Aise Johan de Jong -- Homepage

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

Current going ons

The algebraic geometry seminar at Columbia is organized by Micheal Thaddeus (and Friedman/me).

This semester the graduate student algebraic geometry seminar will be a literature seminar.
Please look at this page for a list of papers.
We meet on Fridays from 3:30-4:30 (+/- 15 minutes) in Room 312 of the Mathematics Building.
The first organizational meeting was a short meeting on Friday, January 22 where we divided up the talks.
List of talks with names and topics is here.
If you would like to be on the mailing list please email me.

This semester I am teaching an undergraduate course on algebraic curves.
Please visit its webpage to learn more.

The stacks project

The stacks project is an open source text book about algebraic stacks and
the algebraic geometry that is needed to define them. It is a resource for algebraic geometers
on foundational questions regarding schemes, topologies on schemes, algebraic spaces,
algebraic stacks, and more. It is being written collaboratively and you can be part of it!
Please visit its project page to learn more. There is a stacks project blog.
As a fun use of the tags system of the stacks project, here is a graph representing the logical
implications in the computation of the cohomology groups of the Serre twists of the structure sheaf on projective space:
The alphanumeric codes refer to tags in the project, explained in the tags system. Other images (some of them very large):

• A picture of the logical implications in the chapter on schemes: plaatje.png.
• Proof of Chow's lemma (Noetherian case): chows_lemma.png.
• All implications in older version of the algebra chapter (without tags): algebra.jpg.
• Proof of the Cohen structure theorem: cohen_structure_theorem.png.
• Etale equivalence relations give algebraic spaces: algebraic_spaces.png (4.1 MB).

Past projects

Here is the webpage for courses taught.

Here is the webpage for previous graduate student algebraic geometry seminars.

Last summer I organized a conference on ``Spaces of curves and their interaction with diophantine problems''. Here is a link to the web page for the conference. Here is the schedule of talks. Here is the list of abstracts.

Frobenius matrix computation project: see Frobenius. There is now an excecutable (compiled for generic x86 Linux machines) that you can try. Brief instructions: Download the executable, and save it as ``filename'' in a directory, change mode to executable, type ``./filename'' on the command line. Enjoy. Three versions:

• Frobenius matrices of surfaces in weighted projective 3-spaces: excecutable
• Frobenius matrices of curves in weighted projective 2-spaces: excecutable
• Frobenius matrices of surfaces which are double covers of weighted projective 2-spaces branched along curves: excecutable

Here is a list of papers/projects with links to the .dvi-files of some of the papers. Note that the .dvi files are not always identical with the published version (but they are very close), as in some cases editing was done after proof reading the proofs from the journal in question.

