Negative Values of Wahl's Second Chern Number for Normal Surface Singularities
Abstract
We disprove Wahl's 1993 vanishing conjecture, open for more than thirty years, which asserts that c2w(X)=0 for every nonquotient normal surface singularity. Let C be a smooth projective curve with a nontrivial involution having r>0 fixed points, and put d=2g(C)−2. If d>r, the union of the diagonal and the graph of the involution in C× C has negative definite intersection matrix and can be analytically contracted to a normal surface singularity (X,o). A direct relative Chern-class computation and Wahl's residue formula give c2w(X)=−r(d+r)/(d−r). An explicit curve of genus four gives c2w(X)=−4. Branched double covers of a fixed curve of genus two yield examples with c2w(Xm)=−2m(m+1), so these values are unbounded below. We give the construction and intersection calculations in detail, as well as a line-bundle filtration realizing the negative value.
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BibTeX
@misc{du2026negative,
title = {{Negative Values of Wahl's Second Chern Number for Normal Surface Singularities}},
author = {Rong Du},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25606},
eprint = {2610.25606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25606},
}
RIS
TY - UNPB
TI - Negative Values of Wahl's Second Chern Number for Normal Surface Singularities
AU - Rong Du
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25606
UR - https://arxiv.org/abs/2610.25606
AB - We disprove Wahl's 1993 vanishing conjecture, open for more than thirty years, which asserts that c_2^w(X)=0 for every nonquotient normal surface singularity. Let C be a smooth projective curve with a nontrivial involution having r>0 fixed points, and put d=2g(C)−2. If d>r, the union of the diagonal and the graph of the involution in C× C has negative definite intersection matrix and can be analytically contracted to a normal surface singularity (X,o). A direct relative Chern-class computation and Wahl's residue formula give c_2^w(X)=−r(d+r)/(d−r). An explicit curve of genus four gives c_2^w(X)=−4. Branched double covers of a fixed curve of genus two yield examples with c_2^w(X_m)=−2m(m+1), so these values are unbounded below. We give the construction and intersection calculations in detail, as well as a line-bundle filtration realizing the negative value.
ER -