Autoequivalences of blow-ups of minimal surfaces

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-08-08 DOI:10.1112/blms.13131

Xianyu Hu, Johannes Krah

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Abstract

Let $X$ be the blow-up of $P_{C}^{2}$ in a finite set of very general points. We deduce from the work of Uehara [Trans. Amer. Math. Soc. 371 (2019), no. 5, 3529–3547] that $X$ has only standard autoequivalences, no non-trivial Fourier–Mukai partners, and admits no spherical objects. If $X$ is the blow-up of $P_{C}^{2}$ in 9 very general points, we provide an alternate and direct proof of the corresponding statement. Further, we show that the same result holds if $X$ is a blow-up of finitely many points in a minimal surface of non-negative Kodaira dimension which contains no $(- 2)$ -curves. Independently, we characterize spherical objects on blow-ups of minimal surfaces of positive Kodaira dimension.

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极小曲面炸开的自等价性

让 X $X$ 是 P C 2 $\mathbb {P}^2_\mathbb {C}$ 在一个有限的非常一般的点集中的炸开。我们从上原的研究[Trans. Amer. Math. Soc. 371 (2019), no. 5, 3529-3547]中推导出，X $X$只有标准的自等价性，没有非三维的傅立叶-穆凯伙伴，也不允许球面对象。如果 X $X$ 是 P C 2 $\mathbb {P}^2_\mathbb {C}$ 在 9 个非常一般的点上的炸开，我们提供了相应声明的另一种直接证明。此外，我们还证明了如果 X $X$ 是不包含 ( - 2 ) $(-2)$ 曲线的非负柯达伊拉维度的最小曲面中有限多个点的炸开，则同样的结果成立。另外，我们还描述了科代拉维度为正的极小曲面炸开后的球面对象的特征。

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