An answer for a Mistretta-Stoppino's conjecture

Abstract: We study the relation between linear stability of generated linear series on smooth curves and slope stability of their associated syzygy bundles. Motivated by conjectures of Mistretta and Stoppino, we establish new cases in which linear stability implies slope stability, focusing first on generated linear series over general curves and then on curves lying on polarized K3 surfaces. In the case of general curves, we use Brill-Noether-theoretic arguments to relate the numerical conditions on the linear series to the semi-stability of the syzygy bundle. For curves on K3 surfaces, we combine Lazarsfeld-Mukai bundles with Bridgeland stability conditions and restriction techniques to obtain slope-stability results under explicit degree bounds. These results provide further evidence for the expected equivalence between linear stability of linear series and slope stability of syzygy bundles.

• The paper proves that linear stability of generated, possibly incomplete, linear series on general curves is equivalent to slope stability of their syzygy bundles when codim(V) ≤ h¹(L), or equivalently d ≤ g+r.
• It characterizes the strictly semistable cases through precise numerical conditions and divisor data, and shows that slope, linear, and cohomological stability coincide under the stated general-curve hypotheses.
• For curves on polarized K3 surfaces satisfying a divisibility condition, Bridgeland stability, Lazarsfeld–Mukai bundles, and restriction theorems show that linear stability implies slope stability when 1 < r < d ≤ min{g−1, kr}.

This paper by Erick David Luna Núñez addresses two conjectures of Mistretta and Stoppino concerning the relationship between linear stability of generated linear series on smooth curves and slope stability of their associated syzygy (kernel) bundles (2608.16809). The author establishes new positive cases: for generated linear series on general curves under a codimension condition, and for curves lying on polarized K3 surfaces satisfying a divisibility condition, using Brill–Noether theory in the first setting and Bridgeland stability combined with Lazarsfeld–Mukai bundles and restriction theorems in the second.

Background and main statements

Let $C$ be a smooth projective curve over $\mathbb{C}$ and $(L,V)$ a generated linear series of type $(d,r+1)$, i.e., $V \subset H^0(L)$ generates $L$. The syzygy bundle $M_{V,L}$ is defined as the kernel of the evaluation map $V \otimes \mathcal{O}_C \to L$, so that $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$; when $V = H^0(L)$ one writes $\mathbb{C}$0. Stoppino extended Mumford's linear stability to such pairs, and Mistretta–Stoppino conjectured:

• Conjecture 1: if $\mathbb{C}$1, where $\mathbb{C}$2, then linear (semi)stability of $\mathbb{C}$3 with $\mathbb{C}$4 is equivalent to slope-(semi)stability of $\mathbb{C}$5.
• Conjecture 2: for any curve $\mathbb{C}$6 and line bundle $\mathbb{C}$7, linear (semi)stability of $\mathbb{C}$8 is equivalent to slope-(semi)stability of $\mathbb{C}$9.

Conjecture 2 was already known for general and hyperelliptic curves via Castorena–Torres-López. The present work extends these techniques to the incomplete case ($(L,V)$0), yielding two headline results. First, for a general curve $(L,V)$1 of genus $(L,V)$2: if $(L,V)$3 — equivalently, by Riemann–Roch, $(L,V)$4, which for general curves implies $(L,V)$5 — then linear (semi)stability of $(L,V)$6 is equivalent to slope-(semi)stability of $(L,V)$7. Second, for a curve $(L,V)$8 on a polarized K3 surface $(L,V)$9 satisfying the divisibility condition that $(d,r+1)$0 for all curve classes $(d,r+1)$1: if $(d,r+1)$2 and $(d,r+1)$3 is linearly stable, then $(d,r+1)$4 is slope-stable.

The incomplete case on general curves

The argument adapts the Castorena–Torres-López strategy to non-complete linear series, and the author is careful to flag where the adaptation genuinely breaks. Dualizing the first row of Butler's diagram for $(d,r+1)$5 by a subbundle $(d,r+1)$6 and twisting by $(d,r+1)$7 yields a multiplication map $(d,r+1)$8. A key observation is that, unlike the complete case, surjectivity of $(d,r+1)$9 does not follow from vanishing of $V \subset H^0(L)$0 where $V \subset H^0(L)$1: the map $V \subset H^0(L)$2 fails to be an isomorphism precisely because $V \subset H^0(L)$3. Consequently, results for complete series cannot be transferred verbatim, and the author supplies replacement arguments.

