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Syzygies of curves in products of projective spaces

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Abstract

Motivated by toric geometry, we lift machinery for understanding the syzygies of curves in projective space to the setting of products of projective spaces. Using this machinery, we show an analogue of an influential result of Gruson, Peskine, and Lazarsfeld that gives a bound on the regularity of a possibly singular curve given its degree and the dimension of the ambient projective space. To do so, we show new results linking the shape of multigraded resolutions of a sheaf to its regularity region.

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Notes

  1. One can compute using Macaulay2 that the actual regularity region is \((2,2) + {{\mathbb {N}}}^2\).

  2. The main bounds in [46] are for curves embedded in \({{\mathbb {P}}}^{d_1} \times {{\mathbb {P}}}^{d_2}\) where \(d_1,d_2 > 1\) with birational projections onto each factor.

References

  1. Almousa, A., Bruce, J., Loper, M., Sayrafi, M.: The virtual resolutions package for Macaulay. J. Softw. Algebra Geomet. 10(1), 51–60 (2020)

    Article  Google Scholar 

  2. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves Volume i, Grundlehren der mathematischen Wissenschaften. Springer New York, New York (1985)

    Google Scholar 

  3. Botbol, N., Chardin, M.: Castelnuovo mumford regularity with respect to multigraded ideals. J. Algebra 474, 361–392 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bruce, J., Heller, L.C., Sayrafi, M.: Characterizing multigraded regularity on products of projective spaces (2021). arXiv:2110.10705

  5. Bruce, J., Heller, L.C., Sayrafi, M.: Bounds on multigraded regularity (2022). arXiv:2208.11115

  6. Brown, M.K., Erman, D.: Tate resolutions on toric varieties (2021). arXiv:2108.03345

  7. Brown, M.K., Erman, D.: Linear syzygies of curves in weighted projective space (2023). arXiv:2301.09150

  8. Brown, M.K., Erman, D.: A short proof of the hanlon-hicks-lazarev theorem (2023). arXiv:2303.14319

  9. Beilinson, A.A.: Coherent sheaves on \(\mathbb{P} ^n\) and problems of linear algebra. Funct. Anal. Appl. 12, 214–216 (1978)

    Article  Google Scholar 

  10. Berkesch, C., Erman, D., Smith, G.G.: Virtual resolutions for a product of projective spaces. Algebra Geom. 7(4), 460–481 (2020)

    MathSciNet  Google Scholar 

  11. Berkesch, C., Klein, P., Loper, M.C., Yang, J.: Homological and combinatorial aspects of virtually Cohen-Macaulay sheaves. Trans. Lond. Math. Soc. 8(1), 413–434 (2021)

  12. Booms-Peot, C., Cobb, J.: Virtual criterion for generalized Eagon-Northcott complexes. J. Pure Appl. Algebra 226(12), 107–138 (2022)

  13. Bruce, J.: Asymptotic syzygies in the setting of semi-ample growth (2019). arXiv:1904.04944

  14. Bruce, J.: The quantitative behavior of asymptotic syzygies for Hirzebruch surfaces. J. Commut. Algebra 14(1), 19–26 (2022)

    Article  MathSciNet  Google Scholar 

  15. Brown, M.K., Sayrafi, M.: A short resolution of the diagonal for smooth projective toric varieties of picard rank 2 (2022). arXiv:2208.00562

  16. Bayer, D., Stillman, M.E.: On the complexity of computing syzygies. J. Symb. Comput. 6, 135–147 (1988)

    Article  MathSciNet  Google Scholar 

  17. Chardin, M., Holanda, R.: Multigraded tor and local cohomology (2022). arXiv:2211.14357

  18. Costa, L., Miró-Roig, R.M.: m-blocks collections and Castelnuovo-Mumford regularity in multiprojective spaces. Nagoya Math. J. 186, 119–155 (2006)

    Article  MathSciNet  Google Scholar 

  19. Chardin, M., Nemati, N.: Multigraded regularity of complete intersections (2020). arXiv:2012.14899

  20. Eisenbud, D., Erman, D., Schreyer, F.-O.: Tate resolutions for products of projective spaces. Acta Math. Vietnam 40, 5–36 (2014)

    Article  MathSciNet  Google Scholar 

  21. Eisenbud, D.: Geometry of Syzygies. Graduate Texts in Mathematics. Springer New York, New York (2005)

    Google Scholar 

  22. Ein, L., Lazarsfeld, R.: Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. Invent. Math. 111, 51–67 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  23. Ein, L., Niu, W., Park, J.: Singularities and syzygies of secant varieties of nonsingular projective curves. Invent. Math. 222(2), 615–665 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  24. Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S.L., Nitsure, N., Vistoli, A.: Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, RI (2005). Grothendieck’s FGA explained. MR2222646

  25. Fulton, W.: Intersection Theory. Springer New York, New York (1998)

    Book  Google Scholar 

  26. Green, M., Lazarsfeld, R.: A simple proof of petri’s theorem on canonical curves. Prog. Math. 60, 129–142 (1985)

    MathSciNet  Google Scholar 

  27. Gao, J., Li, Y., Loper, M.C., Mattoo, A.: Virtual complete intersections in \({\mathbb{P} }^1\times {\mathbb{P} }^1\). J. Pure Appl. Algebra 225(1), 106473 (2021)

    Article  Google Scholar 

  28. Gruson, L., Lazarsfeld, R., Peskine, C.: On a theorem of Castelnuovo, and the equations defining space curves. Invent. Math. 72(3), 491–506 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  29. Gotzmann, G.: Eine bedingung für die flachheit und das hilbertpolynom eines graduierten ringes. Math. Z. 158, 61–70 (1978)

