Effective non-vanishing of asymptotic adjoint syzygies

Zhou, Xin

doi:10.1090/S0002-9939-2014-11947-2

Effective non-vanishing of asymptotic adjoint syzygies
HTML articles powered by AMS MathViewer

by Xin Zhou PDF

Proc. Amer. Math. Soc. 142 (2014), 2255-2264 Request permission

Abstract:

The purpose of this paper is to establish an effective non-vanishing theorem for the syzygies of an adjoint-type line bundle on a smooth variety as the positivity of the embedding increases. Our purpose here is to show that for an adjoint-type divisor $B = K_X+ bA$ with $b \geq n+1$, one can obtain an effective statement for arbitrary $X$ which specializes to the statement for Veronese syzygies in the paper “Asymptotic Syzygies of Algebraic Varieties” by Ein and Lazarsfeld. We also give an answer to Problem 7.9 in that paper in this setting.

References

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
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, DOI 10.1112/S0010437X05001776
Lawrence Ein and Robert Lazarsfeld, Asymptotic syzygies of algebraic varieties, Invent. Math. 190 (2012), no. 3, 603–646. MR 2995182, DOI 10.1007/s00222-012-0384-5
Lawrence Ein and Robert Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993), no. 1, 51–67. MR 1193597, DOI 10.1007/BF01231279
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
Igor Netay, Syzygy algebras for the Segre embedding, arXiv:1108.3733.
Giorgio Ottaviani and Raffaella Paoletti, Syzygies of Veronese embeddings, Compositio Math. 125 (2001), no. 1, 31–37. MR 1818055, DOI 10.1023/A:1002662809474
Elena Rubei, On syzygies of Segre embeddings, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3483–3493. MR 1918824, DOI 10.1090/S0002-9939-02-06597-8
Andrew Snowden, Syzygies of Segre embeddings and Delta-modules, to appear.