The core semistability result proceeds as follows. For a proper subbundle $V \subset H^0(L)$4 of rank $V \subset H^0(L)$5, generality of $V \subset H^0(L)$6 forces the Brill–Noether number of $V \subset H^0(L)$7 to be non-negative, giving $V \subset H^0(L)$8 and hence

$V \subset H^0(L)$9

where $L$0. Since $L$1, this is non-positive, so $L$2 is slope-semistable. Moreover, strict semistability forces very rigid numerics: $L$3, $L$4, and $L$5 with $L$6. A further Brill–Noether computation shows $L$7 for any slope-equal subbundle, which in turn implies $L$8 when $L$9. Combining these facts, any strictly slope-semistable $M_{V,L}$0 yields a strictly linearly semistable pair, establishing the equivalence between linear and slope stability. The author also characterizes exactly when stability fails: $M_{V,L}$1 is not stable if and only if $M_{V,L}$2, $M_{V,L}$3 with $M_{V,L}$4, and there exists an effective divisor $M_{V,L}$5 of degree $M_{V,L}$6 with $M_{V,L}$7 and $M_{V,L}$8.

The paper also treats cohomological stability in the sense of Ein–Lazarsfeld. Using exterior power sequences associated to kernel bundles of restricted series $M_{V,L}$9 and bounds on $V \otimes \mathcal{O}_C \to L$0 from Proposition-type Brill–Noether estimates, the author proves that $V \otimes \mathcal{O}_C \to L$1 is cohomologically semistable whenever $V \otimes \mathcal{O}_C \to L$2 induces a birational morphism on a general curve, and cohomologically stable if either $V \otimes \mathcal{O}_C \to L$3 or ($V \otimes \mathcal{O}_C \to L$4 and $V \otimes \mathcal{O}_C \to L$5). Furthermore, strict slope-semistability is characterized by the existence of a line bundle $V \otimes \mathcal{O}_C \to L$6 of degree $V \otimes \mathcal{O}_C \to L$7 with $V \otimes \mathcal{O}_C \to L$8 and $V \otimes \mathcal{O}_C \to L$9. Thus, for general curves under the standing hypothesis, linear stability, slope stability, and cohomological stability of $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$0 all coincide.

Bridgeland stability machinery

The middle portion of the paper develops the required framework: tilting $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$1 by the torsion pair $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$2 defined via the shifted slope $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$3, producing the heart $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$4 of two-term complexes, and recalling Bridgeland's stability conditions $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$5 on $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$6, which exist for $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$7. The wall-and-chamber structure of the $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$8-plane is described following Bayer–Macrì and Macrì–Schmidt: walls are semicircles centered on the $$0 \to M_{V,L} \to V \otimes \mathcal{O}_C \to L \to 0$$9-axis or vertical rays, semicircular walls are nested around a unique vertical wall at $V = H^0(L)$0, and stability within a chamber is independent of the choice of $V = H^0(L)$1. Moduli spaces $V = H^0(L)$2 of $V = H^0(L)$3-stable objects are smooth projective irreducible holomorphic symplectic varieties, non-empty exactly when $V = H^0(L)$4, and for large $V = H^0(L)$5 they coincide with Gieseker moduli spaces.

For the Mukai vector $V = H^0(L)$6 parametrizing sheaves supported on curves in $V = H^0(L)$7, the Gieseker chamber is bounded by a wall where $V = H^0(L)$8 aligns with $V = H^0(L)$9; the sheaves destabilized there are exactly those with sections, via sequences $\mathbb{C}$00. When $\mathbb{C}$01 is globally generated, $\mathbb{C}$02, the shift of the Lazarsfeld–Mukai bundle, and Bayer's criterion shows $\mathbb{C}$03 is stable off the wall. For an incomplete series $\mathbb{C}$04, the analogous object is $\mathbb{C}$05, which is strictly semistable on the wall and stable on the side where $\mathbb{C}$06 is stable; notably, $\mathbb{C}$07, so incompleteness produces precisely the expected number of destabilizing subobjects.

