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 $C$. 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.

• The paper proves that Feyzbakhsh–Novik stability conditions on coherent systems degenerate at w = 2 to a weak stability condition that descends through a Verdier quotient to the Kuznetsov component of a nodal cubic threefold.
• For a general genus four curve, the bound Φ_C(x) ≤ (x−3)² + 1.9 establishes genuine stability for w > 3 and supports the limiting construction at the critical value w = 2.
• The paper shows that wall-crossing identifies objects parametrized by Sym²(C) precisely when their associated secant lines share a point on C, recovering the normalization of the cubic threefold’s Fano variety of lines.

Overview

This note by Nicolás Vilches studies the Bridgeland stability conditions constructed by Feyzbakhsh and Novik on the derived category $D^b(T_C)$ of generalized coherent systems on a smooth projective curve $C$, in the special case where $C$ is a general curve of genus four. The main result establishes that these stability conditions degenerate, as the parameter $w$ approaches a critical value, to a weak stability condition $\sigma_{3,2}$ that descends to an honest Bridgeland stability condition $\overline{\sigma}_{3,2}$ on the Kuznetsov component $\Ku(Y)$ of the associated nodal cubic threefold (2602.12076). This provides a concrete instance of the general principle—pursued in several recent works—that degenerations of stability conditions on a triangulated category should converge to stability conditions on suitable Verdier quotients.

Coherent systems and their derived category

For a smooth projective curve $C$ of genus $g$, the paper works with the abelian category $T_C$ of triples $C$0, where $C$1, $C$2 is a vector space, and $C$3. Relaxing the classical inclusion $C$4 to an arbitrary morphism is what makes $C$5 abelian. Two exact adjoint pairs $C$6 (with vector spaces) and $C$7 (with coherent sheaves) yield a semi-orthogonal decomposition

$C$8

with gluing functor given, up to shift, by $C$9. The Euler pairing is computed by an explicit matrix: writing $C$0, one has $C$1, so $C$2. Moreover, $C$3 embeds as an admissible subcategory of $C$4 for a smooth proper variety $C$5, which ensures it is well-behaved from the categorical standpoint.

A key ingredient is the class of Brill–Noether–Petri extremal line bundles $C$6—those for which the Petri map $C$7 is an isomorphism. To each such $C$8 one associates the object $C$9, which Alexeev–Kuznetsov show is exceptional in $w$0; these are the BN-exceptional objects.

The Feyzbakhsh–Novik construction

The stability conditions are built via tilting with respect to the slope function $w$1 (with value $w$2 when $w$3). For each real $w$4, the torsion pair $w$5 determined by $w$6-semistable objects yields the tilted heart $w$7, and for $w$8 the pair $w$9 is a Bridgeland stability condition, where

$\sigma_{3,2}$0

The constraint on $\sigma_{3,2}$1 is governed by the Brill–Noether function

$\sigma_{3,2}$2

which encodes higher-rank Brill–Noether data of $\sigma_{3,2}$3: the bound $\sigma_{3,2}$4 is exactly what forces $\sigma_{3,2}$5 to have positive real part on phase-one objects arising from evaluation maps of semistable bundles of slope $\sigma_{3,2}$6. The paper also records a quadratic form $\sigma_{3,2}$7 providing the support property uniformly for all $\sigma_{3,2}$8, under a strict inequality $\sigma_{3,2}$9; this uniformity is essential for taking limits later.

Genus four geometry and the nodal cubic threefold

A genus four curve $\overline{\sigma}_{3,2}$0 is called general here if it is non-hyperelliptic and its unique ambient quadric $\overline{\sigma}_{3,2}$1 (under the canonical embedding as a $\overline{\sigma}_{3,2}$2-complete intersection) is smooth. The two rulings of $\overline{\sigma}_{3,2}$3 induce two non-isomorphic degree-three line bundles $\overline{\sigma}_{3,2}$4 with $\overline{\sigma}_{3,2}$5, and these are precisely the BNP-extremal line bundles of $\overline{\sigma}_{3,2}$6.

