs, that is

${\displaystyle 0\longrightarrow \operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-qd_{i}){\stackrel {f_{1}^{q},\ldots ,f_{n}^{q}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0}$

(where ${\displaystyle {}q=p^{e}}$), and to relate the asymptotic behavior of

${\displaystyle {}\operatorname {length} \,(R/I^{[q]})=\operatorname {dim} _{K}\,(R/I^{[q]})=\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(R/I^{[q]})_{m}\,}$

to the asymptotic behavior of the global sections of the Frobenius pull-backs

${\displaystyle {}(F^{e*}(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})(m)=\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\,.}$

What we want to compute is just the cokernel of the complex of global sections of the above sequence, namely

${\displaystyle \operatorname {dim} _{K}\,(R/I^{[q]})_{m}=h^{0}(C,{\mathcal {O}}_{C}(m))-\sum _{i=1}^{n}h^{0}(C,{\mathcal {O}}_{C}(m-qd_{i}))+h^{0}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))\,.}$

The summation over ${\displaystyle {}m}$ is finite (but the range depends on ${\displaystyle {}q}$), and the terms

${\displaystyle {}h^{0}(C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,\Gamma (C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,R_{m}\,}$

are easy to control, so we have to understand the behavior of the global syzygies

${\displaystyle H^{0}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))}$

for all ${\displaystyle {}q}$ and ${\displaystyle {}m}$, 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 ${\displaystyle {}H^{0}-H^{1}}$ and then use semistability properties to show that ${\displaystyle {}H^{0}}$ or ${\displaystyle {}H^{1}}$ are ${\displaystyle {}0}$ in certain ranges.

We need the concept of (strong) semistability.

## Definition

Let ${\displaystyle {}{\mathcal {S}}}$ be a vector bundle on a smooth projective curve ${\displaystyle {}C}$. It is called semistable, if ${\displaystyle {}\mu ({\mathcal {T}})={\frac {\deg({\mathcal {T}})}{\operatorname {rk} ({\mathcal {T}})}}\leq {\frac {\deg({\mathcal {S}})}{\operatorname {rk} ({\mathcal {S}})}}=\mu ({\mathcal {S}})}$ for all subbundles ${\displaystyle {}{\mathcal {T}}\subseteq {\mathcal {S}}}$.

Suppose that the base field has positive characteristic ${\displaystyle {}p>0}$. Then ${\displaystyle {}{\mathcal {S}}}$ is called strongly semistable, if all (absolute)

Frobenius pull-backs ${\displaystyle {}F^{e*}({\mathcal {S}})}$ are semistable.

The rational number ${\displaystyle {}\mu ({\mathcal {S}})={\frac {\operatorname {deg} ({\mathcal {S}})}{\operatorname {rk} ({\mathcal {S}})}}}$ is called the slope of a vector bundle.

## Definition

Let ${\displaystyle {}{\mathcal {S}}}$ be a vector bundle on a smooth projective curve ${\displaystyle {}C}$ over an algebraically closed field ${\displaystyle {}K}$. Then the (uniquely determined) filtration

${\displaystyle {}0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}={\mathcal {S}}\,}$

of subbundles such that all quotient bundles ${\displaystyle {}{\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}}$ are semistable with decreasing slopes ${\displaystyle {}\mu _{k}=\mu ({\mathcal {S}}_{k}/{\mathcal {S}}_{k-1})}$,

is called the Harder-Narasimhan filtration of ${\displaystyle {}{\mathcal {S}}}$.

## Theorem

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

${\displaystyle {}0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}=F^{e*}({\mathcal {S}})\,}$

has the property that the quotients ${\displaystyle {}{\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}}$ are strongly semistable.

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

${\displaystyle {}e_{HK}(I)={\frac {\operatorname {deg} \,(C)}{2}}{\left(\sum _{k=1}^{t}r_{k}\nu _{k}^{2}-\sum _{i=1}^{n}d_{i}^{2}\right)}\,.}$
In particular, it is a rational number.

## Corollary

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

${\displaystyle {}e_{HK}(R)=\delta {\left(\nu _{2}^{2}-3\nu _{2}+3\right)}\,.}$