On rank $3$ quadratic equations of projective varieties
HTML articles powered by AMS MathViewer
- by Euisung Park;
- Trans. Amer. Math. Soc. 377 (2024), 2049-2064
- DOI: https://doi.org/10.1090/tran/9083
- Published electronically: December 22, 2023
- HTML | PDF | Request permission
Abstract:
Let $X \subset \mathbb {P}^r$ be a linearly normal variety defined by a very ample line bundle $L$ on a projective variety $X$. Recently it is shown by Kangjin Han, Wanseok Lee, Hyunsuk Moon, and Euisung Park [Compos. Math. 157 (2021), pp. 2001–2025] that there are many cases where $(X,L)$ satisfies property $\mathsf {QR} (3)$ in the sense that the homogeneous ideal $I(X,L)$ of $X$ is generated by quadratic polynomials of rank $3$. The locus $\Phi _3 (X,L)$ of rank $3$ quadratic equations of $X$ in $\mathbb {P}\left ( I(X,L)_2 \right )$ is a projective algebraic set, and property $\mathsf {QR} (3)$ of $(X,L)$ is equivalent to that $\Phi _3 (X)$ is nondegenerate in $\mathbb {P}\left ( I(X)_2 \right )$.
In this paper we study geometric structures of $\Phi _3 (X,L)$ such as its minimal irreducible decomposition. Let \begin{equation*} \Sigma (X,L) \!=\! \{ (A,B) \mid A,B \!\in \! {Pic}(X),~L \!=\! A^2 \otimes B,~h^0 (X,A) \!\geq \! 2,~h^0 (X,B) \!\geq \! 1 \}. \end{equation*} We first construct a projective subvariety $W(A,B) \subset \Phi _3 (X,L)$ for each $(A,B)$ in $\Sigma (X,L)$. Then we prove that the equality \begin{equation*} \Phi _3 (X,L) ~=~ \bigcup _{(A,B) \in \Sigma (X,L)} W(A,B) \end{equation*} holds when $X$ is locally factorial. Thus this is an irreducible decomposition of $\Phi _3 (X,L)$ when ${Pic} (X)$ is finitely generated and hence $\Sigma (X,L)$ is a finite set. Also we find a condition that the above irreducible decomposition is minimal. For example, it is a minimal irreducible decomposition of $\Phi _3 (X,L)$ if ${Pic}(X)$ is generated by a very ample line bundle.
References
- Enrico Arbarello and Joseph Harris, Canonical curves and quadrics of rank $4$, Compositio Math. 43 (1981), no. 2, 145–179. MR 622446
- I. V. Arzhantsev, On the factoriality of Cox rings, Mat. Zametki 85 (2009), no. 5, 643–651 (Russian, with Russian summary); English transl., Math. Notes 85 (2009), no. 5-6, 623–629. MR 2572855, DOI 10.1134/S0001434609050022
- Florian Berchtold and Jürgen Hausen, Homogeneous coordinates for algebraic varieties, J. Algebra 266 (2003), no. 2, 636–670. MR 1995130, DOI 10.1016/S0021-8693(03)00285-0
- Alessandra Bernardi, Ideals of varieties parameterized by certain symmetric tensors, J. Pure Appl. Algebra 212 (2008), no. 6, 1542–1559. MR 2391665, DOI 10.1016/j.jpaa.2007.10.022
- David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875
- David Eisenbud, Jee Koh, and Michael Stillman, Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), no. 3, 513–539. MR 944326, DOI 10.2307/2374621
- Lawrence Ein and Robert Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993), no. 1, 51–67. MR 1193597, DOI 10.1007/BF01231279
- E. Javier Elizondo, Kazuhiko Kurano, and Kei-ichi Watanabe, The total coordinate ring of a normal projective variety, J. Algebra 276 (2004), no. 2, 625–637. MR 2058459, DOI 10.1016/j.jalgebra.2003.07.007
- F. J. Gallego and B. P. Purnaprajna, Projective normality and syzygies of algebraic surfaces, J. Reine Angew. Math. 506 (1999), 145–180. MR 1665689, DOI 10.1515/crll.1999.506.145
- Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
- Mark L. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), no. 1, 279–289. MR 772134
- M. L. Green, Quadrics of rank four in the ideal of a canonical curve, Invent. Math. 75 (1984), no. 1, 85–104. MR 728141, DOI 10.1007/BF01403092
- M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), no. 3, 301–314. MR 959214
- Huy Tài Hà, Box-shaped matrices and the defining ideal of certain blowup surfaces, J. Pure Appl. Algebra 167 (2002), no. 2-3, 203–224. MR 1874542, DOI 10.1016/S0022-4049(01)00032-9
- Kangjin Han, Wanseok Lee, Hyunsuk Moon, and Euisung Park, Rank 3 quadratic generators of Veronese embeddings, Compos. Math. 157 (2021), no. 9, 2001–2025. MR 4298641, DOI 10.1112/S0010437X2100748X
- Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157, DOI 10.1007/978-1-4757-3849-0
- S. P. Inamdar, On syzygies of projective varieties, Pacific J. Math. 177 (1997), no. 1, 71–76. MR 1444773, DOI 10.2140/pjm.1997.177.71
- Hyunsuk Moon and Euisung Park, On the Rank Index of Some Quadratic Varieties, Mediterr. J. Math. 20 (2023), DOI 10.1007/s00009-023-02460-9
- Shigeru Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, Cambridge, 2003. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. MR 2004218, DOI 10.1017/CBO9781316257074
- Euisung Park, On surfaces of minimal degree in $\Bbb {P}^5$, J. Symbolic Comput. 109 (2022), 116–123. MR 4306939, DOI 10.1016/j.jsc.2021.08.001
- Euisung Park, On the rank of quadratic equations for curves of high degree, Mediterr. J. Math. 19 (2022), no. 6, Paper No. 244, 9. MR 4496651, DOI 10.1007/s00009-022-02170-8
- Euisung Park and Saerom Shim, On rank 3 quadratic equations of rational normal curves, in preparation.
- K. Petri, Über die invariante Darstellung algebraischer Funktionen einer Veränderlichen, Math. Ann. 88 (1923), no. 3-4, 242–289 (German). MR 1512130, DOI 10.1007/BF01579181
- Mario Pucci, The Veronese variety and catalecticant matrices, J. Algebra 202 (1998), no. 1, 72–95. MR 1614174, DOI 10.1006/jabr.1997.7190
- Bernard Saint-Donat, Sur les équations définissant une courbe algébrique, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A324–A327 (French). MR 289516
- B. Saint-Donat, On Petri’s analysis of the linear system of quadrics through a canonical curve, Math. Ann. 206 (1973), 157–175. MR 337983, DOI 10.1007/BF01430982
- Jessica Sidman and Gregory G. Smith, Linear determinantal equations for all projective schemes, Algebra Number Theory 5 (2011), no. 8, 1041–1061. MR 2948471, DOI 10.2140/ant.2011.5.1041
Bibliographic Information
- Euisung Park
- Affiliation: Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea
- MR Author ID: 760202
- Email: euisungpark@korea.ac.kr
- Received by editor(s): May 30, 2022
- Received by editor(s) in revised form: August 30, 2023
- Published electronically: December 22, 2023
- Additional Notes: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (No. 2022R1A2C1002784).
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 2049-2064
- MSC (2020): Primary 14E25, 13C05, 14M15, 14A10
- DOI: https://doi.org/10.1090/tran/9083
- MathSciNet review: 4744750