Examples in Algebraic Geometry, Fall 2017

Professor A.J. de Jong, Columbia university, Department of Mathematics.

Basic setup: every week one of the graduate students picks a topic from the list below and explains what new example is given in the paper(s).


  1. Please email me if you want to be on the associated mailing list.
  2. The talks will be 2x45 minutes with a short break.
  3. Time and place: Room 407, Fridays 10:30 -- 12:00 AM.
  4. First meeting: September 8. Everybody interested please attend.

List of lectures: TBA

  1. Sep 15. Qixiao Ma. Del Pezzo surfaces with irregularity
  2. Sep 22. Carl Lian. An example of unirational surfaces in characteristic p.
  3. Sep 29. Shizhang Li. A nontrivial etale covering space of P^1.
  4. Oct 6. Remy van Dobben de Bruyn. Conjugate varieties with nonisomorphic pi_1.
  5. Oct 13. No meeting because of AGNES.
  6. Oct 20. Monica Marinescu. Varieties with big discrete fundamental groups.
  7. Oct 27. Marco Castronovo. Infinitely many derived equivalent nonisomorphic varieties.
  8. Nov 3. Raymond Cheng. Moving codimension 1 subvarieties over finite fields.
  9. Nov 10. Tim Large. Kahler spaces not defo equivalent to varieties.
  10. Nov 17. Gerhardt Hinkle. Patterns of dependence among powers of polynomials.
  11. Nov 24. University Holiday.
  12. Dec 1. Daniel Gulotta. Nonliftable varieties.
  13. Dec 8. TBA

