In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Weierstrass point
on a nonsingular
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
defined over the complex numbers is a point such that there are more functions on
, with their
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
s restricted to
only, than would be predicted by the
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
.
The concept is named after
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
.
Consider the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s
:
where
is the space of
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
s on
whose order at
is at least
and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on
; 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
is the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
of
, the dimension from the
-th term is known to be
:
for
Our knowledge of the sequence is therefore
:
What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument:
has dimension as most 1 because if
and
have the same order of pole at
, then
will have a pole of lower order if the constant
is chosen to cancel the leading term). There are
question marks here, so the cases
or
need no further discussion and do not give rise to Weierstrass points.
Assume therefore
. There will be
steps up, and
steps where there is no increment. A non-Weierstrass point of
occurs whenever the increments are all as far to the right as possible: i.e. the sequence looks like
:
Any other case is a Weierstrass point. A Weierstrass gap for
is a value of
such that no function on
has exactly a
-fold pole at
only. The gap sequence is
:
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
gaps.)
For
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
s, for example, we may have a function
with a double pole at
only. Its powers have poles of order
and so on. Therefore, such a
has the gap sequence
:
In general if the gap sequence is
:
the weight of the Weierstrass point is
:
This is introduced because of a counting theorem: on a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
the sum of the weights of the Weierstrass points is
For example, a hyperelliptic Weierstrass point, as above, has weight
Therefore, there are (at most)
of them.
The
ramification points of the
ramified covering
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
of degree two from a hyperelliptic curve to the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
are all hyperelliptic Weierstrass points and these exhausts all the Weierstrass points on a hyperelliptic curve of genus
.
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
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
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
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
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
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
defined over an algebraically closed field
of characteristic
, the gap numbers for all but finitely many points is a fixed sequence
These points are called non-Weierstrass points.
All points of
whose gap sequence is different are called Weierstrass points.
If
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
by equation
, where
is a prime power.
Notes
References
*
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*
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