TY - JOUR
T1 - On the Rank of Quadratic Equations for Curves of High Degree
AU - Park, Euisung
N1 - Funding Information:
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2022R1A2C1002784). The author is also grateful to the referees for the valuable comments and helpful corrections.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/12
Y1 - 2022/12
N2 - Let C⊂ Pr be a linearly normal curve of arithmetic genus g and degree d. In Saint-Donat (CR Acad Sci Paris Ser A 274: 324–327, 1972), B. Saint-Donat proved that the homogeneous ideal I(C) of C is generated by quadratic equations of rank at most 4 whenever d≥ 2 g+ 2. Also, in Eisenbud et al. (Amer J Math 110: 513–539, 1988) Eisenbud, Koh and Stillman proved that I(C) admits a determinantal presentation if d≥ 4 g+ 2. In this paper, we will show that I(C) can be generated by quadratic equations of rank 3 if either g= 0 , 1 and d≥ 2 g+ 2 or else g≥ 2 and d≥ 4 g+ 4.
AB - Let C⊂ Pr be a linearly normal curve of arithmetic genus g and degree d. In Saint-Donat (CR Acad Sci Paris Ser A 274: 324–327, 1972), B. Saint-Donat proved that the homogeneous ideal I(C) of C is generated by quadratic equations of rank at most 4 whenever d≥ 2 g+ 2. Also, in Eisenbud et al. (Amer J Math 110: 513–539, 1988) Eisenbud, Koh and Stillman proved that I(C) admits a determinantal presentation if d≥ 4 g+ 2. In this paper, we will show that I(C) can be generated by quadratic equations of rank 3 if either g= 0 , 1 and d≥ 2 g+ 2 or else g≥ 2 and d≥ 4 g+ 4.
KW - homogeneous ideal
KW - Projective curve
KW - property QR(3)
UR - http://www.scopus.com/inward/record.url?scp=85139846903&partnerID=8YFLogxK
U2 - 10.1007/s00009-022-02170-8
DO - 10.1007/s00009-022-02170-8
M3 - Article
AN - SCOPUS:85139846903
VL - 19
JO - Mediterranean Journal of Mathematics
JF - Mediterranean Journal of Mathematics
SN - 1660-5446
IS - 6
M1 - 244
ER -