Year: 2026
Type: preprint
Abstract: Let \(X\) be a log del Pezzo surface with cyclic quotient singularities and \(\mathcal{X}\) its canonical stack. We define the categorical genus of \(\mathcal{X}\) as half the rank of its antisymmetric numerical Euler pairing. Using Serre duality and inertial Riemann--Roch, we compute \[ g_{\mathrm{cat}}(\mathcal{X})=1+\frac{1}{2}\sum_j w_j(\ell_j-1), \] where \(w_j\) and \(\ell_j\) are the local widths and Gorenstein indices of the singularities, and identify the local contributions with twisted sectors of age less than one. We determine how categorical genus changes under general \(\mathbb{Q}\)-Gorenstein deformation and express the resulting decrease in terms of the smoothable \(T\)-content. When \(X\) admits a toric \(\mathbb{Q}\)-Gorenstein degeneration, the categorical genus of a general deformation agrees with Tveiten's mutable genus, given by the residual interior lattice points of the corresponding Fano polygon. Under additional hypotheses, we recover this genus formula through the singularities of compactified mirror curves.
Source: arXiv (Cornell University)
Author: Saltachin, Alex Junior Gomez
Institution: Pontifícia Universidade Católica do Rio de Janeiro
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212717163