The generalized Plücker–Klein map

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Аннотация

The intersection of two quadrics is called a biquadric. If we mark a non-singular quadricin the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein, a Kummer surfacewas canonically associated with every three-dimensional marked biquadric (that is, witha quadratic line complex provided that the Plücker–Klein quadric is marked).In Reid's thesis, this correspondence was generalizedto odd-dimensional marked biquadrics of arbitrary dimension $\ge 3$. In this case,a Kummer variety of dimension $g$ corresponds to every biquadric of dimension $2g-1$.Reid only constructed the generalized Plücker–Klein correspondence. This map was notstudied later. The present paper is devoted to a partial solution of the problem of creatingthe corresponding theory.

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Авторлар туралы

Vyacheslav Krasnov

P.G. Demidov Yaroslavl State University

Email: vakras@yandex.ru
Doctor of physico-mathematical sciences, Associate professor

Әдебиет тізімі

  1. Ф. Гриффитс, Дж. Харрис, Принципы алгебраической геометрии, Мир, М., 1982, 864 с.
  2. J. W. S. Cassels, E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2, London Math. Soc. Lecture Note Ser., 230, Cambridge Univ. Press, Cambridge, 1996, xiv+219 pp.
  3. I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge Univ. Press, Cambridge, 2012, xii+639 pp.
  4. M. Reid, The complete intersection of two or more quadrics, Ph.D. thesis, Univ. of Cambridge, Cambridge, 1972, 94 pp.
  5. А. Н. Тюрин, “О пересечении квадрик”, УМН, 30:6(186) (1975), 51–99
  6. R. Donagi, “Group law on the intersection of two quadrics”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7:2 (1980), 217–239
  7. F. Klein, “Zur Theorie der Liniencomplexe des ersten und zweiten Grades”, Math. Ann., 2:2 (1870), 198–226
  8. C. M. Jessop, A treatise on the line complex, Cambridge Univ. Press, Cambridge, 1903, xv+362 pp.
  9. R. W. H. T. Hudson, Kummer's quartic surface, Cambridge Univ. Press, Cambridge, 1905, xi+219 pp.
  10. W. Fulton, J. Harris, Representation theory. A first course, Grad. Texts in Math., 129, Springer-Verlag, New York, 1991, xvi+551 pp.
  11. U. V. Desale, S. Ramanan, “Classification of vector bundles of rank 2 on hyperelliptic curves”, Invent. Math., 38:2 (1976), 161–185
  12. R. E. Kutz, “Cohen–Macaulay rings and ideal theory in rings of invariants of algebraic groups”, Trans. Amer. Math. Soc., 194 (1974), 115–129
  13. M. M. Kapranov, “Veronese curves and Grothendieck–Knudsen moduli space $overline{M}_{0,n}$”, J. Algebraic Geom., 2:2 (1993), 239–262
  14. P. E. Newstead, “Stable bundles of rank 2 and odd degree over a curve of genus 2”, Topology, 7:3 (1968), 205–215
  15. M. S. Narasimhan, S. Ramanan, “Moduli of vector bundles on a compact Riemann surface”, Ann. of Math. (2), 89:1 (1969), 14–51
  16. Д. Мамфорд, Лекции о тета-функциях, Мир, М., 1988
  17. S. Mukai, “Curves and symmetric spaces. I”, Amer. J. Math., 117:6 (1995), 1627–1644
  18. I. Dolgachev, D. Ortland, Point sets in projective spaces and theta functions, Asterisque, 165, Soc. Math. France, Paris, 1988, 210 pp.
  19. E. Spanier, “The homology of Kummer manifolds”, Proc. Amer. Math. Soc., 7 (1956), 155–160
  20. A. Grothendieck, “Sur quelques points d'algèbre homologique”, Tohoku Math. J. (2), 9:2 (1957), 119–221
  21. Wei-Liang Chow, “On the geometry of algebraic homogeneous spaces”, Ann. of Math. (2), 50 (1949), 32–67
  22. М. Холл, Теория групп, ИЛ, М., 1962, 468 с.
  23. S. Ramanan, “The theory of vector bundles on algebraic curves with some applications”, Moduli spaces and vector bundles, London Math. Soc. Lecture Note Ser., 359, Cambridge Univ. Press, Cambridge, 2009, 165–209
  24. D. Avritzer, H. Lange, “Moduli spaces of quadratic complexes and their singular surfaces”, Geom. Dedicata, 127 (2007), 177–197
  25. K. Rohn, “Die verschiedenen Gestalten der Kummer'schen Fläche”, Math. Ann., 18 (1881), 99–159
  26. W. Barth, I. Nieto, “Abelian surfaces of type $(1,3)$ and surfaces with $16$ skew lines”, J. Algebraic Geom., 3:2 (1994), 173–222
  27. A. Coble, “Point sets and allied Cremona groups. I”, Trans. Amer. Math. Soc., 16:2 (1915), 155–198
  28. В. А. Краснов, “Вещественные кубики Сегре, квартики Игузы и квартики Куммера”, Изв. РАН. Сер. матем., 84:3 (2020), 71–118
  29. G. van der Geer, “On the geometry of a Siegel modular threefold”, Math. Ann., 260:3 (1982), 317–350
  30. В. А. Краснов, “Вещественные куммеровы квартики и их гейзенберг-инвариантность”, Изв. РАН. Сер. матем., 84:1 (2020), 105–162

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