Weak stability conditions on coherent systems of genus four curves

The derived category of coherent systems is an interesting triangulated category associated with a smooth, projective curve $C$. These categories admit Bridgeland stability conditions, as recently shown by Feyzbakhsh and Novik. Their construction depends explicitly on the higher rank Brill-Noether theory of $C$. In this short note, we study the Feyzbakhsh–Novik stability conditions for a general curve of genus four. We show that these stability conditions degenerate to a stability condition on the Kuznetsov component of the corresponding nodal cubic threefold, using a result of Alexeev-Kuznetsov.

💡 Research Summary

The paper investigates Bridgeland stability conditions on the derived category of coherent systems associated to a smooth projective curve (C) of genus 4, building on the recent construction of Feyzbakhsh and Novik. The authors first recall the abelian category (\mathcal T_C) of generalized coherent systems, whose objects are triples ((V,E;\varphi)) with (V) a finite‑dimensional vector space, (E) a coherent sheaf on (C), and (\varphi:\mathcal O_C\otimes V\to E) a morphism. Two exact adjoint pairs ((i^,i_)) and ((j^,j_)) embed (\operatorname{Vect}) and (\operatorname{Coh}(C)) into (\mathcal T_C), yielding a semi‑orthogonal decomposition \