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May 24th, 2013
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  1. Considering the projective variety of dimension 1 in PG(2, Q) defined by
  2. x^3 - y^2*z + z^3
  3. The same equations, all of the same degree (d = 3)
  4. x^3 - y^2*z + z^3
  5. The dimension vector is (1, 9, 6)
  6. The 1 morphism(s) defining the variety (i.e. the maps 2->1) are
  7. x0 - x7 + x9
  8. The 3 morphisms defining the d-uple embedding (i.e. the maps 2->3) are described by
  9. (x0, x1, x2, x3, x4, x5)
  10. (x1, x3, x4, x6, x7, x8)
  11. (x2, x4, x5, x7, x8, x9)
  12.  
  13. Considering the projective variety of dimension 1 in PG(2, Q) defined by
  14. x^3*y + y^3*z + x*z^3
  15. The same equations, all of the same degree (d = 4)
  16. x^3*y + y^3*z + x*z^3
  17. The dimension vector is (1, 14, 10)
  18. The 1 morphism(s) defining the variety (i.e. the maps 2->1) are
  19. x1 + x9 + x11
  20. The 3 morphisms defining the d-uple embedding (i.e. the maps 2->3) are described by
  21. (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)
  22. (x1, x3, x4, x6, x7, x8, x10, x11, x12, x13)
  23. (x2, x4, x5, x7, x8, x9, x11, x12, x13, x14)
  24.  
  25. Considering the projective variety of dimension 2 in PG(3, Q) defined by
  26. x^4 + y^4 + x*y*z*w + z^2*w^2 - w^4
  27. The same equations, all of the same degree (d = 4)
  28. x^4 + y^4 + x*y*z*w + z^2*w^2 - w^4
  29. The dimension vector is (1, 34, 20)
  30. The 1 morphism(s) defining the variety (i.e. the maps 2->1) are
  31. x0 + x14 + x20 + x32 - x34
  32. The 4 morphisms defining the d-uple embedding (i.e. the maps 2->3) are described by
  33. (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)
  34. (x1, x4, x5, x6, x10, x11, x12, x13, x14, x15, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29)
  35. (x2, x5, x7, x8, x11, x13, x14, x16, x17, x18, x21, x23, x24, x26, x27, x28, x30, x31, x32, x33)
  36. (x3, x6, x8, x9, x12, x14, x15, x17, x18, x19, x22, x24, x25, x27, x28, x29, x31, x32, x33, x34)
  37.  
  38. Considering the projective variety of dimension 2 in PG(4, Q) defined by
  39. x0 + x1 + x2 + x3 + x4
  40. x0^3 + x1^3 + x2^3 + x3^3 + x4^3
  41. The same equations, all of the same degree (d = 3)
  42. x0^3 + x0^2*x1 + x0^2*x2 + x0^2*x3 + x0^2*x4
  43. x0^3 + x1^3 + x2^3 + x3^3 + x4^3
  44. The dimension vector is (1, 34, 15)
  45. The 2 morphism(s) defining the variety (i.e. the maps 2->1) are
  46. x0 + x1 + x2 + x3 + x4
  47. x0 + x15 + x25 + x31 + x34
  48. The 5 morphisms defining the d-uple embedding (i.e. the maps 2->3) are described by
  49. (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)
  50. (x1, x5, x6, x7, x8, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)
  51. (x2, x6, x9, x10, x11, x16, x19, x20, x21, x25, x26, x27, x28, x29, x30)
  52. (x3, x7, x10, x12, x13, x17, x20, x22, x23, x26, x28, x29, x31, x32, x33)
  53. (x4, x8, x11, x13, x14, x18, x21, x23, x24, x27, x29, x30, x32, x33, x34)
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