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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)