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

$L(0),L(P),L(2P),L(3P),\dots$

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 of $C$, the dimension from the $k$-th term is known to be

$l(kP)=k-g+1,$ for $k\ge 2g-1.$

Our knowledge of the sequence is therefore

$1,?,?,\dots ,?,g,g+1,g+2,\dots .$

What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument: $L(nP)/L((n-1)P)$ has dimension as most 1 because 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 $g\ge 2$. 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

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

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

$1,2,\dots ,g$

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

$1,3,5,\dots ,2g-1.$

In general if the gap sequence is

$a,b,c,\dots$

the weight of the Weierstrass point is

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

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

For example, a hyperelliptic Weierstrass point, as above, has weight $g(g-1)/2.$ Therefore, there are (at most) $2(g+1)$ of them. The $2g+2$ 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 $g$.

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 ). 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 $C$ defined over an algebraically closed field $k$ of characteristic $p\ge 0$, the gap numbers for all but finitely many points is a fixed sequence ${ϵ}_1,...,{ϵ}_g.$ These points are called non-Weierstrass points. All points of $C$ whose gap sequence is different are called Weierstrass points.

If ${ϵ}_1,...,{ϵ}_g=1,...,g$ 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 $GF(q^2)$ by equation $y^q+y=x^{q+1}$, where $q$ 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