Tangent and Cotangent modules of a morphism

We are still working towards a formulation of the problem we are interested in.

Exercise 34: Let G_0, …, G_n be homogeneous elements of K[S, T] of degree d with gcd(G_0, …, G_n) = 1. So these define a morphism φ : P^1 —> P^n of degree d. Compute the Hilbert polynomial of

Ω(φ) = KERNEL( R(-d) ⊕ … ⊕ R(-d) — G_0, …, G_n —> R )

Definition: Let φ = (G_0, …, G_n) : P^1 —> P^n be a morphism. The module Ω(φ) of Exercise 34 is called the pullback of the cotangent bundle.

Exercise 33 shows that the pullback of the cotangent bundle is graded free.

Definition: Let φ = (G_0, …, G_n) : P^1 —> P^n be a morphism. The splitting type of φ is the splitting type of the pullback of the cotangent bundle.

Examples of splitting types:

1. The splitting type of a degree d morphism P^1 —> P^1 should be -2d.
2. The splitting type of a line in P^n should be -2, -1, …, -1 with -1 repeated (n - 2) times.
3. The splitting type of a conic in P^2 should be -3, -3. (Conic = degree 2 map which does not map into a line.)
4. The splitting type of a conic in P^3 should be -3, -3, -2.
5. The splitting type of a conic in P^4 should be -3, -3, -2, -2.
6. The splitting type of a degree 3 rational curve in P^2 should be -5, -4 unless the morphism maps into a line in which case you get -6, -3.
7. The splitting type of a degree 3 rational curve in P^3 should be -4,-4,-4 or -5, -4, -3 or -6, -3, -3 depending on whether it is general or maps into a plane or maps into a line.

Exercise 35: Discuss efficient algorithms to compute the splitting type of a morphism φ = (G_0, …, G_n) : P^1 —> P^n over Z/pZ. We are especially interested in the case n = 5 and p = 2.

Exercise 36: Let G_0, …, G_n be homogeneous elements of K[S, T] of degree d with gcd(G_0, …, G_n) = 1. So these define a morphism φ : P^1 —> P^n of degree d. Assume that φ maps into the nonsingular hypersurface X : F = 0 of degree e. Let F_i = ∂F/∂X_i and denote F_i(G) = F_i(G_0, …, G_n). Compute the Hilbert polynomial of the module

E_X(φ) = KERNEL( R(d) ⊕ … ⊕ R(d) — F_0(G), …, F_n(G) —> R(ed) )

Definition: Let φ = (G_0, …, G_n) : P^1 —> P^n be a morphism mapping into the nonsingular hypersurface X. The module E_X(φ) of Exercise 36 is called the pullback of the extended tangent bundle of X.

Exercise 33 shows that the pullback of the extended tangent bundle is graded free.

Exercise 37: Relate the splitting types of E_X(φ) and Ω(φ) in the case n = 5, K = Z/2Z, and X is the degree 5 Fermat, i.e., given by X_0^5 + X_1^5 + X_2^5 + X_3^5 + X_4^5 + X_5^5 = 0.

GOAL OF THE PROJECT: In the situation of Exercise 37 find a morphism φ such that the splitting type of E_X(φ) consists entirely of nonnegative integers, or prove that such a morphism cannot exist.

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