TY - JOUR

T1 - On higher syzygies of ruled surfaces III

AU - Choi, Youngook

AU - Park, Euisung

N1 - Funding Information:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( NRF-2011-0014465 ) and by the 2013 Yeungnam University Research Grant for the first named author. This research was supported by the Korea Research Foundation Grant funded by the Korean Government ( NRF-2013R1A1A2008445 ) for the second named author.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - In this paper, we study the minimal free resolution of homogeneous coordinate rings of a ruled surface S over a curve of genus g with the numerical invariant e<0 and a minimal section C0. Let L∈PicX be a line bundle in the numerical class of aC0+bf such that a≥1 and 2b-ae=4g-1+k for some k≥max(2, -e). We prove that the Green-Lazarsfeld index index(S, L) of (S, L), i.e. the maximum p such that L satisfies condition N2,p, satisfies the inequalitiesk2-g≤index(S,L)≤k2-ae+32+max(0,⌈2g-3+ae-k4⌉). Also if S has an effective divisor D≡2C0+ef, then we obtain another upper bound of index(S, L), i.e., index(S,L)≤k+max(0,⌈2g-4-k2⌉). This gives a better bound in case b is small compared to a. Finally, for each e∈{-g, . ., -1} we construct a ruled surface S with the numerical invariant e and a minimal section C0 which has an effective divisor D≡2C0+ef.

AB - In this paper, we study the minimal free resolution of homogeneous coordinate rings of a ruled surface S over a curve of genus g with the numerical invariant e<0 and a minimal section C0. Let L∈PicX be a line bundle in the numerical class of aC0+bf such that a≥1 and 2b-ae=4g-1+k for some k≥max(2, -e). We prove that the Green-Lazarsfeld index index(S, L) of (S, L), i.e. the maximum p such that L satisfies condition N2,p, satisfies the inequalitiesk2-g≤index(S,L)≤k2-ae+32+max(0,⌈2g-3+ae-k4⌉). Also if S has an effective divisor D≡2C0+ef, then we obtain another upper bound of index(S, L), i.e., index(S,L)≤k+max(0,⌈2g-4-k2⌉). This gives a better bound in case b is small compared to a. Finally, for each e∈{-g, . ., -1} we construct a ruled surface S with the numerical invariant e and a minimal section C0 which has an effective divisor D≡2C0+ef.

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U2 - 10.1016/j.jpaa.2015.02.037

DO - 10.1016/j.jpaa.2015.02.037

M3 - Article

AN - SCOPUS:84929518650

VL - 219

SP - 4653

EP - 4666

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 10

ER -