Weierstrass point

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. That is, looking at the vector spaces

L(0), L(P), L(2P), L(3P), ...,

where L(kP) is the space of meromorphic functions on C whose order at P is at least −k and with no other poles.

The concept is named after Karl Weierstrass.

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(kP) = k − g + 1, for k ≥ 2g − 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 + cg will have a pole of lower order if the constant c is chosen to cancel the leading term). There are

2g − 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, ..., 2g − 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(g2 − 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; as those can be found (for example, the six points of ramification when g = 2 and C is presented as a ramified covering of the projective line) this exhausts all the Weierstrass points on C.

Further information on the gaps comes from applying Clifford's theorem. Multiplication of functions gives the non-gaps a semigroup structure, and an old question of Adolf Hurwitz asked for a characterization of the semigroups occurring. A new necessary condition was found by 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. A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic was given by F. K. Schmidt in 1939.

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