Weierstrass point

In mathematics, a Weierstrass point ${\displaystyle P}$ on a nonsingular algebraic curve ${\displaystyle C}$ defined over the complex numbers is a point such that there are more functions on ${\displaystyle C}$, with their poles restricted to ${\displaystyle P}$ 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 ${\displaystyle C}$ whose order at ${\displaystyle P}$ is at least ${\displaystyle -k}$ and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on ${\displaystyle C}$; 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 ${\displaystyle g}$ is the genus of ${\displaystyle C}$, the dimension from the ${\displaystyle k}$-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: ${\displaystyle L(nP)/L((n-1)P)}$ has dimension as most 1 because if ${\displaystyle f}$ and ${\displaystyle g}$ have the same order of pole at ${\displaystyle P}$, then ${\displaystyle f+cg}$ will have a pole of lower order if the constant ${\displaystyle c}$ is chosen to cancel the leading term). There are ${\displaystyle 2g-2}$ question marks here, so the cases ${\displaystyle g=0}$ or ${\displaystyle 1}$ need no further discussion and do not give rise to Weierstrass points.

Assume therefore ${\displaystyle g\geq 2}$. There will be ${\displaystyle g-1}$ steps up, and ${\displaystyle g}$ steps where there is no increment. A non-Weierstrass point of ${\displaystyle C}$ 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 ${\displaystyle P}$ is a value of ${\displaystyle k}$ such that no function on ${\displaystyle C}$ has exactly a ${\displaystyle k}$-fold pole at ${\displaystyle P}$ 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 ${\displaystyle g}$ gaps.)

For hyperelliptic curves, for example, we may have a function ${\displaystyle F}$ with a double pole at ${\displaystyle P}$ only. Its powers have poles of order ${\displaystyle 4,6}$ and so on. Therefore, such a ${\displaystyle P}$ 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. The ${\displaystyle 2g+2}$ 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 ${\displaystyle g}$.

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 ^[1]). A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic was given by F. K. Schmidt in 1939.

More generally, for a nonsingular algebraic curve ${\displaystyle C}$ defined over an algebraically closed field ${\displaystyle k}$ of characteristic ${\displaystyle p\geq 0}$, the gap numbers for all but finitely many points is a fixed sequence ${\displaystyle \epsilon _{1},...,\epsilon _{g}.}$ These points are called non-Weierstrass points. All points of ${\displaystyle C}$ whose gap sequence is different are called Weierstrass points.

If ${\displaystyle \epsilon _{1},...,\epsilon _{g}=1,...,g}$ 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 ${\displaystyle GF(q^{2})}$ by equation ${\displaystyle y^{q}+y=x^{q+1}}$, where ${\displaystyle q}$ is a prime power.

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