A simpler proof of the Gieseker-Petri theorem on special divisors. (English) Zbl 0533.14012

The Gieseker-Petri theorem is equivalent to the following statement, which climaxes the theory of special divisors on general curves: Theorem: If C is a general curve of genus g, then the variety \(G^ r\!_ d(C)\) of linear series of dimension r and degree d on C is smooth and of dimension \(g-(r+1)(g-d+r).\)
This theorem was first proven by Gieseker in 1982 [D. Gieseker, Invent. Math. 66, 251-275 (1982; Zbl 0522.14015)] using a degeneration to a kind of reducible curves. The paper under review replaces the limit curves used by Gieseker by a different type, and simplifies a number of Gieseker’s arguments, resulting in a far simpler proof of this central result (in characteristic 0). The techniques which the authors use here have been further developed by them, and lead to many new results in the theory of general curves; see for example the authors’ announcement in Bull. Am. Math. Soc., New Ser. 10, 277-280 (1984; see the following review).


14H99 Curves in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
14H10 Families, moduli of curves (algebraic)
14H40 Jacobians, Prym varieties
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