Hilbert-Kunz theory/Two-dimensional graded/Relative situation/Section

Theorem

Let

$C\longrightarrow \operatorname {Spec} {\left(\mathbb {Z} _{g}\right)}$

be a smooth projective relative curve and let ${}{\mathcal {T}}$ be a vector bundle over ${}C$. For the generic fiber ${}C_{0}$ let

${}\mu _{HK}({\mathcal {T}}_{0})=\sum _{\ell }r_{\ell }\mu _{\ell }^{2}\,,$

where the ${}\mu _{\ell }$ are the slopes (and ${}r_{\ell }$ the ranks) in the Harder-Narasimhan filtration of ${}{\mathcal {T}}_{0}$, and for every prime number let

${}{\bar {\mu }}_{HK}({\mathcal {T}}_{p})={\frac {\sum _{k}r_{k}\mu _{k}^{2}}{p^{e}}}\,,$

where the ${}\mu _{k}$ are the slopes in the strong Harder-Narasimhan filtration of ${}F^{e*}({\mathcal {T}}_{p})$ on the fiber ${}C_{p}$. Then

${}\lim _{p\rightarrow \infty }{\bar {\mu }}_{HK}({\mathcal {T}}_{p})=\mu _{HK}({\mathcal {T}}_{0})\,$

In charcteristic zero and a given homogeneous ideal, we use the slopes from the Harder-Narasimhan filtration of the syzygy bundle to define the Hilbert-Kunz multiplicity by

${}e_{HK}^{0}(I)={\frac {1}{2\operatorname {deg} \,C}}{\left(\mu _{HK}(\operatorname {Syz} )-(\operatorname {deg} \,C)^{2}\sum _{i=1}^{n}d_{i}^{2}\right)}\,.$

In the relative situation, an ideal gives a syzygy bundle over the relative curve and there we can compare the slopes coming from a strong Harder-Narasimhan filtration in the fibers to get the following Corollary.

Corollary

Let ${}\mathbb {Z} \subseteq S$ be a finitely generated graded domain of relative dimension two with normal fibers and let ${}I\subseteq S$ be a homogeneous ideal such that ${}I_{p}$ is an ${}(S\otimes _{\mathbb {Z} }\mathbb {Z} /(p))_{+}$-primary ideal for all prime reductions. Then the limit

$\lim _{p\rightarrow \infty }e_{HK}(I_{p})$

exists and equals the Hilbert-Kunz multiplicity ${}e_{HK}^{0}(I_{0})$ in characteristic zero.

It is also known by a result of Brenner, Li, Miller, that it is enough to compute in every positive characteristic ${}p$ just the first Frobenius pull-back (not all Frobenius powers) and then let ${}p$ go to infinity to get the same limit.

Example

The Fermat quartic ${}x^{4}+y^{4}+z^{4}=0$ is the easiest example where the Hilbert-Kunz multiplicity of the maximal ideal ${}(x,y,z)$ fluctuates with the characteristic. We have

${}e_{HK}(R_{p})={\begin{cases}3{\text{ for }}p=\pm 1\,\operatorname {mod} \,8\\3+{\frac {1}{p^{2}}}{\text{ for }}p=\pm 3\,\operatorname {mod} \,8\,.\end{cases}}\,$

The limit is of course ${}3$, which corresponds to the fact that the syzygy bundle is semistable in characteristic zero. The syzygy bundle is semistable for all prime characteristics ${}p\geq 3$, but not strongly semistable for the prime numbers ${}p=\pm 3\,\operatorname {mod} \,8$.