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) → (V,0) of a normal complex surface singularity other than a rational double point, the Wahl defect is defined by Δ(V,0) := h1(𝒪X)−h1(SX)+h1(bigwedge2SX), where E is reduced and SX = ΘX(−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 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.
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BibTeX
@misc{park2026nonlinear,
title = {{Nonlinear neighborhoods of smooth curves and arbitrarily negative Wahl defects}},
author = {Heesang Park and Dongsoo Shin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21093},
eprint = {2610.21093},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21093},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonlinear neighborhoods of smooth curves and arbitrarily negative Wahl defects
AU - Heesang Park
AU - Dongsoo Shin
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.21093
UR - https://arxiv.org/abs/2610.21093
AB - 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) → (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(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 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.
ER -