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:
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.
Back to the start page.