Weierstrass point explained

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( 2 P), L( 3 P), ...

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 bel(kP) = k - g + 1 , for k \geq 2 g - 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:

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 2 g - 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 \geq 2 . There will be g - 1 steps up, and g 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 like1 , 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 is1 , 2 , ... , 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 sequence1 , 3 , 5 , ... , 2 g - 1 .

In general if the gap sequence is

a, b, c, ...

the weight of the Weierstrass point is

(a - 1 ) + (b - 2 ) + (c - 3 ) + ... .

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 2 g +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

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

If

\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

GF(q ^{2)} by equation y ^{q} + y = x ^{q+1} , where q is a prime power.

References

• Book: Phillip Griffiths

. P. Griffiths . Phillip Griffiths . J. Harris . Joe Harris (mathematician) . Principles of Algebraic Geometry . Wiley Classics Library . Wiley Interscience . 1994 . 0-471-05059-8 . 273–277 .

• Book: Farkas . Kra . Riemann Surfaces . limited . Graduate Texts in Mathematics . Springer-Verlag . 1980 . 0-387-90465-4 . 76–86 .
• Eisenbud. David. Harris. Joe. Existence, decomposition, and limits of certain Weierstrass points. Invent. Math.. 1987. 87. 3 . 495–515. 10.1007/bf01389240. 122385166 .
• Garcia. Arnaldo. Viana. Paulo. Weierstrass points on certain non-classical curves. Archiv der Mathematik. 1986. 46. 4 . 315–322. 10.1007/BF01200462. 120983683 .