  We study Hilbert-Kunz multiplicity of non-singular curves in positive
characteristic. We analyse the relationship between the Frobenius semistability
of the kernel sheaf associated with the curve and its ample line bundle, and
the HK multiplicity. This leads to a lower bound, achieved iff the kernel sheaf
is Frobenius semistable, and otherwise to formulas for the HK multiplicity in
terms of parameters measuring the failure of Frobenius semistability. As a
byproduct, an explicit example of a vector bundle on a curve is given whose
$n$-th iterated Frobenius pullback is not semistable, while its $(n-1)$-th such
pullback is semistable, where $n>0$ is arbitrary.
