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

${\displaystyle L(0),L(P),L(2P),L(3P),\dots }$

where ${\displaystyle L(kP)}$ 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

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

Our knowledge of the sequence is therefore

${\displaystyle 1,?,?,\dots ,?,g,g+1,g+2,\dots .}$

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

Assume therefore $g\geq 2$. There will be ${\displaystyle g-1}$ steps up, and ${\displaystyle g-1}$ 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

${\displaystyle 1,1,\dots ,1,2,3,4,\dots ,g-1,g,g+1,\dots .}$

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

${\displaystyle 1,2,\dots ,g}$

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 ${\displaystyle 4,6}$ and so on. Therefore, such a $P$ has the gap sequence

${\displaystyle 1,3,5,\dots ,2g-1.}$

In general if the gap sequence is

${\displaystyle a,b,c,\dots }$

the weight of the Weierstrass point is

${\displaystyle (a-1)+(b-2)+(c-3)+\dots .}$

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

For example, a hyperelliptic Weierstrass point, as above, has weight ${\displaystyle g(g-1)/2.}$ Therefore, there are (at most) ${\displaystyle 2(g+1)}$ of them; as those can be found (for example, the six points of ramification when ${\displaystyle g=2}$ 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

• 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.