Categorical genus for log del Pezzo surfaces with cyclic quotient singularities

Source abstract

For the canonical stack $\mathcal{X}$ of a log del Pezzo surface with cyclic quotient singularities, we compute its categorical genus as $$g_{\mathrm{cat}}(\mathcal{X})= 1+\frac{1}{2}\sum_j w_j(\ell_j-1),$$ where $w_j=\gcd(n_j,q_j+1)$ and $\ell_j=n_j/w_j$ are the local widths and Gorenstein indices of the singularities $\frac{1}{n_j}(1,q_j)$. Serre duality and inertial Riemann--Roch separate the smooth contribution from the local canonical characters, and identify categorical genus with $1+\dim H_{\mathcal{X}}^{\mathrm{age}<1}$. Passing to a sufficiently general $\mathbb{Q}$-Gorenstein deformation defines the residual invariant $g_{\mathrm{cat}}^{\mathrm{qG}}(X)$; its difference from $g_{\mathrm{cat}}(\mathcal{X})$ is exactly the weighted T-content contribution. For any surface admitting a toric $\mathbb{Q}$-Gorenstein degeneration, this identifies the residual categorical invariant with Tveiten's mutable genus, the genus of a general maximally mutable Laurent-polynomial fiber for the chosen degeneration. The locally $\mathbb{Q}$-Gorenstein rigid comparison is the special case in which the drop vanishes. For toric surfaces, the categorical genus of the given canonical stack counts all interior lattice points and equals the genus of a general Laurent-polynomial fiber. Explicit Oneto--Petracci mirrors give direct checks.