Special Linear System on an Algebraic Curve and its Application

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Let W^r_d(C) be a scheme of line bundles defined by W^r_d(C)={L|L∈Pic^d(C),dimΓ(C,L)【greater than or equal】r+1} (usually, W^r_d(C) can be defined as a subscheme of Pic^d(C)). Kempf and Kleiman-Laksov prove that the variety W^r_d(C) has dimension at least p=g-(r+l)(g-d+r). Griffith-Harris and Fulton-Lazarsfeld prove that W^r_d(C) is smooth of dimension p=g-(r+1)(g-d+r) when C is a general curve in the moduli spaceM_g. "A general curve" means there is an open subset U⊂M_g inM_g, W^r_d(C) is smooth of dimension p=g-(r+1)(g-d+r) for any curve C∈U. So it is natural to ask to classify which curve belongs to this open subset U⊂M_g. As for this problem, we can give good sufficient conditions (No.2 and No.3).For a finitely generated numerical semigroup which start from 4, Komeda proved that every such numerical semigroup H is Weierstrass, i.e. there is a pointed curve (C,P) such that the semigroup of non-gaps of P is just H. Unfortunatelly, in Komeda's argument, (C,P) is not constructive for some special type numerical semigroups, i.e. he uses some general theory of torus embedding and proved only the existence of (C,P). So it is very natural to ask whether a pointed curve (C,P) can be constructed concretely for this special type numerical semigroups. Our result is that we can construct (C,P) for such special type semigroup by using a double covering of hyperelliptic curve which is n-sheeted covering of another hyperelliptic curve.(No.1)