Hilbert-Kunz function/Projective curve/Global sections and Frobenius pull-back/Section

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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.