发表机构
Dartmouth College（达特茅斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究3×3平方幻方参数化代数曲面的几何，计算其自同构群、拓扑不变量及Picard群子格，并探讨相关曲面。
AI 中文摘要
是否存在具有不同整数元素的3×3平方幻方的问题自18世纪以来一直悬而未决。我们研究了参数化3×3平方幻方的代数曲面的几何，该曲面是射影8空间中六个二次曲面的奇异完全交。我们通过涉及Gale对偶性的论证计算了其几何自同构群。我们计算了其解的基本拓扑不变量和Hodge菱形。我们提供了其几何Picard群的一个显式秩518子格，接近Hodge理论上限544。最后，我们研究了作为坐标投影出现的del Pezzo、K3和Enriques曲面的几何与算术。
英文摘要
The question of whether a 3-by-3 magic square of squares with distinct integer entries exists has been open since the 18th century. We study the geometry of the algebraic surface parameterizing 3-by-3 magic squares of squares, which is a singular complete intersection of six quadrics in projective 8-space. We compute its geometric automorphism group via an argument involving Gale duality. We compute the basic topological invariants and Hodge diamond of its resolution. We provide an explicit rank 518 sublattice of its geometric Picard group, close to the Hodge-theoretic upper bound of 544. Finally, we study the geometry and arithmetic of del Pezzo, K3, and Enriques surfaces that arise as coordinate projections.
Comments43 pages, 2 figures, 2 tables. Comments welcome!
