Betti tables of reducible algebraic curves
Authors:
David J. Bruce, Pin-Hung Kao, Evan D. Nash, Ben Perez and Peter Vermeire
Journal:
Proc. Amer. Math. Soc. 142 (2014), 4039-4051
MSC (2010):
Primary 13D02; Secondary 14N05, 14H99, 14N20, 14Q05
DOI:
https://doi.org/10.1090/S0002-9939-2014-12158-7
Published electronically:
August 14, 2014
MathSciNet review:
3266976
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses. We also give explicit formulas for the entries of the Betti tables for all curves of genus zero and one. Last, we give formulas for the graded Betti numbers for a class of curves of higher genus.
- [1] E. Ballico, Arithmetically Cohen-Macaulay reducible curves in projective spaces, Bull. Belg. Math. Soc. Simon Stevin 10 (2003), no. 1, 43–47. MR 2032323
- [2] Dave Bayer and David Eisenbud, Graph curves, Adv. Math. 86 (1991), no. 1, 1–40. With an appendix by Sung Won Park. MR 1097026, https://doi.org/10.1016/0001-8708(91)90034-5
- [3] Gregory Burnham, Zvi Rosen, Jessica Sidman, and Peter Vermeire, Line arrangements modeling curves of high degree: Equations, syzygies and secants, arXiv:1201.5010 (2012).
- [4] David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875
- [5] David Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu, Restricting linear syzygies: algebra and geometry, Compos. Math. 141 (2005), no. 6, 1460–1478. MR 2188445, https://doi.org/10.1112/S0010437X05001776
- [6] Tom Fisher, The higher secant varieties of an elliptic normal curve, Preprint (2006).
- [7] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
- [8] Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
- [9] Robert Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 500–559. MR 1082360
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13D02, 14N05, 14H99, 14N20, 14Q05
Retrieve articles in all journals with MSC (2010): 13D02, 14N05, 14H99, 14N20, 14Q05
Additional Information
David J. Bruce
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
djbruce@umich.edu
Pin-Hung Kao
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859
Email:
kao1p@cmich.edu
Evan D. Nash
Affiliation:
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Email:
e{\textunderscore}nash@cox.net
Ben Perez
Affiliation:
Department of Mathematics, St. Olaf College, Northfield, Minnesota 55057
Email:
perez@stolaf.edu
Peter Vermeire
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859
Email:
p.vermeire@cmich.edu
DOI:
https://doi.org/10.1090/S0002-9939-2014-12158-7
Received by editor(s):
October 17, 2012
Received by editor(s) in revised form:
January 24, 2013
Published electronically:
August 14, 2014
Additional Notes:
The first, third, and fourth authors were supported by NSF grant DMS-1156890.
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2014
American Mathematical Society
