Abstract
We study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows. The main result asserts that the syzygy modules are non-zero in almost all degrees allowed by Castelnuovo–Mumford regularity. We also give an effective statement for Veronese varieties that we conjecture to be optimal.
Similar content being viewed by others
References
Aprodu, M.: Green–Lazarsfeld gonality conjecture for a generic curve of odd genus. Int. Math. Res. Not. 63, 3409–3416 (2004)
Aprodu, M., Voisin, C.: Green–Lazarsfeld’s conjecture for generic curves of large gonality. C. R. Math. Acad. Sci. Paris 336, 335–339 (2003)
Bombieri, E.: Canonical models of surfaces of general type. Publ. Math. IHÉS 42, 171–219 (1973)
Bruns, W., Conca, A., Römer, T.: Koszul homology and syzygies of Veronese subalgebras. Math. Ann. 351, 761–779 (2011)
Bruns, W., Conca, A., Römer, T.: Koszul cycles. In: Combinatorial Aspects of Commutative Algebra and Algebraic Geometry. Proceedings of the Abel Symposium 2009, pp. 17–32 (2011)
Castelnuovo, G.: Sui multipli di uni serie lineare di gruppi di punti apparetmenente as una curva algebrica. Rend. Circ. Mat. Palermo 7, 99–119 (1893)
Catanese, F.: Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications. Invent. Math. 63, 433–465 (1981)
Catanese, F.: Commutative algebra methods and equations of regular surfaces. In: Algebraic Geometry, Bucharest 1982. Lecture Notes in Math., vol. 1056, pp. 30–50. Springer, Berlin (1983)
Ein, L., Lazarsfeld, R.: Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. Invent. Math. 111, 51–67 (1993)
Eisenbud, D.: The Geometry of Syzygies. Graduate Texts in Math., vol. 229. Springer, Berlin (2005)
Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Restricting linear syzygies: algebra and geometry. Compos. Math. 141, 1460–1478 (2005)
Eisenbud, D., Schreyer, F.: Betti numbers of graded modules and cohomology of vector bundles. J. Am. Math. Soc. 22, 859–888 (2009)
Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984)
Green, M.: Koszul cohomology and the geometry of projective varieties, II. J. Differ. Geom. 20, 279–289 (1984)
Green, M., Lazarsfeld, R.: A simple proof of Petri’s theorem on canonical curves. In: Geometry Today (Rome 1984). Progr. Math., vol. 60. Birkhäuser, Basel (1985)
Green, M., Lazarsfeld, R.: Some results on the syzygies of finite sets and algebraic curves. Compos. Math. 67, 301–314 (1988)
Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1985)
Hering, M., Schenck, H., Smith, G.: Syzygies, multigraded regularity and toric varieties. Compos. Math. 142, 1499–1506 (2006)
Lazarsfeld, R.: A sampling of vector bundle techniques in the study of linear series. In: Lectures on Riemann Surfaces, pp. 500–559. World Scientific, Singapore (1989)
Maclagan, D., Smith, G.: Multigraded Castelnuovo–Mumford regularity. J. Reine Angew. Math. 571, 179–212 (2004)
Mumford, D.: On the equations defining abelian varieties. Invent. Math. 1, 287–354 (1966)
Mumford, D.: Varieties defined by quadratic equations. In: Questions on Algebraic Varieties, Corso CIME 1969, Rome, pp. 30–100 (1970)
Ottaviani, G., Paoletti, R.: Syzygies of Veronese embeddings. Compos. Math. 125, 31–37 (2001)
Pareschi, G.: Syzygies of abelian varieties. J. Am. Math. Soc. 13, 651–664 (2000)
Rubei, E.: A result on resolutions of Veronese embeddings. Ann. Univ. Ferrara, Sez. 7 50, 151–165 (2004)
Schreyer, F.: Syzygies of canonical curves and special linear series. Math. Ann. 275, 105–137 (1986)
Sidman, J., Van Tuyl, A., Wang, H.: Multigraded regularity: coarsenings and resolutions. J. Algebra 301, 703–727 (2006)
Snowden, A.: Syzygies of Segre varieties and Δ functors. To appear
Zhou, X.: Thesis in preparation
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of the first author partially supported by NSF grant DMS-1001336.
Research of the second author partially supported by NSF grant DMS-0652845.
Rights and permissions
About this article
Cite this article
Ein, L., Lazarsfeld, R. Asymptotic syzygies of algebraic varieties. Invent. math. 190, 603–646 (2012). https://doi.org/10.1007/s00222-012-0384-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-012-0384-5