Abstract
Let
C be a characteristic
p irreducible projective plane curve defined by a degree
d form
f, and
n
→
e
n
(
f
)
be the Hilbert–Kunz function of
f.
e
n
=
μ
p
2
n
−
R
n
with
μ
⩾
3
d
4
and
R
n
=
O
(
p
n
)
.
When
C is smooth,
R
n
=
O
(
1
)
; Brenner has shown the
R
n
to be eventually periodic when one further assumes
C defined over a finite field. We generalize these results, dropping smoothness. An additional term, (periodic)
p
n
now appears in
R
n
, with the periodic function taking values in
1
d
⋅
Z
[
1
p
]
. We describe it using 1-dimensional Hilbert–Kunz theory in the local rings of the singular points of
C, together with sheaf theory on
C, and work explicit examples.