发表机构
Pontifícia Universidade Católica do Rio de Janeiro（里约热内卢天主教大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文计算具有循环商奇点的log del Pezzo曲面的典范栈的范畴亏格，通过Serre对偶和惯性Riemann–Roch得到闭式公式，并证明其与Tveiten可变亏格及toric退化下的Laurent多项式纤维亏格一致。
AI 中文摘要
对于具有循环商奇点的log del Pezzo曲面的典范栈$\mathcal{X}$，我们计算其范畴亏格为 \\[ g_{\mathrm{cat}}(\mathcal{X})= 1+\frac{1}{2}\sum_j w_j(\ell_j-1), \\] 其中$w_j=\gcd(n_j,q_j+1)$和$\ell_j=n_j/w_j$分别是奇点$\frac{1}{n_j}(1,q_j)$的局部宽度和Gorenstein指数。Serre对偶性和惯性Riemann–Roch将光滑贡献与局部典范特征分开，并将范畴亏格等同于$1+\dim H_{\mathcal{X}}^{\mathrm{age}<1}$。通过一个足够一般的$\mathbb{Q}$-Gorenstein形变定义了剩余不变量$g_{\mathrm{cat}}^{\mathrm{qG}}(X)$；它与$g_{\mathrm{cat}}(\mathcal{X})$的差恰好是加权T-内容贡献。对于任何具有toric $\mathbb{Q}$-Gorenstein退化的曲面，这将其剩余范畴不变量等同于Tveiten的可变亏格，即所选退化的一般最大可变Laurent多项式纤维的亏格。局部$\mathbb{Q}$-Gorenstein刚性比较是下降为零的特殊情况。对于toric曲面，给定典范栈的范畴亏格计算所有内部格点，并等于一般Laurent多项式纤维的亏格。显式的Oneto–Petracci镜像提供了直接检验。
英文摘要
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.
Comments45 pages, 6 figures
