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 Incoming links: Picard group

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 4 / ii / Solution Created 2026-09-24 Updated 2026-09-24

Define the homomorphism associated to a line bundle on an abelian variety

ϕ_{L} : X (k) \to Pic X, ϕ_{L} (x) = T_{x}^{*} L \otimes L^{- 1} .

The Theorem of the square gives

ϕ_{L} (x + y) = ϕ_{L} (x) \otimes ϕ_{L} (y)

, so this is a homomorphism.

Tensor products satisfy

ϕ_{L \otimes M} = ϕ_{L} \otimes ϕ_{M}

and

ϕ_{L^{- 1}} = ϕ_{L}^{- 1}

. Therefore

Pic^{0} X = {L : ϕ_{L} = 0}

is a subgroup of the Picard group. If

M = ϕ_{L} (x)

, translations commute and the theorem of the square gives

T_{y}^{*} M \otimes M^{- 1} ≅ O_{X}

for every

y

. Hence

M \in Pic^{0} X

, proving

im ϕ_{L} \subseteq Pic^{0} X

.

Solved by gpt-5.6-sol high.

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 2 / a / Solution Created 2026-09-24 Updated 2026-09-24

Write the invariant prime divisors in the order of the rays

v_{1} = (1, 0), v_{2} = (0, 1), v_{3} = (- 1, k), v_{4} = (0, - 1)

D_{1}, \dots, D_{4}

. The toric divisor class sequence is

0 ⟶ M ⟶ Z^{4} ⟶ Cl (F_{k}) ⟶ 0,

and the two characters in the standard basis of

M

give

D_{1} - D_{3} = 0, D_{2} + k D_{3} - D_{4} = 0.

Thus

Cl (F_{k}) ≅ Z [S] \oplus Z [F],

where

S = D_{2}

,

F = D_{1} = D_{3}

, and

D_{4} = S + k F

. Since the fan is smooth, this is also the Picard group.

The orbit-cone correspondence decomposes

F_{k}

into one two-dimensional torus, four one-dimensional torus orbits, and four fixed points. Only the zero-dimensional orbits contribute to the compactly supported Euler characteristic, so

χ_{top} (F_{k}) = 4.

Solved by gpt-5.6-sol high.

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