On the syzygies of quasi-complete intersection space curves

Choi, Youngook

doi:10.1090/S0002-9939-09-09996-1

On the syzygies of quasi-complete intersection space curves
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Proc. Amer. Math. Soc. 137 (2009), 3999-4006 Request permission

Abstract:

In this paper, we discuss minimal free resolutions of the homogeneous ideals of quasi-complete intersection space curves. We show that if $X$ is a quasi-complete intersection curve in $\mathbb P^3$, then $I_X$ has a minimal free resolution \[ 0\to \oplus _{i=1}^{\mu -3} S(d_{i+3}+c_1)\to \oplus _{i=1}^{2\mu -4}S(-e_i)\to \oplus _{i=1}^\mu S(-d_i)\to I_X\to 0, \] where $d_i,e_i\in \mathbb Z$ and $c_1=-d_1-d_2-d_3$. Therefore the ranks of the first and the second syzygy modules are determined by the number of elements in a minimal generating set of $I_X$. Also we give a relation for the degrees of syzygy modules of $I_X$. Using this theorem, one can construct a smooth quasi-complete intersection curve $X$ such that the number of minimal generators of $I_X$ is $t$ for any given positive integer $t\in \mathbb Z^+$.

References

V. Beorchia and Ph. Ellia, On the equations defining quasicomplete intersection space curves, Arch. Math. (Basel) 70 (1998), no. 3, 244–249. MR 1604080, DOI 10.1007/s000130050191
Henrik Bresinsky, Minimal free resolutions of monomial curves in $\textbf {P}^{3}_{k}$, Linear Algebra Appl. 59 (1984), 121–129. MR 743050, DOI 10.1016/0024-3795(84)90163-0
H. Bresinsky, P. Schenzel, and J. Stückrad, Quasi-complete intersection ideals of height $2$, J. Pure Appl. Algebra 127 (1998), no. 2, 137–145. MR 1620704, DOI 10.1016/S0022-4049(97)00085-6
Henrik Bresinsky, Peter Schenzel, and Wolfgang Vogel, On liaison, arithmetical Buchsbaum curves and monomial curves in $\textbf {P}^{3}$, J. Algebra 86 (1984), no. 2, 283–301. MR 732252, DOI 10.1016/0021-8693(84)90034-6
Wolfram Decker, Monads and cohomology modules of rank $2$ vector bundles, Compositio Math. 76 (1990), no. 1-2, 7–17. Algebraic geometry (Berlin, 1988). MR 1078855
Davide Franco, Steven L. Kleiman, and Alexandru T. Lascu, Gherardelli linkage and complete intersections, Michigan Math. J. 48 (2000), 271–279. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786490, DOI 10.1307/mmj/1030132718
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
Robin Hartshorne, Stable vector bundles of rank $2$ on $\textbf {P}^{3}$, Math. Ann. 238 (1978), no. 3, 229–280. MR 514430, DOI 10.1007/BF01420250
G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 689–713. MR 169877, DOI 10.1112/plms/s3-14.4.689
Mireille Martin-Deschamps and Daniel Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990), 208 (French). MR 1073438
Juan C. Migliore, Introduction to liaison theory and deficiency modules, Progress in Mathematics, vol. 165, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1712469, DOI 10.1007/978-1-4612-1794-7
Prabhakar Rao, A note on cohomology modules of rank-two bundles, J. Algebra 86 (1984), no. 1, 23–34. MR 727366, DOI 10.1016/0021-8693(84)90053-X
A. Prabhakar Rao, Liaison among curves in $\textbf {P}^{3}$, Invent. Math. 50 (1978/79), no. 3, 205–217. MR 520926, DOI 10.1007/BF01410078