Deforming a Canonical Curve Inside a Quadric

Let C ⊂ ℙg^-1 be a canonically embedded nonsingular nonhyperelliptic curve of genus g and let X ⊂ ℙg^-1 be a quadric containing C. Our main result states among other things that the Hilbert scheme of X is at [C ⊂ X] a local complete intersection of dimension g² - 1 and is smooth when X is. It also includes the assertion that the minimal obstruction space for this deformation problem is in fact the full associated Ext¹-group and that in particular the deformations of C in X are obstructed in case C meets the singular locus of X. Applications will be given in a forthcoming paper.