In mathematics, a Weierstrass point
on a nonsingular algebraic curve
defined over the complex numbers is a point such that there are more functions on
The concept is named after Karl Weierstrass.
is the space of meromorphic functions on
We know three things: the dimension is at least 1, because of the constant functions on
; it is non-decreasing; and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right.
entries is that they can increment by at most 1 each time (this is a simple argument:
will have a pole of lower order if the constant
is chosen to cancel the leading term).
question marks here, so the cases
need no further discussion and do not give rise to Weierstrass points.
occurs whenever the increments are all as far to the right as possible: i.e. the sequence looks like Any other case is a Weierstrass point.
The gap sequence is 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
Its powers have poles of order
has the gap sequence In general if the gap sequence is the weight of the Weierstrass point is This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is
For example, a hyperelliptic Weierstrass point, as above, has weight
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
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.
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 [1]).
A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic was given by F. K. Schmidt in 1939.
More generally, for a nonsingular algebraic curve
defined over an algebraically closed field
, the gap numbers for all but finitely many points is a fixed sequence
whose gap sequence is different are called Weierstrass points.
In characteristic zero, all curves are classical.
These are projective curves defined over finite field