Nonlinear neighborhoods of smooth curves and arbitrarily negative Wahl defects

Abstract

We develop a theory of nonlinear neighborhoods of smooth projective curves of genus at least two, constructed by gluing products of open subsets of a curve with a disc using nonlinear changes of the disc coordinate. Their zero sections have normal bundles of negative degree and contract to normal surface singularities. For the minimal good resolution \((X,E) \to (V,0)\) of a normal complex surface singularity other than a rational double point, the Wahl defect is defined by \(Δ(V,0) := h^1(\mathcal{O}_X)-h^1(S_X)+h^1(\bigwedge^2S_X)\), where \(E\) is reduced and \(S_X = Θ_X(-\log E)\). We give a finite formula for this defect in terms of extension obstructions when \(E\) is a single smooth curve. For our constructed neighborhoods, the formula involves extensions of functions alone, includes obstructions at every order, and gives a sharp lower bound depending only on the genus. We apply the theory to hyperelliptic curves and smooth plane curves. Both families consist of singularities that are neither \(\mathbb{Q}\)-Gorenstein nor quasi-homogeneous. The hyperelliptic examples realize every nonpositive integer as a Wahl defect and attain the genus bound in every genus at least two. They disprove Wahl's Main Conjecture, which asserts nonnegativity for non-Gorenstein singularities, with equality precisely in the quasi-homogeneous case.