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.
