Seminar "Algebra/Algebraische Geometrie"
(Winter 2018/19, Do 14:15-15:45 Arnim 3, Raum 119)
http://www.math.fu-berlin.de/altmann/LEHRE/xx18_WS_Algeom_SEM/agSem_WWW
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Dear friends of the AG-seminar,
not all of you have registered for this seminar - but I hope that some
will nevertheless decide to come.
First, after a small discussion about distributing the first talks
for the new subject "Abelian Varieties",
we will start with concluding the subject "Derived Categories" from
the summer semester. It begins with a talk from Sebastian Posur
about how to bring these gadgets on a computer. Please note that
this talk will be within the research seminar, i.e. it starts at 4pm.
In the two weeks to follow, Dominic Bunnett will explain the McKay
correspondence from the derived point of vies.
Then, after the NoGags workshop in Leipzig, we really start with
the new subject "Abelian Varieties" on November 11. If you know other
potentially interested people - please spread the word.
Distributing the talks for the "Abelian Varieties" part of the seminar
works by firstCome/firstServed. I.e., when you have a special wish
to deliver a certain talk - the please write me an email. I will try
to update the web page all the time.
Best regards,
Klaus Altmann
REFERENCES [Abelian Varieties]:
[Mu] Mumford, David: Abelian varieties
[Mi] Milne, James: Abelian varieties
[Ca] Cais, Bryden: Abelian varieties
REFERENCES [Derivierte Kategorien]
[DoDer] Dolgachev: http://www.math.lsa.umich.edu/~idolga/derived9.pdf
[Huy] Daniel Huybrechts: Fourier-Mukai Transforms in Algebraic Geometry
(Oxford 2006)
[GM] Gelfand-Manin
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Do, 18.10.: Introduction + Discussion + 4pm: Sebastian Posur
Do, 25.10.: Dominic Bunnett: McKay correspondence (Bridgeland-King-Reid)
Do, 1.11.: Dominic II
Do, 8.11.: NoGags Leipzig
Do, 15.11.: (1) Benjamin Kaiser
Do, 22.11.: (2) Yumeng Li
Do, 29.11.: (3) Lars Ran (Erasmus)
Do, 6.12.: (4) Anna-Lena Winz
Do, 13.12.: (5)
Do, 20.12.: (7) "Noah Gießing"
Do, 10. 1.: (6)
Do, 17. 1.:
Do, 24. 1.:
Do, 31. 1.:
Do, 7. 2.:
Do, 14. 2.:
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Talks concerning Abelian Varieties:
1) Basics of abelian varieties
(Definition; rigidity; morphisms between abelian varieties;
commutativity; (co) tangent sheaf is trivial)
[Mu, II.4]; [Mi, \S1]
2) Abelian Varieties over the Complex Numbers
(complex tori; cohomology; polarization; Weil pairing)
[Mi, \S2]
3) Rational Maps Into Abelian Varieties
(extending up to codim 2 and then even everywhere defined; rigidity;
birational equivalence implies isomorphism; no rational curves)
[Mi, \S3]
4) Cohomology and base change.
(including the Seesaw-Theorem)
[Mu, II.5]
5) The Theorem of the Cube
(Criterion for triviality of LB on XxYxZ; pullbacks of LB from
abelian varieties; Corollaries 3+4 = formulae for n*(L) and
the translation (+x)^*(L). The map \Phi_L; ampleness criterion
in Application 1. Abelian varieties are divisible groups)
[Mu, II.6] - until Application 2; [Mi, \S5]
6) The n-torsion
(n_X:=#{n-torsion elements}=n^{2g} from [Mu, II.6] Application 3
using intersection theory;
+ Appendix II.6 for an alternative proof)
7) Abelian Varieties are Projective
(recalling basic concepts: base points, linear systems; Theorem 6.4 on
p.29 in [Mi, \S6];
maybe: recalling facts about ampleness => get rid of "alg closed")
8) Quotients by finite groups, Isogenies
('etale maps; show that (Spec A)/G is an algebraic variety with
structure sheaf A^G. Free actions provide 'etale quotients.
[Example of two-dimensional CQS - with fixed point 0.]
Coherent sheaves along free actions - Prop 2 on p.70.)
[Mu, II.7], [Mi, \S7]
9) The dual abelian variety
(the map \Phi_L from [Mu], II.6; its kernel; Pic^0(A))
[Mu, II.8], [Mi, \S8]