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Weak stability conditions on coherent systems of genus four curves

Nicolás Vilches

Abstract

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.

Weak stability conditions on coherent systems of genus four curves

Abstract

The derived category of coherent systems is an interesting triangulated category associated with a smooth, projective curve . 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 . 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.
Paper Structure (11 sections, 23 theorems, 21 equations, 1 figure)

This paper contains 11 sections, 23 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgements
  4. Coherent systems
  5. Brill--Noether function
  6. Tilting stability conditions
  7. Genus four curves
  8. Two semi-orthogonal decompositions
  9. Proof of Theorem \ref{['teo:intro_stab']}
  10. A moduli space
  11. Questions

Key Result

Theorem 1.1

Let Then there exists a family of stability conditions on $\mathrm{D}^b(\mathcal{T}_C)$, depending of two parameters $b, w$ satisfying $w>\Phi_C(b)$.

Figures (1)

  • Figure 1: A bound and a partial bound of $y=\Phi_C(x)$.

Theorems & Definitions (43)

  • Theorem 1.1: Feyzbakhsh--Novik, FN25*§ 3
  • Theorem 1.2: Alexeev--Kuznetsov
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1: FN25*p. 5
  • Lemma 2.2: FN25*§ 2.1
  • Claim 2.3
  • Lemma 2.4: FN25*Proposition 2.2
  • Corollary 2.5: cf. FN25*p. 7
  • Proposition 2.6: AK25*3.12
  • ...and 33 more