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

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be a smooth projective relative curve and let be a vector bundle over . For the generic fiber let

where the are the slopes (and the ranks) in the Harder-Narasimhan filtration of , and for every prime number let

where the are the slopes in the strong Harder-Narasimhan filtration of on the fiber . Then

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

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.

Let be a finitely generated graded domain of relative dimension two with normal fibers and let be a homogeneous ideal such that is an -primary ideal for all prime reductions. Then the limit

exists and equals the Hilbert-Kunz multiplicity in characteristic zero.

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

The Fermat quartic is the easiest example where the Hilbert-Kunz multiplicity of the maximal ideal fluctuates with the characteristic. We have

The limit is of course , which corresponds to the fact that the syzygy bundle is semistable in characteristic zero. The syzygy bundle is semistable for all prime characteristics , but not strongly semistable for the prime numbers .