Vol. 101, No. 2, 1982

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Independence of normal Weierstrass points under deformation

Robert F. Lax

Vol. 101 (1982), No. 2, 393–398

Let X denote a compact Riemann surface of genus g and suppose P X. The Weierstrass nongaps at P are those positive integers n such that there exists a meromorphic function on X which has a pole of order n at P and is holomorphic everywhere else. The Weierstrass semigroup at P, which we denote by Γ(P), is the additive semigroup consisting of 0 and the nongaps. A point P is a Weierstrass point if there exists a nongap less than g + 1 at P and is a normal Weierstrass point if Γ(P) = {0,g,g + 2,g + 3,g + 4,}. We consider here the following problem: Given a collection P1,,Pn of points on X, describe the infinitesimal variations of complex structure on X which preserve the Weierstrass semigroups at P1,,Pn. Our main result says, roughly speaking, that normal Weierstrass points deform as independently of each other as possible.

Mathematical Subject Classification 2000
Primary: 14H15
Secondary: 30F10, 32G15, 14F07
Received: 28 December 1980
Revised: 25 February 1981
Published: 1 August 1982
Robert F. Lax