HEXAGONbeta

Natural Cohomology of General Steiner Bundles

Primarily AI-generated textHuman understanding: some partsmath.AC — Commutative Algebra

Contributed by Vlado Vojdanovski ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We prove that a general Steiner bundle on projective space over a field of characteris- tic zero has natural cohomology. The proof constructs a matrix of linear forms from multiplication of sections on a curve. When the rank exceeds the dimension of the projective space, a finite étale isogeny of elliptic curves splits the bundles controlling the graded kernels and cokernels into line bundles of equal degree. Choosing the auxiliary line bundle outside a finite set gives maximal rank in every degree. A rational curve provides the boundary case. Two open rank conditions then yield the assertion for general matrices and all twists.

Provenance statement

The specific additions of this paper are understood, the constructions of the cited papers are not and rely on the provided lean formalization.

Formalizations

References

  1. M. Aprodu, G. Farkas, and A. Ortega, Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces , J. Reine Angew. Math. 730 (2017), 225--249. doi:10.1515/crelle-2014-0124 (https://doi.org/10.1515/crelle-2014-0124).DOI
  2. E. Arrondo, Schwarzenberger bundles of arbitrary rank on the projective space , J. Lond. Math. Soc. (2) 82 (2010), no. 3, 697--716. doi:10.1112/jlms/jdq056 (https://doi.org/10.1112/jlms/jdq056).DOI
  3. H. Brenner, Looking out for stable syzygy bundles , with an appendix, Semistability of the general syzygy bundle , by G. Hein, Adv. Math. 219 (2008), no. 2, 401--427. doi:10.1016/j.aim.2008.04.009 (https://doi.org/10.1016/j.aim.2008.04.009).DOI
  4. I. Coskun, J. Huizenga, and G. Smith, Stability and cohomology of kernel bundles on $\PP^n$ , Michigan Math. J. 75 (2025), no. 1, 173--198. doi:10.1307/mmj/20226238 (https://doi.org/10.1307/mmj/20226238). arXiv:2204.10247 (https://arxiv.org/abs/2204.10247).arXivDOI
  5. A. Kaid and R. Kasprowitz, Semistable vector bundles and Tannaka duality from a computational point of view , Exp. Math. 21 (2012), no. 2, 171--188. doi:10.1080/10586458.2011.594665 (https://doi.org/10.1080/10586458.2011.594665).DOI
  6. R. Lazarsfeld and J. Sheridan, Torelli theorems for some Steiner bundles , Math. Z. 306 (2024), no. 1, Paper No. 19. doi:10.1007/s00209-023-03399-z (https://doi.org/10.1007/s00209-023-03399-z).DOI
  7. T. Oda, Vector bundles on an elliptic curve , Nagoya Math. J. 43 (1971), 41--72. doi:10.1017/S0027763000014367 (https://doi.org/10.1017/S0027763000014367).DOI
  8. The Stacks Project Authors, The Stacks Project , Tag 01XT (https://stacks.math.columbia.edu/tag/01XT) (Cohomology of projective space) and Tag 0BVH (https://stacks.math.columbia.edu/tag/0BVH) (Discriminant of a finite locally free morphism).link

Version history

  1. v1Submitted by Vlado VojdanovskiInitial depositCurrentOct 28, 2026