List of topics

  1. Nonliftable varieties, see this beautiful paper: Serre, Jean-Pierre, Exemples de variétés projectives en caractéristique $p$ non relevables en caractéristique zéro, Proc. Nat. Acad. Sci. U.S.A. 47, 1961, pp. 108--109.
  2. Counterexamples to vanishing theorems in positive characteristic, see Raynaud, M., Contre-exemple au ``vanishing theorem'' en caractéristique $p > 0$, in C. P. Ramanujam---a tribute, Tata Inst. Fund. Res. Studies in Math. 8, pp. 273--278, 1978.
  3. Clemens' result (improving on a result of Griffiths) that homological equivalence modulo algebraic equivalence is not finitely generated, via 1-cycles on quintic 3-folds Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated, Inst. Hautes Études Sci. Publ. Math. 58, 1983, pp. 19--38 (1984)
  4. Counterexamples to Hilbert's 14th problem (see e.g. this paper by Totaro
  5. Recent examples starting with Borisov of pairs of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of L, see Lev Borisov's page and look for Grothendieck. There is also a paper by Nicolas Martin.
  6. Recent examples showing that rationality is not deformation invariant using quadric bundles (original examples by Hassett-Pirutka-Tschinkel, and newer work by Stefan Schreieder). For example, see ArXiv paper by Schreieder
  7. The Hodge conjecture is false for trivial reasons, see Alexander Grothendieck, Hodge's general conjecture is false for trivial reasons, Topology 8, 1969, pp. 299--303.
  8. J.P. Serre Exemples de variétés projective conjugées non homémorphes C.r. Acad. Sci. Paris, 258 (1964), pp. 4194-4196.
  9. varieties with infinitely many Fourier-Mukai partners, see Lesieutre's paper
  10. Varieties with non-finitely generated automorphism group, see Lesieutre's paper
  11. Counterexamples to integral Hodge/integral Tate conjectures, see Kollar in "Trento Examples" in the book "Classification of Irregular Varieties" edited by Ballico, Catanese, and Ciliberto.
  12. Kahler manifolds not deformation equivalent to a variety, see Voisin-1 and Voisin
  13. Non-isomorphic varieties with isomorphic analytification, start looking on mathoverflow
  14. Curves with more than 84(g-1) automorphisms in characteristic p. There are many papers you can look at for this. It appears from the review of one of these papers that the first example in the literature was given by Hermann Ludwig Schmid in 1938. There are papers by Hans-Wolfgang Henn and Henning Stichtenoth bounding the numbers. Then there is a zoo of examples...
  15. Burkhardt quartic (threefold of degree 4 in P^4 with max nr nodes), see H.F. Baker, A Locus with 25920 Linear Self-Transformations, Cambridge Tracts in Mathematics and Mathematical Physics, no. 39. Cambridge, at the University Press; New York, The Macmillan Company, 1946. xi+107 pp. See also: A.J. de Jong, N.I. Shepherd-Barron, and A. Van de Ven, On the Burkhardt quartic, Math. Ann. 286, 1990, no 1-3, pp. 309--328.
  16. surfaces in P^3 with maximal number of nodes, not quite sure where to look... but one absolutely stunning paper is: Beauville, Arnaud Sur le nombre maximum de points doubles d'une surface dans P3 (¿(5)=31). (French) Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 207 -- 215, Sijthoff and Noordhoff, Alphen aan den Rijn -- Germantown, Md., 1980. It seems the same technique can be used to prove something similar about sextics, see this paper by Wahl
  17. curves in P^2 with maximal number of cusps (don't know records off the top of my head), not quite sure where to look... but start with the paper by Alberto Calabri, Diego Paccagnan, and Ezio Stagnaro, Plane algebraic curves with many cusps, with an appendix by Eugenii Shustin
  18. Smooth surfaces in P^3 with high Picard number, look at this paper by Beauville for the sextic case and look at Section 6 of this paper by Shioda for the quintic case.
  19. Surfaces in characteristic p with "too many" divisors, see Shioda, Tetsuji, An example of unirational surfaces in characteristic p, Math. Ann. 211 (1974), p. 233 -- 236.
  20. Explicit surfaces of prime degree having Picard number 1, in Shioda, Tetsuji, On the Picard number of a complex projective variety, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, p. 303 -- 321 (same Shioda paper as above). Please read the review of this paper on mathscinet!
  21. High rank elliptic curves over function fields, see D. Ulmer, Elliptic curves with large rank over function fields, Ann. of Math. (2) 155 (2002), p. 295 -- 315.
  22. Moving codimension-one subvarieties over finite fields by Burt Totaro, see this arxiv paper.
  23. nontrivial etale covering pace of P^1 (rigid analytic), see this paper by de Jong
  24. stacky curve which is not a K(pi, 1) and more fun things in Behrend, Kai; Noohi, Behrang Uniformization of Deligne-Mumford curves. J. Reine Angew. Math. 599 (2006), p. 111 -- 153. Another thing worth reading in relation to this is the paper Noohi, B. Fundamental groups of algebraic stacks. J. Inst. Math. Jussieu 3 (2004), no. 1, p. 69 -- 103.
  25. Non-projective nonsingular proper threefold and a non-projective singular proper surface. Look in your copy of Hartshorne or make them yourself.
  26. Curves with maximal number of rational points over finite fields. survey paper by Voight
  27. Mumford, Fake projectve planes --- this is hard, there are other constructions
  28. Bruce Resnick, Patterns of dependence among powers of polynomials, Bruce Resnick's paper
  29. Construction of low rank vector bundles on P4 and P5, by N. Mohan Kumar, Chris Peterson, and A. Prabhakar Rao. Abstract: We describe a technique which permits a uniform construction of a number of low rank bundles, both known and new. In characteristic two, we obtain rank two bundles on P5 . In characteristic p, we obtain rank two bundles on P4 and rank three bundles on P5 . In arbitrary characteristic, we obtain rank three bundles on P4 and rank two bundles on the quadric S5 in P6.
  30. Mumford pathologies IV
  31. Mumford pathologies III
  32. Mumford pathologies II
  33. Mumford pathologies I
  34. Murphy's law.... Ravi Vakil: this is really fun and what you should do here is to prove the universality for configuration spaces of lines and points in P^2 yourself without looking in any papers! I can help!
  35. Zachary Maddock, Del Pezzo surfaces with irregularity link to paper