Obstructions to deforming a space curve

Abstract:

Mumford described a curve, $\gamma$, in ${{\mathbf {P}}^3}$ that has obstructed infinitesimal deformations (in fact the Hilbert scheme of the curve is generically nonreduced). This paper studies $\gamma ’{\text {s}}$ Hilbert scheme by studying deformations of $\gamma$ in ${{\mathbf {P}}^3}$ over parameter spaces of the form $\operatorname {Spec} (k[t]/({t^n})), n = 2, 3, \ldots$. Given a deformation of $\gamma$ over $\operatorname {Spec} (k[t]/({t^n}))$ one attempts to extend it to a deformation of $\gamma$ over $\operatorname {Spec} (k[t]/({t^{n + 1}}))$. If it will not extend, this deformation is said to be obstructed at the nth order. I show that on a generic version of Mumford’s curve, an infinitesimal deformation (i.e., a deformation over $\operatorname {Spec} (k[t]/({t^2}))$) is either obstructed at the second order, or at no order, in which case we say it is unobstructed.