# 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 be a two-dimensional standard-graded normal domain over an algebraically closed field . Let

be the corresponding smooth projective curve and let

be an -primary homogeneous ideal with generators of degrees . Then we get on the short exact sequence

Here is a vector bundle, called the *syzygy bundle*, of rank and of degree

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

and twists of its -th Frobenius pull-backs, that is

(where ), and to relate the asymptotic behavior of

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

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

The summation over is finite (but the range depends on ), and the terms

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

for all and , 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 and then use semistability properties to show that or are in certain ranges.

We need the concept of (strong) semistability.

Let be a vector bundle on a smooth projective curve . It is called *semistable*, if
for all subbundles
.

Suppose that the base field has positive characteristic
.
Then is called *strongly semistable*, if all
(absolute)

The rational number
is called the *slope* of a vector bundle.

Let be a vector bundle on a smooth projective curve over an algebraically closed field . Then the (uniquely determined) filtration

of subbundles such that all quotient bundles are semistable with decreasing slopes ,

is called the*Harder-Narasimhan filtration*of .

Let denote a smooth projective curve over an algebraically closed field of positive characteristic , and let be a vector bundle on . Then there exists a natural number such that the Harder-Narasimhan filtration of the th Frobenius pull-back , say

has the property that the quotients 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.

Let be a two-dimensional standard-graded normal domain over an algebraically closed field of positive characteristic. Let be a homogeneous -primary ideal with homogeneous generators of degree . Let be the syzygy bundle on and suppose that the Harder-Narasimhan filtration of is strong, and let , , be the corresponding slopes. We set and . Then the Hilbert-Kunz multiplicity of is

Let be a normal homogeneous hypersurface domain of dimension two and degree over an algebraically closed field of positive characteristic. Then there exists a rational number , , such that the Hilbert-Kunz multiplicity of is