 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-18/4/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

Pull

L

a : C \times X ⟶ X, (c, x) ⟼ φ (c) + x .

The fiber of this line bundle over

x

φ^{*} T_{x}^{*} L

. An abelian variety is connected, so the preceding constancy of line bundle degree in a family applies. Consequently $de g (φ^{*} T_{x}^{*} L) = de g (φ^{*} L)$ for every $x \in X$ .

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