arXiv 2403.096532024-03math.AG

光滑射影环面簇的对角线分解

A Resolution of the Diagonal for Smooth Projective Toric Varieties

University of California, Irvine（加州大学尔湾分校）

Reginald Anderson

备注：: 24 pages, 12 figures. Submitted. Revised version: the main theorem now includes the central symmetry hypothesis, and all figures have been redrawn in TikZ. Added AI acknowledgments. arXiv admin note: text overlap with arXiv:2303.17497

AI总结：

研究光滑射影环面簇在主除子格条件不成立时的对角线分解问题，通过阶梯函数标记顶点并施加扇形对称假设，得到$X_\Sigma\times X_\Sigma$上$\mathcal{O}_\Delta$的局部自由分解，还探讨了变形参数$\epsilon$对分解族的影响。

摘要：

拜尔 - 波佩斯库 - 斯特姆费尔斯的胞腔构造将贝林森的对角线分解从射影空间扩展到主除子格为单模的射影环面簇。我们研究主除子格条件不成立的光滑射影情形。此时周期排列包含格外顶点，相关有限胞腔复形在0度及更高维有同调。我们用阶梯函数标记为顶点分配洛朗单项式，使其与周期超平面排列兼容。一个反例表明，无进一步假设时，未变形复形不必分解对角线。因此我们对扇形施加对称假设（关于原点中心对称），在此假设下该构造给出$X_\Sigma\times X_\Sigma$上$\mathcal{O}_\Delta$的局部自由分解。我们还讨论了变形参数$\epsilon$如何导出更广泛的分解族。

英文摘要：

The cellular construction of Bayer-Popescu-Sturmfels extends Beilinson's diagonal resolution from projective space to projective toric varieties whose lattice of principal divisors is unimodular. We investigate the smooth projective case in which this lattice condition fails. The periodic arrangement then contains vertices outside the lattice, and the associated finite cellular complex can acquire homology in degree $0$ and in higher degrees. We use a floor-function labeling to assign Laurent monomials to vertices in a way compatible with the periodic hyperplane arrangement. A counterexample shows that, without further hypotheses, the undeformed complex need not resolve the diagonal. We therefore impose a symmetry hypothesis on the fan (central symmetry across the origin) under which the construction yields a locally free resolution of $\mathcal{O}_Δ$ on $X_Σ\times X_Σ$. We also discuss how a deformation parameter $ε$ should lead to a broader family of resolutions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文