Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

On syzygies of linear sections


Author: Euisung Park
Journal: Proc. Amer. Math. Soc. 143 (2015), 1831-1836
MSC (2010): Primary 14N05, 13D02
DOI: https://doi.org/10.1090/S0002-9939-2015-12130-2
Published electronically: January 9, 2015
MathSciNet review: 3314094
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study how the minimal free resolution of a closed subscheme $ X \subset \mathbb{P}^r$ relates to that of a linear section $ X \cap \Lambda \subset \Lambda = \mathbb{P}^s$ $ (0 < s <r)$. Our main result implies that the shape of the final non-zero row of the Betti diagram of $ X$ is preserved under taking the zero-dimensional and one-dimensional linear sections.


References [Enhancements On Off] (What's this?)

  • [E] David Eisenbud, The geometry of syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. MR 2103875 (2005h:13021)
  • [EGHP] 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 (2006m:14072), https://doi.org/10.1112/S0010437X05001776
  • [GL] M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), no. 3, 301-314. MR 959214 (90d:14034)
  • [L] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471 (2005k:14001a)
  • [NP] Uwe Nagel and Yves Pitteloud, On graded Betti numbers and geometrical properties of projective varieties, Manuscripta Math. 84 (1994), no. 3-4, 291-314. MR 1291122 (95i:13017), https://doi.org/10.1007/BF02567458
  • [P] Euisung Park, Higher syzygies of hyperelliptic curves, J. Pure Appl. Algebra 214 (2010), no. 2, 101-111. MR 2559684 (2010i:14057), https://doi.org/10.1016/j.jpaa.2009.04.006

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14N05, 13D02

Retrieve articles in all journals with MSC (2010): 14N05, 13D02


Additional Information

Euisung Park
Affiliation: Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea
Email: euisungpark@korea.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-2015-12130-2
Keywords: Minimal free resolution, Linear section
Received by editor(s): October 31, 2012
Received by editor(s) in revised form: January 3, 2013
Published electronically: January 9, 2015
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society