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Mistretta–Stoppino conjecture

Let CC be a smooth projective curve and let (L,V)(L,V) be a generated linear series with deg⁡L=d\deg L=d and dim⁡V=r+1\dim V=r+1, where r≥2r\ge 2. Let ML,VM_{L,V} be its syzygy bundle, defined by the exact sequence 0→ML,V→V⊗OC→L→00\to M_{L,V}\to V\otimes\mathcal O_C\to L\to 0. The Mistretta–Stoppino conjecture asks whether (L,V)(L,V) is linearly stable if and only if ML,VM_{L,V} is slope-stable. Here linear stability means that for every subspace W⊊VW\subsetneq V with dim⁡W≥2\dim W\ge 2, if LWL_W is the line bundle generated by the image of W⊗OC→LW\otimes\mathcal O_C\to L, then deg⁡LWdim⁡W−1>dr\frac{\deg L_W}{\dim W-1}>\frac{d}{r}; slope stability means that every nonzero proper subbundle F⊊ML,VF\subsetneq M_{L,V} satisfies μ(F)<μ(ML,V)=−dr\mu(F)<\mu(M_{L,V})=-\frac{d}{r}.

Progress summary

Partially solved

A 2020 counterexample disproved the unrestricted claim, while work in 2025 and 2026 established additional special cases.

The conjecture asks whether linear stability is equivalent to stability of the associated syzygy bundle, for suitable linear series on curves. Mistretta and Stoppino formulated broad complete and non-complete versions in their 2012 paper, which proved several restricted cases.

Known results

  • Mistretta–Stoppino, 2012: equivalence under Clifford-index, degree, codimension, and completeness hypotheses.
  • Mistretta–Stoppino, 2012: non-general non-complete systems can be linearly stable while their syzygy bundles are unstable.
  • Butler: broad semistability results under high-degree hypotheses, without resolving this conjecture.
  • Farkas–Ortega, 2020: a smooth plane curve of degree 77 gives a linearly stable complete series whose rank-22 syzygy bundle is not semistable, disproving the unrestricted complete-case conjecture.

2025–2026 restricted advances

A 2025 paper obtained positive cases for a generalized conjecture via rank-two Clifford indices and Lazarsfeld–Mukai bundles. In August 2026, Erick Luna proved equivalence for generated series when d≤krd\leq kr, for general curves under a codimension bound, and for curves on polarized K3K3 surfaces under an additional degree bound; these are not a general resolution.

Current status (as of August 2026): The unrestricted complete-case conjecture is disproved, while the restricted non-complete and geometric versions remain only partially established.

Sources
Sources & referencesView supporting material

Primary source

arXiv

Additional references

algebraic-geometry

Solutions 0

No solutions have been posted yet.