Curves on K3 surfaces

The K3 argument combines three ingredients. First, Feyzbakhsh's effective restriction theorem: for a $\mathbb{C}$08-stable reflexive sheaf $\mathbb{C}$09 of rank $\mathbb{C}$10 on a surface, $\mathbb{C}$11 remains $\mathbb{C}$12-(semi)stable for $\mathbb{C}$13 provided $\mathbb{C}$14. For $\mathbb{C}$15 with Chern character $\mathbb{C}$16 and $\mathbb{C}$17, the discriminant computes to $\mathbb{C}$18, and the restriction bound evaluates to less than $\mathbb{C}$19 for all $\mathbb{C}$20; hence $\mathbb{C}$21 suffices and $\mathbb{C}$22 is slope-stable. This is a clean quantitative payoff of the wall-crossing formalism.

Second, the exact sequence $\mathbb{C}$23 gives a bijection between subbundles $\mathbb{C}$24 and subbundles $\mathbb{C}$25 containing $\mathbb{C}$26, via pullback. Third, Russo–Teixidor's lemma (stability of $\mathbb{C}$27 forces $\mathbb{C}$28 for any extension $\mathbb{C}$29) controls section spaces along this correspondence.

Assuming $\mathbb{C}$30 is not stable, let $\mathbb{C}$31 be a maximal destabilizing subbundle of rank $\mathbb{C}$32 and degree $\mathbb{C}$33. A slope comparison shows $\mathbb{C}$34, so $\mathbb{C}$35. On the other hand, a claim established via stability of $\mathbb{C}$36 shows $\mathbb{C}$37, whence Koszul cohomology computations (Aprodu–Nagel) give $\mathbb{C}$38, identified with a cokernel whose dimension is controlled by Euler characteristics $\mathbb{C}$39. A chain of inequalities using $\mathbb{C}$40, $\mathbb{C}$41, $\mathbb{C}$42, and $\mathbb{C}$43 shows this cokernel is nonzero whenever $\mathbb{C}$44 — contradicting the vanishing above. Therefore $\mathbb{C}$45 must be slope-stable.

Two contextual points deserve emphasis. The result does not contradict the counterexample of Castorena–Mistretta–Torres-López, which uses a smooth plane septic: Martens' theorem forbids a K3 surface from containing plane curves of degree $\mathbb{C}$46, so that counterexample lies outside the geometric scope of the theorem. This delineates sharply where the equivalence holds and fails. Additionally, the author notes in a final remark that if $\mathbb{C}$47 were known to be slope-stable for $\mathbb{C}$48, the proof could be adapted with modified numerical inequalities — but this extension is asserted only conditionally, as the restriction bound used here requires $\mathbb{C}$49.

Limitations and open questions

Several restrictions bound the applicability of the results. On general curves, everything hinges on the codimension condition $\mathbb{C}$50 (equivalently $\mathbb{C}$51); outside this range neither semistability nor the equivalence is established. The birationality assumption on the morphism induced by $\mathbb{C}$52 is needed for the cohomological stability statements. On the K3 side, the divisibility condition $\mathbb{C}$53 for all curve classes $\mathbb{C}$54 is essential to the argument (it is automatic when $\mathbb{C}$55), and the degree bound $\mathbb{C}$56 excludes both higher degrees and low rank ($\mathbb{C}$57 is required). The final remark leaves open whether the restriction theorem can be pushed to cover $\mathbb{C}$58, which would extend the K3 equivalence accordingly. More broadly, Conjecture 1 remains open for arbitrary curves and for degrees beyond those treated here.

Conclusion

The paper extends the Mistretta–Stoppino program in two directions: it proves the full linear/slope stability equivalence for generated linear series on general curves under the condition $\mathbb{C}$59, with a complete numerical characterization of the strictly semistable locus, and it proves that linear stability implies slope stability for generated series with $\mathbb{C}$60 on curves of genus $\mathbb{C}$61 lying on K3 surfaces satisfying a Picard divisibility condition. The methods — Brill–Noether numerics in the first case, and the combination of Lazarsfeld–Mukai bundles, Bridgeland wall-crossing, and effective restriction theorems in the second — are deployed consistently with the known counterexamples, which fall outside both settings. The remaining cases, particularly higher degrees on K3 curves and arbitrary curves beyond the gonality bound, constitute the natural open territory for this line of inquiry.