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.

The concept is named after Karl Weierstrass.

Consider the vector spaces


 * $$L(0), L(P), L(2P), L(3P), \dots$$

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


 * $$l(kP) = k - g + 1,$$ for $$k \geq 2g - 1.$$

Our knowledge of the sequence is therefore


 * $$1, ?, ?, \dots, ?, g, g + 1, g + 2, \dots.$$

What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument: $$L(nP)/L((n-1)P)$$ 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


 * $$1, 1, \dots, 1, 2, 3, 4, \dots, g - 1, g, g + 1, \dots.$$

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


 * $$1, 2, \dots, g$$

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


 * $$1, 3, 5, \dots, 2g - 1.$$

In general if the gap sequence is


 * $$a, b, c, \dots$$

the weight of the Weierstrass point is


 * $$(a - 1) + (b - 2) + (c - 3) + \dots.$$

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 $$\epsilon_1, ..., \epsilon_g.$$ These points are called non-Weierstrass points. All points of $$C$$ whose gap sequence is different are called Weierstrass points.

If $$\epsilon_1, ..., \epsilon_g = 1, ..., g$$ 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 $$y^q + y = x^{q+1}$$, where $$q$$ is a prime power.