Questions of Connectedness of the Hilbert Scheme of Curves in ℙ3

Abstract

In studying algebraic curves in projective spaces, our forefathers in the 19th century noted that curves naturally move in algebraic families. In the projective plane, this is a simple matter. A curve of degree d is defined by a single homogeneous polynomial in the homogeneous coordinates X 0,x 1,X 2. The coefficients of this polynomial give a point in another projective space, and in this way curves of degree d in the plane are parametrized by the points of a ℙN with \( N = \tfrac{1} {2}d(d + 3) \). For an open set of ℙN, the corresponding curve is irreducible and nonsingular. The remaining points of ℙN correspond to curves that are singular, or reducible, or have multiple components. In particular, the nonsingular curves of degree d in ℙ2 form a single irreducible family.