Weierstrass point

In mathematics, a Weierstrass point [math]\displaystyle{ P }[/math] on a nonsingular algebraic curve [math]\displaystyle{ C }[/math] defined over the complex numbers is a point such that there are more functions on [math]\displaystyle{ C }[/math], with their poles restricted to [math]\displaystyle{ P }[/math] only, than would be predicted by the Riemann–Roch theorem. The concept is named after Karl Weierstrass.

Consider the vector spaces

[math]\displaystyle{ L(0), L(P), L(2P), L(3P), \dots }[/math]

where [math]\displaystyle{ L(kP) }[/math] is the space of meromorphic functions on [math]\displaystyle{ C }[/math] whose order at [math]\displaystyle{ P }[/math] is at least [math]\displaystyle{ -k }[/math] and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on [math]\displaystyle{ C }[/math]; 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 [math]\displaystyle{ g }[/math] is the genus of [math]\displaystyle{ C }[/math], the dimension from the [math]\displaystyle{ k }[/math]-th term is known to be

[math]\displaystyle{ l(kP) = k - g + 1, }[/math] for [math]\displaystyle{ k \geq 2g - 1. }[/math]

Our knowledge of the sequence is therefore

[math]\displaystyle{ 1, ?, ?, \dots, ?, g, g + 1, g + 2, \dots. }[/math]

What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument: [math]\displaystyle{ L(nP)/L((n-1)P) }[/math] has dimension as most 1 because if [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] have the same order of pole at [math]\displaystyle{ P }[/math], then [math]\displaystyle{ f+cg }[/math] will have a pole of lower order if the constant [math]\displaystyle{ c }[/math] is chosen to cancel the leading term). There are [math]\displaystyle{ 2g - 2 }[/math] question marks here, so the cases [math]\displaystyle{ g = 0 }[/math] or [math]\displaystyle{ 1 }[/math] need no further discussion and do not give rise to Weierstrass points.

Assume therefore [math]\displaystyle{ g \geq 2 }[/math]. There will be [math]\displaystyle{ g - 1 }[/math] steps up, and [math]\displaystyle{ g - 1 }[/math] steps where there is no increment. A non-Weierstrass point of [math]\displaystyle{ C }[/math] occurs whenever the increments are all as far to the right as possible: i.e. the sequence looks like

[math]\displaystyle{ 1, 1, \dots, 1, 2, 3, 4, \dots, g - 1, g, g + 1, \dots. }[/math]

Any other case is a Weierstrass point. A Weierstrass gap for [math]\displaystyle{ P }[/math] is a value of [math]\displaystyle{ k }[/math] such that no function on [math]\displaystyle{ C }[/math] has exactly a [math]\displaystyle{ k }[/math]-fold pole at [math]\displaystyle{ P }[/math] only. The gap sequence is

[math]\displaystyle{ 1, 2, \dots, g }[/math]

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 [math]\displaystyle{ g }[/math] gaps.)

For hyperelliptic curves, for example, we may have a function [math]\displaystyle{ F }[/math] with a double pole at [math]\displaystyle{ P }[/math] only. Its powers have poles of order [math]\displaystyle{ 4, 6 }[/math] and so on. Therefore, such a [math]\displaystyle{ P }[/math] has the gap sequence

[math]\displaystyle{ 1, 3, 5, \dots, 2g - 1. }[/math]

In general if the gap sequence is

[math]\displaystyle{ a, b, c, \dots }[/math]

the weight of the Weierstrass point is

[math]\displaystyle{ (a - 1) + (b - 2) + (c - 3) + \dots. }[/math]

This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is [math]\displaystyle{ g(g^2 - 1). }[/math]

For example, a hyperelliptic Weierstrass point, as above, has weight [math]\displaystyle{ g(g - 1)/2. }[/math] Therefore, there are (at most) [math]\displaystyle{ 2(g + 1) }[/math] of them. The [math]\displaystyle{ 2g+2 }[/math] 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 [math]\displaystyle{ g }[/math].

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 [math]\displaystyle{ C }[/math] defined over an algebraically closed field [math]\displaystyle{ k }[/math] of characteristic [math]\displaystyle{ p \geq 0 }[/math], the gap numbers for all but finitely many points is a fixed sequence [math]\displaystyle{ \epsilon_1, ..., \epsilon_g. }[/math] These points are called non-Weierstrass points. All points of [math]\displaystyle{ C }[/math] whose gap sequence is different are called Weierstrass points.

If [math]\displaystyle{ \epsilon_1, ..., \epsilon_g = 1, ..., g }[/math] 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 [math]\displaystyle{ GF(q^2) }[/math] by equation [math]\displaystyle{ y^q + y = x^{q+1} }[/math], where [math]\displaystyle{ q }[/math] is a prime power.

References

• P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. pp. 273–277. ISBN 0-471-05059-8. 
• Farkas; Kra (1980). Riemann Surfaces. Graduate Texts in Mathematics. Springer-Verlag. pp. 76–86. ISBN 0-387-90465-4. https://archive.org/details/riemannsurfaces00fark_192. 
• Eisenbud, David; Harris, Joe (1987). "Existence, decomposition, and limits of certain Weierstrass points". Invent. Math. 87 (3): 495–515. doi:10.1007/bf01389240. 
• Garcia, Arnaldo; Viana, Paulo (1986). "Weierstrass points on certain non-classical curves". Archiv der Mathematik 46 (4): 315–322. doi:10.1007/BF01200462. 
• Hazewinkel, Michiel, ed. (2001), "Weierstrass point", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page