##
**Weierstrass points in characteristic p.**
*(English)*
Zbl 0555.14009

Let C be a non-singular curve with a canonical divisor K. Moreover, let D be a positive divisor on C of degree d and (projective) dimension r. For each point P of C there are exactly \(r+1\) numbers \(g_ 1(P),g_ 2(P),...,g_{r+1}(P)\) such that the inclusions \(h^ 0(C,K-D+(g_ i(P)- 1)P)\subseteq h^ 0(C,K-D+g_ i(P)P)\) are equalities. These numbers are the same for nearly all points of C. The reviewer suggested [see also Astérisque 87/88, 207-219 (1981; Zbl 0489.14007) and Ann. Sci. Éc. Norm. Supér., IV. Sér. 17, 45-66 (1984; see the preceding review)] that the sequence \(g_ 1(P),g_ 2(P),...,g_{r+1}(P)\) for a general point is called the ”gap sequence” of the linear system \(h^ 0(C,D)\) and that the finite number of points with a different sequence are called ”Weierstrass points”. If the ground field has characteristic zero then all linear systems have gap sequence \(1,2,...,r+1\) and the reviewer suggested further that linear systems with this gap sequence are called ”classical”.

In positive characteristic several classes of curves with non-classical linear systems on them are known. It appears however to be difficult to find non-trivial examples of non-classical linear systems and there are strong reasons to believe that such linear systems are extremely scarce. The present article is the first really successful attempt to, on the one hand, find criteria for linear systems to be classical and, on the other hand, give non-trivial examples of non-classical linear systems.

First the author proves, with beautifully simple methods: ”Given a curve C over a finite field. There exist positive integers m and n such that for all non-negative integers k all positive divisors of degree \(g- 1+n+k.m\) have classical gap-sequence.” - With slightly more complicated methods he obtains the stronger result: ”Let a, g, and p be positive integers with p prime. Then, in the above situation, there exist positive integers m and n such that all divisors of degree \(a.n+k.p^ m\) give classical linear systems for all non-negative integers k.”

In the other direction, the author gives highly non-trivial examples of curves with non-classical linear systems of arbitrarily high degree and even of curves for which infinitely many multiples of the canonical linear system are non-classical. On the other hand he shows that on a generic curve of genus at least 2 all multiples of the canonical linear system are classical.

In spite of these beautiful results, the gap left between the ”positive” results and the counterexamples is considerable and, as mentioned above, is probably mostly filled with classical linear systems. As the present article indicates it may be extremely difficult to fill the gap. However, it is still a challenging and interesting problem to narrow the gap.

In positive characteristic several classes of curves with non-classical linear systems on them are known. It appears however to be difficult to find non-trivial examples of non-classical linear systems and there are strong reasons to believe that such linear systems are extremely scarce. The present article is the first really successful attempt to, on the one hand, find criteria for linear systems to be classical and, on the other hand, give non-trivial examples of non-classical linear systems.

First the author proves, with beautifully simple methods: ”Given a curve C over a finite field. There exist positive integers m and n such that for all non-negative integers k all positive divisors of degree \(g- 1+n+k.m\) have classical gap-sequence.” - With slightly more complicated methods he obtains the stronger result: ”Let a, g, and p be positive integers with p prime. Then, in the above situation, there exist positive integers m and n such that all divisors of degree \(a.n+k.p^ m\) give classical linear systems for all non-negative integers k.”

In the other direction, the author gives highly non-trivial examples of curves with non-classical linear systems of arbitrarily high degree and even of curves for which infinitely many multiples of the canonical linear system are non-classical. On the other hand he shows that on a generic curve of genus at least 2 all multiples of the canonical linear system are classical.

In spite of these beautiful results, the gap left between the ”positive” results and the counterexamples is considerable and, as mentioned above, is probably mostly filled with classical linear systems. As the present article indicates it may be extremely difficult to fill the gap. However, it is still a challenging and interesting problem to narrow the gap.

Reviewer: D.Laksov

### MSC:

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

14G15 | Finite ground fields in algebraic geometry |

14C20 | Divisors, linear systems, invertible sheaves |

14H25 | Arithmetic ground fields for curves |

### Keywords:

gap sequence; Weierstrass points; positive characteristic; examples of non-classical linear systems### References:

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[2] | Hartshorne, R.: Algebraic geometry. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0367.14001 |

[3] | Laksov, D.: Weierstrass points on curves. Astérisque87-88, 221-247 (1981) · Zbl 0489.14007 |

[4] | Moret-Bailly, L.: Familles de courbes et de variétés abeliens sur ?1. Astérisque86, 109-140 (1981) · Zbl 0515.14007 |

[5] | Mumford, D.: Abelian varieties. Tata Inst. for Fund. Res. 1970 · Zbl 0223.14022 |

[6] | Weil, A.: Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc.55, 497-508 (1949) · Zbl 0032.39402 |

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