On the Rank of Quadratic Equations for Curves of High Degree

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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.

Original languageEnglish
Article number244
JournalMediterranean Journal of Mathematics
Issue number6
Publication statusPublished - 2022 Dec

Bibliographical note

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.


  • homogeneous ideal
  • Projective curve
  • property QR(3)

ASJC Scopus subject areas

  • General Mathematics


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