Youngook Choi

Parallel machine scheduling considering a job-splitting property

This paper focuses on the problem of scheduling jobs on parallel machines considering a job-splitting property. In this problem, it is assumed that a job can be split into a discrete number of subjobs and they are processed on parallel machines independently. A two-phase heuristic algorithm is suggested for the problem with the objective of minimizing total tardiness. In the first phase, an initial sequence is constructed by an existing heuristic method for the parallel-machine scheduling problem. In the second phase, each job is split into subjobs considering possible results of the split, and then jobs and subjobs are rescheduled on the machines using a certain method. To evaluate performance of the suggested algorithm, computational experiments are performed on randomly generated test problems. Results of the experiments show that the suggested algorithm performs better than an existing one.

Projections of Projective Curves with Higher Linear Syzygies

ABSTRACT In this article, we prove that the inner projection of a projective curve with higher linear syzygies has also higher linear syzygies. Specifically, if a very ample line bundle ℒ on a smooth projective curve X satisfies property N p for p ≥ 1 and H 1 (ℒ ⊗ 2) = 0 , then ℒ( − q ) satisfies property N p − 1 for any point q ∈ X . We also give simple proofs of well-known theorems about syzygies and raise some questions related to the line bundles of degree 2 g which do not satisfy property N 1 .

On the syzygies of quasi-complete intersection space curves

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 ℙ 3 , then I x has a minimal free resolution 0 →⊕ μ-3 / i=1 S(di+3 + c 1 ) → ⊕ 2μ-4 i=1 S(-e i ) → ⊕ μ i=1 S(-d i )→I x → 0, where d i ,e i ∈ 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 Ix is t for any given positive integer t ∈ Z + .

REMARKS ON NONSPECIAL LINE BUNDLES ON GENERAL κ-GONAL CURVES

Abstract. In this work we obtain conditions for nonspecial line bundleson general k-gonal curves failing to be normally generated. Let L be anonspecial very ample line bundle on a general k-gonal curve X with k ≥ 4and degL ≥ 32 g + g−2k +1. If L fails to be normally generated, then L isisomorphic to K X −(ng 1k +B)+R for some n ≥ 1, B and R satisfying (1)h 0 (R) = h 0 (B) = 1, (2) n+3 ≤ degR ≤ 2n+2, (3) deg(R∩F) ≤ 1 for anyF ∈ g 1k . Its converse also holds under some additional restrictions. As acorollary, a very ample line bundle L ≃ K X −g 0d +ξ e is normally generatedif g 0d ∈ X (d) and ξ e ∈ X (e) satisfy d ≤ g2 − g−2k − 3, supp(g 0d ∩ ξ 0 ) = ∅and deg(g 0d ∩ F) ≤ k − 2 for any F ∈ g 1k . 1. IntroductionLet X be a smooth algebraic curve of genus g over an algebraically closedfield of characteristic zero. A very ample line bundle L on X is said to benormally generated if H 0 (P N ,O(m)) → H 0 (X,L m ) is surjective for all m ≥ 0,where P N := PH 0 (L) ∗ . Any line bundle on X of degree at least 2g + 1 isnormally generated [2, 7, 8] and any very ample line bundle on X of degree2g is normally generated unless X is hyperelliptic ([5]). On the other hand, ifX is hyperelliptic, any line bundle of degree ≤ 2g is not normally generated(Corollary 3.4 in [5]). If X is a trigonal curve of genus g > 4 with 2g −m

Remarks on Syzygies of the Section Modules and Geometry of Projective Varieties

Let X ⊂ ℙ(H 0(ℒ)) be a smooth projective variety embedded by the complete linear system associated to a very ample line bundle ℒ on X. We call the section module of ℒ. It has been known that the syzygies of R ℒ as R = Sym(H 0(ℒ))-module play important roles in understanding geometric properties of X [2, 3, 5, 9, 10] even if X is not projectively normal. Generalizing the case of N 2, p [2, 10], we prove some uniform theorems on higher normality and syzygies of a given linearly normal variety X and general inner projections when R ℒ satisfies property N 3, p (Theorems 1.1, 1.2, and Proposition 3.1). In particular, our uniform bounds are sharp as hyperelliptic curves and elementary transforms of elliptic ruled surfaces show.

On quasi-complete intersections of codimension 2

In this paper, we prove that if X ⊂ P n , n > 4, is a locally complete intersection of pure codimension 2 and defined scheme-theoretically by three hypersurfaces of degrees d 1 > d 2 > d 3 , then H 1 (P n ,I X (j)) = 0 for j < d 3 using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold X C P 5 is projectively normal if X is defined by three quintic hypersurfaces.

On the Regularity of Codimension Two Surfaces and Threefolds with Singular Loci

ABSTRACT For projective codimension two surfaces and threefolds whose singular locus is one dimensional, we get the sharp Castelnuovo–Mumford regularity bound in terms of degrees of defining equations and give the classification of nearly extremal cases. This is a generalization of the result of Bertram et al.