Negative Values of Wahl's Second Chern Number for Normal Surface Singularities

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 $C\times C$ 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 $c_2^w(X)=-r(d+r)\mathrm{/}(d-r)$. 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.