Nonlinear neighborhoods of smooth curves and arbitrarily negative Wahl defects

Source 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.