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 extra functions on "C", with their poles restricted to "P" only, than would be predicted by looking at the Riemann-Roch theorem. That is, looking at the vector spaces

:"L"(0), "L"("P"), "L"(2"P"), "L"(3"P"), ...

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 2"g"-th term is known to be

:"l"("kP") = "k" − "g" + 1, for "k" ≥ 2"g" − 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

:2"g" − 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, ..., 2"g" − 1.

In general if the gap sequence is

:"a", "b", "c", ...

the weight of the Weierstrass point is

:("a" − 1) + ("b" − 2) + ("c" − 3) + ... .

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; as those can be found (for example, the six points of ramification 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. Multiplication of functions gives the non-gaps a semigroup structure, and an old question of Adolf Hurwitz 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.

References

*