Rank 3 quadratic generators of Veronese embeddings

Kangjin Han, Wanseok Lee, Hyunsuk Moon, Euisung Park

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


Let be a very ample line bundle on a projective scheme defined over an algebraically closed field with. We say that satisfies property if the homogeneous ideal of the linearly normal embedding can be generated by quadrics of rank less than or equal to. Many classical varieties, such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property. In this paper, we first prove that if then satisfies property for all and. We also investigate the asymptotic behavior of property for any projective scheme. Specifically, we prove that (i) if is-regular then satisfies property for all, and (ii) if is an ample line bundle on then satisfies property for all sufficiently large even numbers. These results provide affirmative evidence for the expectation that property holds for all sufficiently ample line bundles on, as in the cases of Green and Lazarsfeld's condition and the Eisenbud-Koh-Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513-539]. Finally, when we prove that fails to satisfy property for all.

Original languageEnglish
Pages (from-to)2001-2025
Number of pages25
JournalCompositio Mathematica
Issue number9
Publication statusPublished - 2021

Bibliographical note

Funding Information:
We would like to thank Daeyeol Jeon for Example 6.2. Kangjin Han was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science and ICT, no. 2017R1E1A1A03070765) and the DGIST Start-up Fund (Ministry of Science, ICT and Future Planning, no. 2016010066). Wanseok Lee was supported by the Basic Science Research Program through NRF funded by the Ministry of Education (no. 2020R1F1A1A01049999). Hyunsuk Moon was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT, no. 2017R1E1A1A03070765). Euisung Park was supported by a Korea Research Foundation grant funded by the Korean government (no. 2018R1D1A1B07041336). The computer algebra software Macaulay2 [GS10] was very useful for many related experiments. We are grateful to the anonymous referee for careful reading and valuable suggestions.

Publisher Copyright:


  • Veronese re-embedding
  • Veronese variety
  • determinantal presentation
  • low rank quadrics
  • property

ASJC Scopus subject areas

  • Algebra and Number Theory


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