Seminar on Abelian Varieties
Elena Lavanda and Inder Kaur
SS 2014
Introduction
The aim of this seminar is to give the audience an introduction to abelian varieties. In particular we will focus on duality theorems. We will follow the draft
book by Ben Moonen, however, if the speaker wants, he can also add material
taken from other references.
Every talk consists of a main result which is emphasized in bold. The speaker
should double-check that every definition and every theorem, that is used in the
proof of the main result, was previously stated. Moreover, since this result is
the most important, its proof should be as detailed as possible.
In case anything is unclear to you, or you have problems in organising the
time of the talk please contact the organisers, who will be happy to help you.
Program of the talks
1.1
Introduction and first examples (15.04.2014)
Aim: Introduce abelian varieties via important and basic examples.
Details: Define a group variety 1.2, and an abelian variety 1.3. State the
proposition 1.5 and explain the idea behind the proof, then prove corollary 1.6.
Explain the example 1.7 about elliptic curves and the example 1.10 about
the complex torus.
1.2
Rigidity lemma for abelian varieties (22.04.2014)
Aim: Explain the rigidity lemma and apply it in the case of abelian varieties.
Details: State and prove rigidity lemma 1.11. Give definition 1.12 of homomorphism between group varieties. State proposition 1.13 and its corollary 1.14.
State and prove in detail theorem 1.17.
1.3
Group schemes, generalities (29.04.2014)
Aim: Introduce the theory of group schemes.
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Details: Give the definition of a group scheme as in 3.1. State Yoneda’s lemma
3.3. Define a representable functor 3.4 and state 3.6. Define a subgroup scheme
3.7 and the go through first three examples of 3.8. Give the definition of affine
group scheme 3.9 and state 3.11 and 3.12. Explain definition 3.13 and go through
3.14. State proposition 3.15 and if time permits give a proof of it.
1.4
Group schemes, Cartier duality (06.05.2014)
Aim: Define the Carter dual of a group scheme.
Details: Define the identity component 3.16. State proposition 3.17 and give
a sketch of the proof. State and prove in detail theorem 3.20. Explain and
summarize Cartier duality, as in 3.21. State theorem 3.22. Define the Cartier
dual 3.23 and explain the first example in 3.24.
1.5
Quotients of group schemes (13.05.2014)
Aim: Define the quotient by a finite group scheme and its properties.
Details: Define categorical quotients 4.1. Explain the idea and the example in
4.4. Give definition 4.5 and explain it via the examples in 4.6. State and prove
proposition 4.8 and go through the remarks. State and explain theorem
4.16, without the proof. Give the definition of a fppf-quotient and explain
the example 4.29.
1.6
Quotient of group schemes, second part (20.05.2014)
Aim: Explain fppf quotient and finite groups over a field.
Details: Go through section 3 and explain what is needed to state theorem
4.35 and understand the diagram that follows it. State 4.38 and give
example 4.40. Then pass to section 4 and explain theorem 4.41, if time permits
prove it. Give definition 4.42 and the examples in 4.43.
1.7
Theorem of the square and theorem of the cube (27.05.2014)
Aim: Explain the proofs of the results that appear in the title.
Details: State and prove theorem 2.1 and the Seesaw principle 2.2. State
lemma 2.4. State and prove the theorem of the cube 2.7. State corollary
2.8. State and prove the theorem of the square 2.9. State corollary 2.10
and 2.12.
1.8
Projectivity of abelian varieties (03.06.2014)
Aim: Prove in detail that an abelian variety is a projective variety.
Details: State the fact 2.13. Give definition of a Mumford line bundle 2.15 and
definition 2.16. State 2.18 and 2.19. State and prove proposition 2.20. State
and prove theorem 2.25. To conclude state theorem 2.27 without proving it.
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1.9
Isogenies (10.06.2014)
Aim: Give the definition and the first properties of an isogeny.
Details: State lemma 5.1 and proposition 5.2. Give the definition of an isogeny
5.3. State lemma 5.4. State and prove proposition 5.6. Define a separable
isogeny 5.7. State corollary 5.8, proposition 5.9 and corollary 5.10. State and
prove corollary 5.11. State proposition 5.12 and corollary 5.13. Go through the
last example on torsion points of 5.14.
1.10
Frobenius and Verschiebung (17.06.2014)
Aim: Define the Frobenius map and the Verschiebung map and analyse their
properties.
Details: Define relative and absolute Frobenius as in the introduction (page
3-4). State proposition 5.15 and explain the construction in 5.16. State lemma
5.17. Define the Verschiebung map 5.18. State and give a sketch of the proof
of proposition 5.19. Prove in detail Prop 5.20. State proposition 5,22. Give
definition 5.23 and go through the examples 5.28 and 5.29.
1.11
Picard scheme (01.07.2014)
Aim: Explain the construction of the Picard scheme.
Details: Use as reference the article "The Picard scheme" by Kleimann and
cover the main results that appear in it.
1.12
Dual abelian variety (08.07.2014)
Aim: State and prove two duality theorems.
Details: Give the definition of the dual abelian variety 6.9. Explain definition
7.1. State and prove theorem 7.5. State proposition 7.6 and lemma 7.8.
State and prove theorem 7.9 and prove its corollary 7.10. State proposition
7.14. State and prove lemma 7.16. State proposition 7.21 and its corollaries 7.22
and 7.23.
1.13
Neron-Severi group and duality between Frobenius
and Verschiebung morphism (15.07.2014)
Aim: Use the notions we introduced in the seminar to define the Neron-Severi
group and analyse the relation between Frobenius morphism and Verschiebung
morphism.
Details: Define Neron-Severi group 7.24. State and prove 7.25 and 7.26.
Cover section 5 on duality between Frobenius and Verschiebung completely,
emphasizing the summary given in 7.33. Finally state and prove 7.34.
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