The structure of exceptional sequences on toric varieties of Picard rank two

Abstract

For a smooth projective toric variety of Picard rank two we classify all exceptional sequences of invertible sheaves which have maximal
length. In particular, we prove that unlike non-maximal sequences, they

1.  remain exceptional under lexicographical reordering
2.  satisfy strong height constraints in the Picard lattice
3.  are full, that is, they generate the derived category of the variety.

Moreover, we determine the exceptional poset of those sequences and show how this poset governs the property "strongly exceptional".

(This is joint work with Frederik Witt, Stuttgart; see arXiv:2112.14637 [math.AG])

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