The geometric bridge to cubic threefolds is classical: cubics through $\overline{\sigma}_{3,2}$7 define a rational map $\overline{\sigma}_{3,2}$8 resolved by $\overline{\sigma}_{3,2}$9, which factors through a nodal cubic threefold $\Ku(Y)$0 via contraction of the strict transform of $\Ku(Y)$1. On $\Ku(Y)$2 there are two semi-orthogonal decompositions—one from $\Ku(Y)$3, giving a "blown-up" Kuznetsov component $\Ku(Y)$4 whose Verdier quotient to $\Ku(Y)$5 has kernel generated by $\Ku(Y)$6 and $\Ku(Y)$7—and one from Orlov's blow-up formula. Combining them, Alexeev–Kuznetsov obtain an equivalence $\Ku(Y)$8 under which the two kernel objects map exactly to $\Ku(Y)$9 and $C$0. Consequently,

$C$1

The degeneration result

The proof of the main theorem proceeds in two steps. First, using the explicit bound on $C$2 for general genus four curves—derived from Liu–Niu's computation of the higher-rank Clifford-type bounds—the paper shows $C$3 for all $C$4. Hence $C$5 is a genuine stability condition for all $C$6, and setting $C$7 produces a weak pre-stability condition $C$8 with discrete-image central charge. Harder–Narasimhan filtrations exist by the standard argument adapted to weak central charges. The support property at $C$9 is proved by a perturbation argument: any $g$0-stable object with $g$1 remains stable for $g$2 with $g$3 (using discreteness of the image of $g$4), and then satisfies the quadratic inequality by the uniform support property; objects with $g$5 are handled directly, invoking the bound $g$6 for stable rank-$g$7 bundles of slope three.

Second, since $g$8—indeed $g$9 consists exactly of objects whose mass tends to zero as $T_C$0—the heart $T_C$1 descends to a bounded t-structure on the quotient, and $T_C$2 is a Bridgeland stability condition on $T_C$3-graded numerically. The author notes that Bol's limiting-stability framework could alternatively be applied directly, since the limiting support property follows from the same computation.

Moduli consequences

For the class $T_C$4, the objects $T_C$5 are $T_C$6-stable for every $T_C$7, parametrizing an irreducible component of the moduli space $T_C$8 isomorphic to $T_C$9; smoothness of dimension two at these points follows from simplicity and vanishing of $C$00.

The behavior at the critical value $C$01 is the most striking part. Two points $C$02 become S-equivalent after descending to $C$03 if and only if the corresponding secant lines in $C$04 meet $C$05 at a common point $C$06—equivalently, both divisors are contained in one of the two trigonal line bundles $C$07. Concretely, there are short exact sequences

$C$08

so both objects map to $C$09 in the quotient. These identifications reproduce exactly the relation between $C$10 and the Fano variety of lines on the nodal cubic threefold through its normalization, confirming that the degenerate moduli problem on $C$11 matches the geometry of $C$12.

Limitations and open questions

The paper is explicit about its scope. The identification of S-equivalence classes at $C$13 is established only for objects of the specific class $C$14; no full moduli space $C$15 or $C$16 is constructed, and the author deliberately avoids the subtleties involved. The comparison between the descended stability condition and other constructions of stability conditions on $C$17 (via blow-up methods or conic fibrations) remains open, and the known uniqueness results do not apply because $C$18 is not proper when $C$19 is singular. Finally, whether the crepant categorical resolutions $C$20 of $C$21 admit Bridgeland stability conditions is left open—a question of independent interest given expectations that certain similar categories should not admit any.

Conclusion

The paper demonstrates that the Feyzbakhsh–Novik family of stability conditions on $C$22, for a general genus four curve, degenerates at the boundary value permitted by the Brill–Noether function to a weak stability condition that descends along the Alexeev–Kuznetsov quotient to a Bridgeland stability condition on the Kuznetsov component of a nodal cubic threefold. The resulting wall-crossing description recovers the normalization of the Fano variety of lines, illustrating concretely how degenerations of stability conditions interact with Verdier quotients and singular Calabi–Yau categories.