# Graduate student algebraic geometry seminar

The idea of this semesters seminar is that each of the graduate students
lectures on one of the papers below. Each time we try to explain the basic new
result in the paper, as well as sketch the idea of the proof. Of course we have
to accpept some of the material/theory that is used in the paper by the author.

If you have a preference, please email me right away. I will hand out talks
on a first come, first served basis.

Topics:

1. GAGA and FAC:
Serre, Jean-Pierre Geometrie algebrique et geometrie analytique. (French) Ann. Inst. Fourier, Grenoble 6 (1955--1956), 1--42.
Serre, Jean-Pierre Faisceaux algebriques coherents. (French) Ann. of Math. (2) 61, (1955). 197--278.
2. Serre Duality:
Claimed by Thibaut Pugin.
Serre, Jean-Pierre Un theoreme de dualite. Comment. Math. Helv. 29, (1955). 9--26.
3. Borel-Weil-Bott:
Claimed by Ben Elias.
Demazure, Michel A very simple proof of Bott's theorem. Invent. Math. 33 (1976), no. 3, 271--272.
Demazure, Michel Sur la formule des caracteres de H. Weyl. (French) Invent. Math. 9 1969/1970 249--252.
Demazure, Michel Une demonstration algebrique d'un theoreme de Bott. (French) Invent. Math. 5 1968 349--356.
Bott, Raoul Homogeneous vector bundles. Ann. of Math. (2) 66 (1957), 203--248.
4. Reductive linear algebraic groups in characteristic p > 0:
Haboush, W. J. Reductive groups are geometrically reductive. Ann. of Math. (2) 102 (1975), no. 1, 67--83.
5. Fun with semi-simplicity and tensor products:
Serre, Jean-Pierre Sur la semi-simplicite des produits tensoriels de representations de groupes. (French) [On the semisimplicity of the tensor products of group representations] Invent. Math. 116 (1994), no. 1-3, 513--530.
Serre, Jean-Pierre Semisimplicity and tensor products of group representations: converse theorems. With an appendix by Walter Feit. J. Algebra 194 (1997), no. 2, 496--520.
6. Amazing paper whose methods should apply in many other situations:
Claimed by Daniel Disegni.
Faltings, Gerd; Wustholz, Gisbert Diophantine approximations on projective spaces. Invent. Math. 116 (1994), no. 1-3, 109--138.
7. Ofer Gabber's proof of nonnegativity of local intersection numbers:
Berthelot, Pierre Alterations de varietes algebriques (d'apres A.J. de Jong). (French) [Alterations of algebraic varieties (following A. J. de Jong)] Seminaire Bourbaki, Vol. 1995/96. Astrisque No. 241 (1997), Exp. No. 815, 5, 273--311.
8. The title says it all:
Claimed by Alexander Palen Ellis.
Mumford, David The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Etudes Sci. Publ. Math. No. 9 1961 5--22.
9. Fourier-Mukai transform:
Claimed by Qi You.
Mukai, Shigeru Duality between D(X) and D(\hat X) with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153--175.
Mukai, Shigeru Fourier functor and its application to the moduli of bundles on an abelian variety. Algebraic geometry, Sendai, 1985, 515--550, Adv. Stud. Pure Math., 10,
10. Mukai implies stuff about cycles on families of abelian varieties:
Deninger, Christopher(D-MUNS); Murre, Jacob(NL-LEID) Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422 (1991), 201--219.
11. Bundles on P^n which can be extended indefinitively split:
Barth, W.; Van de Ven, A. A decomposability criterion for algebraic 2-bundles on projective spaces. Invent. Math. 25 (1974), 91--106.
12. A very interesting example of a nonsplit bundle:
Claimed by Xia Jie.
Horrocks, G.; Mumford, D. A rank 2 vector bundle on P^{4} with 15,000 symmetries. Topology 12 (1973), 63--81.
13. Boundedness of stable sheaves in positive characteristic:
Langer, Adrian Semistable sheaves in positive characteristic. Ann. of Math. (2) 159 (2004), no. 1, 251--276.
Langer, Adrian Addendum to: Semistable sheaves in positive characteristic'' [Ann. of Math. (2) 159 (2004), no. 1, 251--276; MR2051393]. Ann. of Math. (2) 160 (2004), no. 3, 1211--1213.
14. Irreducibility of M_g in char p > 0.
Claimed by Yanhong Yang.
Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus. Inst. Hautes Etudes Sci. Publ. Math. No. 36 1969 75--109.
W.Fulton: On the irreducibility of the moduli space of curves, Appendix to the paper of Harris and Mumford, Inventiones Mathematicae, Volume 67, Number 1 / February, 1982, pp. 87-88.
15. Difficult but important paper on the Weil conjecture for K3 surfaces. Of course this has been obsoleted by Deligne's proof of the Weil conjecture in general. But on the other hand, this is still the only proof of semi-simplicity for Frobenius on H^2 of a K3 surface.
Deligne, Pierre La conjecture de Weil pour les surfaces K3. (French) Invent. Math. 15 (1972), 206--226.
16. Difficult but very important paper on absolute Hodge cycles for abelian varieties. It is the best evidence in favor of the Hodge conjecture so far (you don't have to agree with this of course):
It is the first paper by Deligne in the book
Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, 1982.
17. Theorem on proper subvarieties of A_g:
Keel, Sean; Sadun, Lorenzo Oort's conjecture for A_g. J. Amer. Math. Soc. 16 (2003), no. 4, 887--900.
18. Existence of quotients wiping out all previously known cases:
Keel, Sean; Mori, Shigefumi Quotients by groupoids. Ann. of Math. (2) 145 (1997), no. 1, 193--213.
19. The GHS theorem (family of rationally connected varieties over a curve have a section):