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 where L(kP) 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 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: has dimension as most 1 because 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 \geq 2. There will be g - 1 steps up, and g 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 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 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 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 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 2g+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.

Positive characteristic

More generally, for a nonsingular algebraic curve C defined over an algebraically closed field k of characteristic p \geq 0, the gap numbers for all but finitely many points is a fixed sequence These points are called non-Weierstrass points. All points of C whose gap sequence is different are called Weierstrass points. If 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 GF(q^2) by equation, where q is a prime power.