×
Bruce, Juliette; Erman, Daniel; Goldstein, Steve; Yang, Jay

Conjectures and computations about Veronese syzygies. (English) Zbl 1460.14134

Exp. Math. 29, No. 4, 398-413 (2020).
Summary: We formulate several conjectures which shed light on the structure of Veronese syzygies of projective spaces. These conjectures are motivated by experimental data that we derived from a high-speed high-throughput computation of multigraded Betti numbers based on numerical linear algebra.

Cited in 5 Documents

MSC:

14Q20 Effectivity, complexity and computational aspects of algebraic geometry
13D02 Syzygies, resolutions, complexes and commutative rings
13P20 Computational homological algebra
14N05 Projective techniques in algebraic geometry

Software:

HTCondor; Open Science Grid; Macaulay2; Matlab
PDF BibTeX XML Cite
\textit{J. Bruce} et al., Exp. Math. 29, No. 4, 398--413 (2020; Zbl 1460.14134)
Full Text: DOI arXiv
  • logo of hbz
  • logo of WorldCat

References:

[1] Castryck, [Castryck et al. XX] W., Cools, F., Demeyer, J., and Lemmens, A.. Computing graded Betti tables of toric surfaces. arXiv:1606.08181. · Zbl 1476.14085
[2] [CHTC XX] CHTC. Center for High Throughput Computing. Information athttp://chtc.cs.wisc.edu/.
[3] Ein, [Ein et al. 15] L.; Erman, D.; Lazarsfeld, R., Asymptotics of Random Betti Tables., J. Reine Angew. Math., 702, 55-75 (2015) · Zbl 1338.13023
[4] Ein, [Ein et al. 16] L., Erman, D., and Lazarsfeld, R.. “A Quick Proof of Nonvanishing for Asymptotic Syzygies.” Algebr. Geom.3(2) (2016), (211-222). · Zbl 1397.14027
[5] Ein, [Ein and Lazarsfeld 93] L. and Lazarsfeld, R.. “Syzygies and Koszul Cohomology of Smooth Projective Varieties of Arbitrary Dimension.” Invent. Math.111(1) (1993), (51-67). · Zbl 0814.14040
[6] Ein, [Ein and Lazarsfeld 12] L. and Lazarsfeld, R.. “Asymptotic Syzygies of Algebraic Varieties.” Invent. Math.190(3) (2012), (603-646). · Zbl 1262.13018
[7] Eisenbud, [Eisenbud et al. 02] D., Grayson, D. R., Stillman, M., and Sturmfels, B. (eds.), “Computations in Algebraic Geometry with Macaulay 2.” in Algorithms and Computation in Mathematics, edited by David Eisenbud, Daniel R. Grayson, Michael Stillman, Bernd Sturmfels. Vol. 8. Berlin: Springer-Verlag. (2002), pp. xvi+329. · Zbl 0973.00017
[8] Eisenbud, [Eisenbud and Schreyer 09] D. and Schreyer, F.-O.. “Betti Numbers of Graded Modules and Cohomology of Vector Bundles.” J. Amer. Math. Soc.22(3) (2009), (859-888). · Zbl 1213.13032
[9] Erman, [Erman 15] D.. “The Betti Table of a High-Degree Curve is Asymptotically Pure.” Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser. Vol. 417. Cambridge: Cambridge Univ. Press. (2015) pp. 200-206. · Zbl 1326.13009
[10] Fløystad, [Fløystad 12] G.. Boij-Söderberg theory: introduction and survey, Progress in commutative algebra 1, de Gruyter, Berlin (2012), pp. 1-54. · Zbl 1260.13020
[11] Fulton, [Fulton and Harris 91] W. and Harris, J.. Representation theory. Graduate Texts in Mathematics. Vol. 129. A first course; Readings in Mathematics. New York: Springer-Verlag. (1991). pp. xvi+551. · Zbl 0744.22001
[12] Gill, [Gill et al. 87] P. E.; Murray, W.; Saunders, M. A.; Wright, M. H., Maintaining LU Factors of a General Sparse Matrix., Linear Algebra Appl., 88/89, 239-270 (1987) · Zbl 0618.65019
[13] Golub, [Golub and Van Loan 96] G. H. and Van Loan, C. F.. “Matrix Computations. in Johns Hopkins Studies in the Mathematical Sciences. edition 3. Baltimore, MD: Johns Hopkins University Press. (1996). · Zbl 0865.65009
[14] Green, [Green 84a] M. L.. “Koszul Cohomology and the Geometry of Projective Varieties.” J. Differential Geom.19(1) (1984a), 125-171. · Zbl 0559.14008
[15] Green, [Green 84b] M. L.. “Koszul Cohomology and the Geometry of Projective Varieties. II.” J. Differential Geom.20(1) (1984), 279-289. · Zbl 0559.14009
[16] Greco, [Greco and Martino 16] O. and Martino, I.. “Syzygies of the Veronese Modules.” Comm. Algebra44(9) (2016), 3890-3906. · Zbl 1365.13021
[17] Gu, [Gu and Eisenstat 96] M. and Eisenstat, S. C.. “Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization.” SIAM J. Sci. Comput.17(4) (1996), 848-869. · Zbl 0858.65044
[18] Grayson, [Grayson and Stillman XX] D. R. and Stillman, M. E.. Macaulay 2, a software system for research in algebraic geometry. Available athttp://www.math.uiuc.edu/Macaulay2/.
[19] [HTCondor Team XX] HTCondor Team, Computer Sciences Department, University of Wisconsin-Madison. HTCondor, open source distributed computing software. Information at https://research.cs.wisc.edu/htcondor.
[20] [MATLAB XX] Massachusetts MATLAB. version 7.10.0 (R2010a). The MathWorks Inc. (2010) Natick.
[21] [Open Science Grid XX] Open Science Grid. Information athttps://www.opensciencegrid.org/.
[22] Sam, [Sam 14] S. V.. “Derived Supersymmetries of Determinantal Varieties.” J. Commut. Algebra6(2) (2014), 261-286. · Zbl 1351.17009
[23] Thain, [Thain 05] D., Tannenbaum, T., and Livny, M.. “Distributed Computing in practice: The Condor experience.” Concurrency - Practice and Experience17(2-4) (2005), 323-356.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.