Semi-orthogonal decomposability of the derived category of a curve

TL;DR

This paper proves that the derived category of coherent sheaves on any smooth projective curve, except the projective line, cannot be decomposed into simpler semi-orthogonal components, highlighting a unique indecomposability property.

Contribution

It establishes the non-existence of non-trivial semi-orthogonal decompositions for derived categories of all smooth projective curves except the projective line.

Findings

Derived category of a curve (not projective line) has no non-trivial semi-orthogonal decompositions.

The projective line is the only curve with a semi-orthogonal decomposition.

The result emphasizes the indecomposability of the derived category for most curves.

Abstract

We show that the bounded derived category of coherent sheaves on a smooth projective curve except the projective line admits no non-trivial semi-orthogonal decompositions.