论文标题
表面上数值特殊集合的突变
Mutations of Numerically Exceptional Collections on Surfaces
论文作者
论文摘要
邦德 - 波利希楚克的猜想指出，特别是对于平滑的投影型变种的有界派生类别的类别，编织组对完整特殊集合的作用是转移到变化的。我们表明，辫子组在有理表面上的最大数值特殊集合中进行过渡性，直到Picard晶格的等法和带线束的曲折。考虑到在非常一般的位置上最多9分的投影平面炸毁，这些结果将提高到派生类别。更确切地说，我们证明，在这些假设下，由线束组成的最大数字集合是一个完整的特殊集合，其中任何两个都与一系列突变和变化相关。前者扩展了Elagin-Lunts的结果，而后者则是Kuleshov-Orlov的结果，均与Del Pezzo表面有关。相比之下，我们在伴随的工作中表明[KRAH，一个理性表面上的幻影（预印本）]表明，在10点中，射击平面的爆炸通常承认了一个非足够的特殊最大长度集合，由线束组成。
A conjecture of Bondal-Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin-Lunts and the latter a result of Kuleshov-Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work [Krah, A Phantom on a Rational Surface, (Preprint)] that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.
