对艾森Bud-施赖尔-魏曼乌尔希存在性问题的光滑反例
Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem
浏览论文内容
中文总结 AI 辅助
该研究针对Eisenbud-Schreyer-Weyman乌尔希存在性问题，构造了光滑反例，证明无需皮卡秩1假设即可得到关键不等式，还得到了无穷多反例族，说明部分Chow形式无对应行列式表示。
中文摘要 AI 辅助
在2003年发表于《美国数学会杂志》的文章中，Eisenbud、Schreyer和Weyman提出问题：每个嵌入射影簇是否都带有一个Ul希层。2017年Beauville提出了一条数值路径，用于寻找不存在Ul希丛的曲面：在皮卡秩1的情况下，Ul希丛的存在性要求满足$H^2\ge K_S^2-8\chi(\mathcal{O}_S)$，建议在Bogomolov-Miyaoka-Yau边界附近进行搜索。我们证明该障碍并不需要皮卡秩1的假设：一个与秩无关的Bogomolov-Hodge论证在每个光滑极化曲面上都给出相同的不等式。利用Hirzebruch-Kummer解像上额外的Néron-Severi方向，指数为3的Hesse曲面具有一个非常丰富的类$H=4A-E$，满足$H^2=7\cdot3^9<16\cdot3^9=K_Y^2-8\chi(\mathcal{O}_Y)$，因此是标准形式下Eisenbud-Schreyer-Weyman问题的光滑反例。因此其Chow形式没有来自嵌入曲面上Ul希层的ESW型线性行列式表示。此外，对所有$n\ge3$，Hesse对$(Y_n,4A_n-E_n)$都是反例，这些曲面两两不同构，且$H_n^2/\sigma(Y_n)=7/(3n^2-11)\to0$，而$K_{Y_n}^2/\chi(\mathcal{O}_{Y_n})\to60/7\approx8.5714$。一般的排列理论Rees代数/Segre机制产生进一步的无限族。
英文摘要
In their 2003 article in the Journal of the American Mathematical Society, Eisenbud, Schreyer and Weyman asked whether every embedded projective variety carries an Ulrich sheaf. In 2017 Beauville proposed a numerical route toward a surface with no Ulrich bundles: in Picard rank one, existence forces $H^2\ge K_S^2-8χ(\mathcal O_S)$, suggesting a search near the Bogomolov--Miyaoka--Yau boundary. We show that the Picard-rank-one hypothesis is not needed for the obstruction: a rank-independent Bogomolov--Hodge argument gives the same inequality on every smooth polarized surface. Using additional Neron--Severi directions on Hirzebruch--Kummer resolutions, the exponent-$3$ Hesse surface admits a very ample class $H=4A-E$ with $H^2=7\cdot3^9<16\cdot3^9=K_Y^2-8χ(\mathcal O_Y)$, hence a smooth counterexample to the Eisenbud--Schreyer--Weyman problem in its standard formulation. Consequently its Chow form has no ESW-type linear determinantal representation arising from an Ulrich sheaf on the embedded surface. Moreover, for every $n\ge3$ the Hesse pair $(Y_n,4A_n-E_n)$ is a counterexample, the surfaces are pairwise non-isomorphic, and $H_n^2/σ(Y_n)=7/(3n^2-11)\to0$, while $K_{Y_n}^2/χ(\mathcal O_{Y_n})\to60/7\approx8.5714$. A general arrangement-theoretic Rees-algebra/Segre mechanism yields further infinite families.
发表机构
- Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
