Weierstrass point

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Weierstrass point P on a nonsingular algebraic curve

Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

 C defined over the complex numbers is a point such that there are extra functions on C, with their poles restricted to P only, than would be predicted by looking at the Riemann–Roch theorem

Riemann–Roch theorem

The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...

. 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.

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

Genus (mathematics)

In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

 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
+ (b − 2) + (c − 3) + ... .

This is introduced because of a counting theorem: on a Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

 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

Ramification

In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...

 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

Clifford's theorem

In mathematics, Clifford's theorem on special divisors is a result of W. K. Clifford on algebraic curves, showing the constraints on special linear systems on a curve C....

. Multiplication of functions gives the non-gaps a semigroup structure, and an old question of Adolf Hurwitz

Adolf Hurwitz

Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...

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.