In mathematics, a Weierstrass point
P
on a nonsingular algebraic curve
C
defined over the complex numbers is a point such that there are more functions on
C
, with their poles restricted to
P
only, than would be predicted by the Riemann–Roch theorem.
The concept is named after Karl Weierstrass.
Consider the vector spaces
L
(
0
)
,
L
(
P
)
,
L
(
2
P
)
,
L
(
3
P
)
,
…
where
L
(
k
P
)
is the space of meromorphic functions on
C
whose order at
P
is at least
−
k
and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on
C
; it is non-decreasing; and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right. In fact if
g
is the genus of
C
, the dimension from the
k
-th term is known to be
l
(
k
P
)
=
k
−
g
+
1
,
for
k
≥
2
g
−
1.
Our knowledge of the sequence is therefore
1
,
?
,
?
,
…
,
?
,
g
,
g
+
1
,
g
+
2
,
…
.
What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument: if
f
and
g
have the same order of pole at
P
, then
f
+
c
g
will have a pole of lower order if the constant
c
is chosen to cancel the leading term). There are
2
g
−
2
question marks here, so the cases
g
=
0
or
1
need no further discussion and do not give rise to Weierstrass points.
Assume therefore
g
≥
2
. There will be
g
−
1
steps up, and
g
−
1
steps where there is no increment. A non-Weierstrass point of
C
occurs whenever the increments are all as far to the right as possible: i.e. the sequence looks like
1
,
1
,
…
,
1
,
2
,
3
,
4
,
…
,
g
−
1
,
g
,
g
+
1
,
…
.
Any other case is a Weierstrass point. A Weierstrass gap for
P
is a value of
k
such that no function on
C
has exactly a
k
-fold pole at
P
only. The gap sequence is
1
,
2
,
…
,
g
for a non-Weierstrass point. For a Weierstrass point it contains at least one higher number. (The Weierstrass gap theorem or Lückensatz is the statement that there must be
g
gaps.)
For hyperelliptic curves, for example, we may have a function
F
with a double pole at
P
only. Its powers have poles of order
4
,
6
and so on. Therefore, such a
P
has the gap sequence
1
,
3
,
5
,
…
,
2
g
−
1.
In general if the gap sequence is
a
,
b
,
c
,
…
the weight of the Weierstrass point is
(
a
−
1
)
+
(
b
−
2
)
+
(
c
−
3
)
+
…
.
This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is
g
(
g
2
−
1
)
.
For example, a hyperelliptic Weierstrass point, as above, has weight
g
(
g
−
1
)
/
2.
Therefore, there are (at most)
2
(
g
+
1
)
of them. The
2
g
+
2
ramification points of the ramified covering of degree two from a hyperelliptic curve to the projective line are all hyperelliptic Weierstrass points and these exhausts all the Weierstrass points on a hyperelliptic curve of genus
g
.
Further information on the gaps comes from applying Clifford's theorem. Multiplication of functions gives the non-gaps a numerical semigroup structure, and an old question of Adolf Hurwitz asked for a characterization of the semigroups occurring. A new necessary condition was found by R.-O. Buchweitz in 1980 and he gave an example of a subsemigroup of the nonnegative integers with 16 gaps that does not occur as the semigroup of non-gaps at a point on a curve of genus 16 (see ). A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic was given by F. K. Schmidt in 1939.