特征零下无算术 Cohen-Macaulay 丛的极化簇及无分次极大 Cohen-Macaulay 模的截面环
Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
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- Institute of Mathematics of the Romanian Academy（罗马尼亚科学院数学研究所）
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中文总结 AI 辅助
本文在特征零下构造了无算术 Cohen-Macaulay 丛的极化曲面，其截面环无分次极大 Cohen-Macaulay 模，通过数值判据和 Harder-Narasimhan 滤过与 Horrocks 分裂的比较，反驳了分次细化猜想。
中文摘要 AI 辅助
我们展示了在 $\mathbf{C}$ 上不存在任何非零秩的算术 Cohen-Macaulay 丛的光滑极化曲面 $(Y,H)$。等价地，它们的截面环是三维正规 $\mathbb{N}$-分次 $\mathbf{C}$-整环，在顶点处具有孤立奇点，且不承认任何非零有限生成分次极大 Cohen-Macaulay 模。据作者所知，此前未知这样的环。它们的存在与 Hartshorne、Hochster 和 Peskine-Szpiro 的定理形成对比，该定理指出在特征 $p>0$ 的完美域上，三维 $\mathbb{N}$-分次整环总是具有这样的模。因此，在特征零下，Hochster 的小 Cohen-Macaulay 猜想不承认分次细化。不存在性判据是数值性的：对于无基点的 $|H|$ 且具有有限态射，$H^2<K_Y^2-8\chi(\mathcal{O}_Y)=\tau(Y)$（$Y$ 在 $\mathbf{C}$ 上的符号差）使得不存在非零 $\mathcal{E}$ 满足对所有 $t$ 有 $H^1(Y,\mathcal{E}(tH))=0$。对于 Ulrich 丛，由于它们是半稳定的，这是 Bogomolov 不等式；新的地方在于不需要稳定性假设，也不需要关于 Hilbert 多项式的条件，而是将 $\mathcal{E}$ 的 Harder-Narasimhan 滤过与其在 $\mathbb{P}^2$ 上的直接像的 Horrocks 分裂进行比较。Hirzebruch 的 Hesse 曲面，其中 $H_n=4A_n-E_n$，$n\ge3$，符合条件。在维度 $m$ 中，障碍读作 $(m+1)H^m\ge(K_X^2-2c_2(X))\cdot H^{m-2}$，这是 Lopez 为 Ulrich 丛证明的不等式；其内的乘积和完全交给出每个维度 $\ge3$ 的例子，而三维例子表明 Shimomoto-Tavanfar 判据中特征 $p$ 假设是必要的。
英文摘要
We exhibit smooth polarized surfaces $(Y,H)$ over $\mathbf{C}$ carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal $\mathbb{N}$-graded $\mathbf{C}$-domains, with an isolated singularity at the vertex, admitting no nonzero finitely generated graded maximal Cohen-Macaulay module. To the author's knowledge no such ring was previously known. Their existence contrasts with the theorem of Hartshorne, Hochster and Peskine-Szpiro that a three-dimensional $\mathbb{N}$-graded domain over a perfect field of characteristic $p>0$ always has one. As a consequence, in characteristic zero Hochster's small Cohen-Macaulay conjecture admits no graded refinement. The nonexistence criterion is numerical: for $|H|$ base-point-free with finite morphism, $H^2<K_Y^2-8χ(\mathcal{O}_Y)=τ(Y)$, the signature of $Y$ over $\mathbf{C}$, leaves no nonzero $\mathcal{E}$ with $H^1(Y,\mathcal{E}(tH))=0$ for all $t$. For Ulrich bundles, being semistable, this is Bogomolov's inequality; what is new is that no stability hypothesis and no condition on the Hilbert polynomial are needed, the Harder-Narasimhan filtration of $\mathcal{E}$ being compared instead with the Horrocks splitting of its direct image on $\mathbb{P}^2$. Hirzebruch's Hesse surfaces with $H_n=4A_n-E_n$, $n\ge3$, qualify. In dimension $m$ the obstruction reads $(m+1)H^m\ge(K_X^2-2c_2(X))\cdot H^{m-2}$, the inequality Lopez proved for Ulrich bundles; products and complete intersections inside them give examples in every dimension $\ge3$, and the threefold examples show that the characteristic-$p$ hypothesis in the criterion of Shimomoto-Tavanfar is essential.
