Article

Annali di Matematica Pura ed Applicata

, Volume 157, Issue 1, pp 183-197

Some remarks on the schemesW d r

  • Marc CoppensAffiliated withKatholieke Industriële Hogeschool der Kempen

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Summary

Let X be an irreducible smooth projective curve of genus g. Let ϱ d r (g) be the Brill-Noether Number. In this paper we prove some results concerning the schemes W d r of special divisors. 1) Suppose dim (W d−1 r )=ϱ d− 1 r (g)⩾0 and ϱ d r (g) < g. If W d− 1 r is a reduced (resp. irreducible) scheme, then W d r is a reduced (resp. irreducible) scheme. 2) Under certain conditions, if Z is a generically reduced irreducible component of W d−1 r then Z ⊕ W 1 0 is a generically reduced irreducible component of W d r . For r=1, we obtain some further results in this direction. 3) As an application of it we are able to prove some dimension theorems for the schemes W d 1 .