Weierstrass point

In mathematics, a Weierstrass point formula_1 on a nonsingular algebraic curve formula_2 defined over the complex numbers is a point such that there are more functions on formula_2, with their poles restricted to formula_1 only, than would be predicted by the Riemann–Roch theorem. 
The concept is named after Karl Weierstrass.
Consider the vector spaces
where formula_6 is the space of meromorphic functions on formula_2 whose order at formula_1 is at least formula_9 and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on formula_2; 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 formula_11 is the genus of formula_2, the dimension from the formula_13-th term is known to be
Our knowledge of the sequence is therefore
What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument: formula_17 has dimension as most 1 because if formula_18 and formula_11 have the same order of pole at formula_1, then formula_21 will have a pole of lower order if the constant formula_22 is chosen to cancel the leading term). There are formula_23 question marks here, so the cases formula_24 or formula_25 need no further discussion and do not give rise to Weierstrass points.
Assume therefore formula_26. There will be formula_27 steps up, and formula_27 steps where there is no increment. A non-Weierstrass point of formula_2 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. A Weierstrass gap for formula_1 is a value of formula_13 such that no function on formula_2 has exactly a formula_13-fold pole at formula_1 only. 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 formula_11 gaps.)
For hyperelliptic curves, for example, we may have a function formula_38 with a double pole at formula_1 only. Its powers have poles of order formula_40 and so on. Therefore, such a formula_1 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 formula_45
For example, a hyperelliptic Weierstrass point, as above, has weight formula_46 Therefore, there are (at most) formula_47 of them.
The formula_48 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 formula_11.
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.
Positive characteristic.
More generally, for a nonsingular algebraic curve formula_2 defined over an algebraically closed field formula_13 of characteristic formula_52, the gap numbers for all but finitely many points is a fixed sequence formula_53 These points are called non-Weierstrass points.
All points of formula_2 whose gap sequence is different are called Weierstrass points.
If formula_55 then the curve is called a classical curve.
Otherwise, it is called non-classical. In characteristic zero, all curves are classical.
Hermitian curves are an example of non-classical curves. These are projective curves defined over finite field formula_56 by equation formula_57, where formula_58 is a prime power.