Weierstrass point について

and with no other poles.
The concept is named after Karl Weierstrass.
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 \geq; 2g- 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
:

2g-2

question marks here, so the cases

g=0

1

need no further discussion and do not give rise to Weierstrass points.
Assume therefore

g \geq 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, \ \ldots, \ 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, \ \ldots

the weight of the Weierstrass point is
:

(a-1)+(b-2)+(c-3)+\ldots

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==

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抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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