Here we compute Hilbert-Kunz functions of any nontrivial ruled surface over ${bf P}^1_k$, with respect to all ample line bundles on it.
Here we prove that for a smooth projective variety $X$ of arbitrary dimension and for a vector bundle $E$ over $X$, the Harder-Narasimhan filtration of a Frobenius pull back of $E$ is a refinement of the Frobenius pull-back of the Harder-Narasimhan f
iltration of $E$, provided there is a lower bound on the characteristic $p$ (in terms of rank of $E$ and the slope of the destabilising sheaf of the cotangent bundle of $X$). We also recall some examples, due to Raynaud and Monsky,to show that some lower bound on $p$ is necessary. We further prove an analogue of this result for principal $G$-bundles over $X$. We also give a bound on the instability degree of the Frobenius pull back of $E$ in terms of the instability degree of $E$ and well defined invariants ot $X$ and $E$.
The paper is withdrawn.
In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$. He remarked that to remov
e this condition, it is enough to answer either of the following questions affirmatively: {it For the syzygy bundle $sV_d$ of ${mathcal O}(d)$, is $sV_d$ semistable for arbitrary $n, d$ and $p = {char} k$?, or is there a good estimate on $mu_{max}(sV_d^*)$?} Here we prove that (1) the bundle $sV_d$ is semistable, for a certain infinite set of integers $dgeq 0$, and (2) for arbitrary $d$, there is a good enough estimate on $mu_{max}(sV_d^*)$ in terms of $d$ and $n$. In particular one obtains Langers theorem, in arbitrary characeristic.