  30. Gruson, L., Peskine, C.: Genre des courbes de l’espace projectif. In: Olson, L.D. (eds) Algebraic Geometry. Lecture Notes in Mathematics, vol 687. Springer, Berlin, Heidelberg. (1978). https://link.springer.com/chapter/10.1007/BFb0062927#citeas

  31. Gruson, L., Peskine, C.: Postulation des courbes gauches, In: Ciliberto, C., Ghione, F., Orecchia, F. (eds) Algebraic Geometry – Open Problems. Lecture Notes in Mathematics, vol 997. Springer, Berlin, Heidelberg. (1983). https://link.springer.com/chapter/10.1007/BFb0061646

  32. Green, M.L.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984)

    Article  MathSciNet  Google Scholar 

  33. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. available at https://macaulay2.com. Accessed 1 Dec 2022

  34. Hà, H.T.: Multigraded regularity, \(\text{ a}^{*}\)-invariant and the minimal free resolution. J. Algebra 310(1), 156–179 (2007)

    Article  MathSciNet  Google Scholar 

  35. Hering, M.: Multigraded regularity and the Koszul property. J. Algebra 323(4), 1012–1017 (2010)

    Article  MathSciNet  Google Scholar 

  36. Hanlon, A., Hicks, J., Lazarev, O.: Resolutions of toric subvarieties by line bundles and applications (2023). arXiv:2303.03763

  37. Hà, H.T., Strunk, B.: Minimal free resolutions and asymptotic behavior of multigraded regularity. J. Algebra 311(2), 492–510 (2007)

    Article  MathSciNet  Google Scholar 

  38. Hering, M., Schenck, H., Smith, G.G.: Syzygies, multigraded regularity and toric varieties. Compos. Math. 142, 1499–1506 (2006)

    Article  MathSciNet  Google Scholar 

  39. Hà, H.T., Van Tuyl, A.: The regularity of points in multi-projective spaces. J. Pure Appl. Algebra 187, 153–167 (2002)

    Article  MathSciNet  Google Scholar 

  40. William Hoffman, J., Wang, H.: Castelnuovo.mumford regularity in biprojective spaces. Adv. Geom. 4, 513–536 (2002)

    Article  MathSciNet  Google Scholar 

  41. Kemeny, M.: Universal secant bundles and syzygies of canonical curves, Invent. Math., pp. 1–32 (2020)

  42. Kwak, S.: Generic projections, the equations defining projective varieties and Castelnuovo regularity. Math. Z. 234, 413–434 (2000)

    Article  MathSciNet  Google Scholar 

  43. Kwak, S.: Castelnuovo regularity for smooth subvarieties of dimensions 3 and 4, Journal of Algebraic. Geometry 7, 195–206 (1998)

    MathSciNet  Google Scholar 

  44. Lazarsfeld, Robert K.: Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear series, vol. 48. Springer (2017)

    Google Scholar 

  45. Loper, M.: What makes a complex a virtual resolution? Trans. Am. Math. Soc. Ser. B 8(28), 885–898 (2021)

    Article  MathSciNet  ADS  Google Scholar 

  46. Lozovanu, V.: Regularity of smooth curves in biprojective spaces. J. Algebra 322, 2355–2365 (2008)

    Article  MathSciNet  Google Scholar 

  47. Lozovanu, V., Smith, G.G.: Vanishing theorems and the multigraded regularity of nonsingular subvarieties, arXiv:1208.0484 (2012)

  48. McCullough, J., Peeva, I.: Counterexamples to the eisenbud-goto regularity conjecture. J. Am. Math. Soc. 31, 473–496 (2017)

    Article  MathSciNet  Google Scholar 

  49. Maclagan, D., Smith, G.G.: Uniform bounds on multigraded regularity, Journal of Algebraic. Geometry 14, 137–164 (2003)

    Google Scholar 

  50. Maclagan, D., Smith, G.G.: Multigraded castelnuovo-mumford regularity. Journal für die reine und angewandteMathematik 571, 179–212 (2004)

    MathSciNet  Google Scholar 

  51. Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces. Birkhäuser, Basel (1980)

    Book  Google Scholar 

  52. Stacks Project Authors, Stacks project

  53. Sidman, J., Van Tuyl, A.: Multigraded regularity: syzygies and fat points. Beitr. Algebra Geom. 47(1), 67–87 (2006)

    MathSciNet  Google Scholar 

  54. Szpiro, L.: Travaux de kempf, kleiman, laksov sur les diviseurs exceptionnels. Séminaire bourbaki 1971(72), 339–353 (1973)

    MathSciNet  Google Scholar 

  55. Weibel, C.A.: An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge (1994). MR1269324

  56. Yang, J.: Virtual resolutions of monomial ideals on toric varieties, Proc. Am. Math. Soc. Ser. B (2019)

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Acknowledgements

Thanks to Daniel Erman for his valuable insight during this project. Thanks also to Michael Kemeny who helped me understand some of the arguments in [28], to Rob Lazarsfeld for alerting me to a different treatment of the \(n=2\) case by his student in [46], and to an anonymous reviewer for their thorough reading and detailed suggestions that greatly improved the paper. The computer algebra system Macaulay2 [33] was used extensively, in particular, the VirtualResolutions package [1].

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    John Cobb

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Cobb, J. Syzygies of curves in products of projective spaces. Math. Z. 306, 27 (2024). https://doi.org/10.1007/s00209-023-03422-3

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