Castelnuovo curves and unobstructed deformations

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Now it is a well-known fact from Kodai ra [8], that if H i ( N ) = 0 , then there exists a non-singular family ~ for which p is surjective. In classical language, this is the "completeness of the characteristic linear series". Alternatively one says that F~---~IP 3 is unobstructed if there exists a family of deformations of F in IP 3 over a non-singular parameter scheme ~ for which the corresponding characteristic linear map is surjective. Mumford in [10] has shown that if for a smooth space curve F, we have that HI(N)+O (here again N denotes the normal bundle to F), then F may be obstructed. In point of fact, Mumford constructs a specific example of an obstructed space curve. In our note we show that for generic Castelnuovo space curves (see our discussion in (1.1) below) of degree d > 8 , we have HI(N)=I=O, but yet the curves are unobstructed explicitly showing the converse to Kodaira ' s aforementioned result is false. We accomplish this by computing the dimension of the component of the Hilbert scheme of space curves consisting of Castelnuovo curves of a given degree, and showing this is equal to d imH~ where N is the normal bundle to a generic Castelnuovo space curve. Since H~ represents the Zariski tangent space to the point of the Hilbert scheme corresponding to the given curve (see e.g. [11]), this means the point is smooth and hence we are done. Finally we make some remarks relating our results tO the classical count ([13]) of parameters on which a given family of space curves depends.