Tilting on non-commutative rational projective curves

TL;DR

This paper introduces a new class of non-commutative projective curves and demonstrates that their derived categories can be embedded into derived categories of finite-dimensional algebra representations, revealing their structure and dimension.

Contribution

It establishes the existence of tilting complexes for certain non-commutative curves and embeds their derived categories into those of finite-dimensional algebras, advancing understanding of their homological properties.

Findings

Derived category of rational projective curves with nodes and cusps embeds into finite-dimensional algebra representations.

The dimension of the derived category for such curves is at most two.

Tilted algebras for Kodaira cycles are gentle algebras, exemplified by the plane nodal cubic.

Abstract

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the plane nodal cubic.