**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:**

- The splitting type of a degree d morphism
**P**^1 —>**P**^1 should be -2d. - The splitting type of a line in
**P**^n should be -2, -1, …, -1 with -1 repeated (n - 2) times. - The splitting type of a conic in
**P**^2 should be -3, -3. (Conic = degree 2 map which does not map into a line.) - The splitting type of a conic in
**P**^3 should be -3, -3, -2. - The splitting type of a conic in
**P**^4 should be -3, -3, -2, -2. - 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. - 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.

Back to the start page.