On nonexistence of full exceptional collections on some relative flags. (English) Zbl 1540.14042
Summary: We show that certain relative flags cannot have full exceptional collections. We also prove that some of these flags are categorical representable in dimension zero if and only if they admit a full exceptional collection. As a consequence, these flags are categorical representable in dimension zero if and only if they have \(k\)-rational points if and only if they are \(k\)-rational. Moreover, we calculate the categorical representability dimension for the flags under consideration.
MSC:
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 14G05 | Rational points |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
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