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**Wronskians and Plücker formulas for linear systems on curves.**
*(English)*
Zbl 0555.14008

The Weierstrass points of a curve C have a geometric interpretation as the points of hyperosculation of the canonical map \(C\to {\mathbb{P}}^{g-1}\). More generally, if \(V\subset H^ 0(C,{\mathcal O}(D))\) is a linear system on C, and if the characteristic is 0 or larger than the degree of D, one can similarly define ”Weierstrass points” of C with respect to V as the scheme of zeros of certain bundle maps and such that these points correspond to the points of hyperosculation of the map \(C\to {\mathbb{P}}(V).\) From this one deduces the Plücker-Cayley-Veronese formulas for \(C\to {\mathbb{P}}(V),\) relating the number and weights of these points to the ranks and genus of the curve. The ranks have various geometric interpretations, e.g. as the degrees of the associated curves to C [see W. F. Pohl, Topology 1, 169-211 (1962; Zbl 0112.366), and the reviewer in Real and Complex Singul., Proc. Nordic Summer Sch., Symp. Math. Oslo 1976, 475-495 (1977; Zbl 0375.14017)].

In small, positive characteristic, however, the classical definition and geometric interpretation of (generalized) Weierstrass points break down. The author of the present paper is able, via a clever construction, to define general Weierstrass points (that he calls Wronskian points) of a curve with respect to a linear system, in arbitrary characteristic. These points are defined as the scheme of zeros of a section of a line bundle, generalizing the classical case. [In the case the linear system is the complete canonical system, this definition is equivalent to the one given by F. K. Schmidt, Math. Z. 45, 75-96 (1939; Zbl 0020.10202).] Hence the author also obtains generalized Plücker-Cayley-Veronese formulas valid in any characteristic and depending also on the so-called (global) gap sequence of the linear system. It turns out, however, that there is no similar geometric interpretation of the algebraic invariants as in the classical case. This is fully illustrated by two quite simple examples - projections of rational normal curves in characteristic 2 and 3.

There is another paper by the author on the same subject [cf. Astérisque 87/88, 221-247 (1981; Zbl 0489.14007)] containing more background and historical remarks. The construction of the present paper allows the author, as he says, to ”generalize and vastly simplify the results of that article”.

In small, positive characteristic, however, the classical definition and geometric interpretation of (generalized) Weierstrass points break down. The author of the present paper is able, via a clever construction, to define general Weierstrass points (that he calls Wronskian points) of a curve with respect to a linear system, in arbitrary characteristic. These points are defined as the scheme of zeros of a section of a line bundle, generalizing the classical case. [In the case the linear system is the complete canonical system, this definition is equivalent to the one given by F. K. Schmidt, Math. Z. 45, 75-96 (1939; Zbl 0020.10202).] Hence the author also obtains generalized Plücker-Cayley-Veronese formulas valid in any characteristic and depending also on the so-called (global) gap sequence of the linear system. It turns out, however, that there is no similar geometric interpretation of the algebraic invariants as in the classical case. This is fully illustrated by two quite simple examples - projections of rational normal curves in characteristic 2 and 3.

There is another paper by the author on the same subject [cf. Astérisque 87/88, 221-247 (1981; Zbl 0489.14007)] containing more background and historical remarks. The construction of the present paper allows the author, as he says, to ”generalize and vastly simplify the results of that article”.

Reviewer: R.Piene

### MSC:

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

14C20 | Divisors, linear systems, invertible sheaves |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14H99 | Curves in algebraic geometry |

### Keywords:

Plücker formulas; Weierstrass points; hyperosculation; positive characteristic; Wronskian points; gap sequence
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\textit{D. Laksov}, Ann. Sci. Éc. Norm. Supér. (4) 17, 45--66 (1984; Zbl 0555.14008)

### References:

[1] | G. GALBURA , Il wronskiano di un sistema di sezioni... (Ann. Math. Pura & Appl., Vol. 98, 1974 , pp. 349-355). MR 50 #2179 | Zbl 0351.14012 · Zbl 0351.14012 |

[2] | P. GRIFFITHS and J. HARRIS , Principles of algebraic geometry . John Wiley & Sons, New York, 1978 . MR 80b:14001 | Zbl 0408.14001 · Zbl 0408.14001 |

[3] | D. LAKSOV , Weierstrass points on curves (Astérisque, Vol. 87-88, 1981 , pp. 221-247). MR 83e:14023 | Zbl 0489.14007 · Zbl 0489.14007 |

[4] | B. H. MATZAT , Ein Vortrag über Weierstrasspunkte . Karlsruhe, 1975 . |

[5] | R. PIENE , Numerical characters of a curve in projective n-space. Real and Complex Singularities , Oslo, 1976 , Sijthoff & Nordhoff, 1977 . Zbl 0375.14017 · Zbl 0375.14017 |

[6] | W. F. POHL , Differential geometry of higher order (Topology, Vol. 1, 1962 , pp. 169-211). MR 27 #4242 | Zbl 0112.36605 · Zbl 0112.36605 |

[7] | F. K. SCHMIDT , Zur arithmetischen Theorie der algebraischen Funktionen II (Math. Zeitschrift, Vol. 45, 1939 , pp. 75-76). Zbl 0020.10202 | JFM 65.0116.01 · Zbl 0020.10202 |

[8] | C. SEGRE , Introduzione alla geometria sopra un ente algebrico simplicemente infinito (Annali di Mat. pura ed. appl., Vol. 22, (Ser. II), 1894 , pp. 41-142). JFM 25.1159.01 · JFM 25.1159.01 |

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