Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 045, 14 pages      arXiv:2405.04002      https://doi.org/10.3842/SIGMA.2025.045

Quadratic Varieties of Small Codimension

Kiwamu Watanabe
Department of Mathematics, Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

Received January 28, 2025, in final form June 10, 2025; Published online June 15, 2025

Abstract
Let $X \subset \mathbb{P}^{n+c}$ be a nondegenerate smooth projective variety of dimension $n$ defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the Hartshorne conjecture on complete intersections, which states that $X$ is a complete intersection provided that $n \geq 2c+1$. As the extremal case, they also classified $X$ with $n=2c$. In this paper, we classify $X$ with $n=2c-1$.

Key words: Hartshorne conjecture; complete intersections; Fano varieties; homogeneous varieties.

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References

  1. Araujo C., Castravet A.M., Polarized minimal families of rational curves and higher Fano manifolds, Amer. J. Math. 134 (2012), 87-107, arXiv:0906.5388.
  2. Bertini E., Introduzione alla geometria proiettiva degli iperspazi con appendice sulle curve algebriche e loro singolaritá, Enrico Spoerri, Pisa, 1907.
  3. Bertram A., Ein L., Lazarsfeld R., Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), 587-602.
  4. Choi Y., Kwak S., Remarks on the defining equations of smooth threefolds in $\mathbb{P}^5$, Geom. Dedicata 96 (2003), 151-159.
  5. Druel S., Classes de Chern des variétés uniréglées, Math. Ann. 335 (2006), 917-935, arXiv:math.AG/0502207.
  6. Ein L., Shepherd-Barron N., Some special Cremona transformations, Amer. J. Math. 111 (1989), 783-800.
  7. Eisenbud D., Harris J., On varieties of minimal degree (a centennial account), in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., Vol. 46, American Mathematical Society, Providence, RI, 1987, 3-13.
  8. Faltings G., Ein Kriterium für vollständige Durchschnitte, Invent. Math. 62 (1981), 393-401.
  9. Fujita T., On the structure of polarized manifolds with total deficiency one. I, J. Math. Soc. Japan 32 (1980), 709-725.
  10. Fujita T., On the structure of polarized manifolds with total deficiency one. II, J. Math. Soc. Japan 33 (1981), 415-434.
  11. Hartshorne R., Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017-1032.
  12. Hartshorne R., Algebraic geometry, Grad. Texts in Math., Vol. 52, Springer, New York, 1977.
  13. Hwang J.-M., Geometry of minimal rational curves on Fano manifolds, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes, Vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 335-393.
  14. Hwang J.-M., Kebekus S., Geometry of chains of minimal rational curves, J. Reine Angew. Math. 584 (2005), 173-194, arXiv:math.AG/0403352.
  15. Ionescu P., Russo F., Conic-connected manifolds, J. Reine Angew. Math. 644 (2010), 145-157, arXiv:math.AG/0701885.
  16. Ionescu P., Russo F., Manifolds covered by lines and the Hartshorne conjecture for quadratic manifolds, Amer. J. Math. 135 (2013), 349-360, arXiv:1209.2047.
  17. Kobayashi S., Ochiai T., Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31-47.
  18. Kollár J., Mori S., Birational geometry of algebraic varieties, Cambridge Tracts in Math., Vol. 134, Cambridge University Press, Cambridge, 1998.
  19. Küchle O., Some remarks and problems concerning the geography of Fano $4$-folds of index and Picard number one, Quaestiones Math. 20 (1997), 45-60.
  20. Kuznetsov A., On linear sections of the spinor tenfold. I, Izv. Math. 82 (2018), 694-751.
  21. Kuznetsov A., Perry A., Derived categories of Gushel-Mukai varieties, Compos. Math. 154 (2018), 1362-1406, arXiv:1605.06568.
  22. Landsberg J.M., Differential-geometric characterizations of complete intersections, J. Differential Geom. 44 (1996), 32-73.
  23. Larsen M.E., On the topology of complex projective manifolds, Invent. Math. 19 (1973), 251-260.
  24. Lichtenstein W., A system of quadrics describing the orbit of the highest weight vector, Proc. Amer. Math. Soc. 84 (1982), 605-608.
  25. Mathematics Stack Exchange, Quadratic equations defining a smooth hyperplane section of the 10-dimensional spinor variety, 2024, available at https://math.stackexchange.com/questions/4899260/quadratic-equations-defining-a-smooth-hyperplane-section-of-the-10-dimensional-sh.
  26. Migliore J.C., An introduction to deficiency modules and liaison theory for subschemes of projective space, Lect. Notes Ser., Vol. 24, Seoul National University, Seoul, 1994.
  27. Mukai S., Biregular classification of Fano $3$-folds and Fano manifolds of coindex $3$, Proc. Nat. Acad. Sci. USA 86 (1989), 3000-3002.
  28. Mumford D., Varieties defined by quadratic equations, in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Centro Internazionale Matematico Estivo (C.I.M.E.), Edizioni Cremonese, Rome, 1970, 29-100.
  29. Netsvetaev N.Yu., Projective varieties defined by small number of equations are complete intersections, in Topology and Geometry – Rohlin Seminar, Lecture Notes in Math., Vol. 1346, Springer, Berlin, 1988, 433-453.
  30. Russo F., On the geometry of some special projective varieties, Lect. Notes Unione Mat. Ital., Vol. 18, Springer, Cham, 2016.
  31. Zak F.L., Tangents and secants of algebraic varieties, Transl. Math. Monogr., Vol. 127, American Mathematical Society, Providence, RI, 1993.

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