Hilbert-Kunz theory/Graded situation/description

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

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

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 (semistable and strongly semistable)

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 ${}{\mathcal {T}}$.

Suppose that the base field has positive characteristic ${}p>0$. Then ${}{\mathcal {S}}$ is called strongly semistable, if all (absolute) Frobenius pull-backs ${}F^{e*}({\mathcal {S}})$ are semistable.

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

Definition (Harder-Narasimhan filtration)

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 ${}\mu _{k}=\mu ({\mathcal {S}}_{k}/{\mathcal {S}}_{k-1})$, is called the Harder-Narasimhan filtration of ${}{\mathcal {S}}$.

Theorem (Langer)

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

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

has the property that the quotients ${}{\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}$ 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 (Brenner, Trivedi)

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

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

In particular, it is a rational number.

Corollary (Brenner, Trivedi)

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

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