• A.J. de Jong, Th. de Jong, The virtual number of D-infinity points II, Topology, 29 (1990), pp. 185-188.
• A.J. de Jong, N.I. Sheperd-Barron, A. Van de Ven, On the Burkhardt quartic, Mathematische Annalen, 286 (1990), pp. 309-328.
• A.J. de Jong, R. Noot, Jacobians with complex multiplications, In: Arithmetic Algebraic Geometry, Eds. G. van der Geer, F. Oort, J.H.M. Steenbrink, Progress in Mathematics, 89 (1991), Birkh\"auser, pp. 177-192.
• A.J. de Jong, J.H.M. Steenbrink, Picard numbers of surfaces in 3-dimensional weighted projective spaces, Mathematisches Zeitschrift, 206 (1991), pp. 341-344.
• A.J. de Jong, The moduli spaces of polarized abelian varieties, Mathematische Annalen, 295 (1993), pp. 485-503. papers/AGEE.dvi
• A.J. de Jong, The moduli spaces of principally polarized abelian varieties with $\Gamma_0(p)$-level structure, Journal of Algebraic Geometry, 2 (1993), pp. 667-688. papers/SGEE.dvi
• A.J. de Jong, Finite locally free group schemes in characteristic $p$ and Dieudonn\'e modules, Inventiones mathematicae, 114 (1993), pp. 89-137. papers/DIEUDONNEE.dvi
• A.J. de Jong, Ample line bundles and intersection theory, In: Diophantine approximation and Abelian varieties, Eds. B. Edixhoven, J.-H. Evertse, Lecture Notes in Mathematics, 1566 (1993), Springer-Verlag, pp. 69-76. papers/SOESTERBERG.dvi
• A.J. de Jong, J.H.M. Steenbrink, Proof of a conjecture of W. Veys, Indagationes Mathematicae, N.S., 6 (1995), pp. 99-104.
• A.J. de Jong, M. van der Put, \'Etale cohomology of rigid spaces, Documenta Mathematica (electronic journal), 1 (1996), pp. 1--56. papers/RIGETALE.dvi
• A.J. de Jong, Crystalline Dieudonn\'e module theory via formal and rigid geometry, Publications Mathematiques I.H.E.S., 82 (1995), pp. 5-96. papers/FORMALRIGID.dvi
  Here is an erratum: /papers/erratum.dvi
• M. Pikaart, A.J. de Jong, Moduli of curves with non-abelian level structures, In: The moduli spaces of curves, Eds. R. Dijkgraaf, C. Faber, G. van der Geer, Progress in Mathematics, 129 (1995), Birkh\"auser, pp. 483-509. papers/MARTIN.dvi
• A.J. de Jong, W. Messing, Crystalline Dieudonn\'e theory over excellent schemes, Bull. Soc. Math. France 127, no. 2, 333--348 (1999). papers/MESSING.dvi
• A.J. de Jong, \'Etale fundamental groups of non-Archimedean analytic spaces, In: Special issue of Compositio in honour of Frans Oort, Compositio Mathematica, 97 (1995), 89-118. papers/ETALEFUN.dvi
• A.J. de Jong, F. Oort, On extending families of curves, Journal of Algebraic Geometry, 6 (1997), pp. 545--562. papers/EXTFAMCURVES.dvi
• A.J. de Jong, A result on formal linearity, Journal of Algebra 225, no. 2, pp. 936--942 (2000). papers/FORMALLINEAR.dvi
• A.J. de Jong, Smoothness, semi-stability and alterations, Publications Mathematiques I.H.E.S., 83(1996), pp. 51-93. papers/ALTERATIONS.dvi
• B. van Geemen, A.J. de Jong, On Hitchin's connection, Journal of the American Mathematical Society, 11 (1998), pp. 189--228. papers/BERT.dvi
• A.J. de Jong, Appendix to a paper by P. Berthelot on finiteness of rigid cohomology, Published as appendix to: P. Berthelot, Finitude et puret\'e cohomologique en cohomologie rigide, Inventiones Mathematicae, 128 (1997), 329--377.
• A.J. de Jong, Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic, Inventiones Mathematicae, 134 (1998). papers/TATE.dvi
  Here is an erratum: papers/ErratumTate.dvi
• F. Oort, A.J. de Jong, Hyperelliptic curves in Abelian Varieties, Published in: "Manin's Festschrift", Journal of mathematical sciences, 82 (1997), pp. 141--166. papers/HYPERELL.dvi
• A.J. de Jong, Families of curves and alterations, Annales de l'institute Fourier, 47 (1997), pp. 599--621. papers/ALTCURVES.dvi
• D. Abramovich, A.J. de Jong, Smoothness, semi-stability and toroidal geometry, Journal of Algebraic Geometry, 6 (1997), pp. 789-801. arXiv
• A.J. de Jong, A conjecture on arithmetic fundamental groups, Israel Journal of Mathematics. {\bf 121}, 61--84 (2001). papers/ARITHFUN.dvi
• A.J. de Jong, Barsotti-Tate groups and crystals, Proceedings ICM 1998 (Berlin), volume II, pp. 259--265. papers/ICM.dvi
• A.J. de Jong, An application of alterations to Dieudonn\'e modules, Resolution of singularities (Obergurgl, 1997), 375--380, Progr. Math., 181, Birkhauser, Basel, 2000. papers/OBERG.dvi
• A.J. de Jong, F. Oort, Purity of the stratification by Newton Polygons, J. Amer. Math. Soc. 13, no. 1, 209--241 (2000). papers/NEWTON.dvi
• A.J. de Jong, N.M. Katz, Monodromy and the Tate conjecture: Picard numbers and Mordell-Weil ranks in families, Israel J. Math. {\bf 120}, part A, 47--79 (2000).
• A.J. de Jong, J. Starr, Every rationally connected variety over the function field of a curve has a rational point, American Journal of Mathematics, {\bf 125}, 567--580 (2003). familyofcurves3.dvi
  Here is the ad hoc argument showing how to obtain reduced fibres after base change which was in theoriginal version of the preprint: alternative.dvi
• A.J. de Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Mathematical Journal, 123, 71--94 (2004). perind.dvi
• A.J. de Jong, Counting elliptic surfaces over finite fields, Moscow Mathematical Journal, 2, 281--381 (2002). surfaces2.dvi
• B. Conrad and A.J. de Jong, Approximation of versal deformations, Journal of Algebra, 255, no. 2, 489--515 (2002). approximation.dvi
• A.J. de Jong, J. Starr, Cubic fourfolds and spaces of rational curves, Illinois Journal of Mathematics, 48 (2004), pp. 415--430.
• A.J. de Jong, A result of Gabber,Preprint with missing references about Br=Br' on quasiprojective schemes. 2-gabber.dvi or 2-gabber.pdf
• J. Starr, A.J. de Jong, A remark on isotrivial families, Preprint with missing references, precursor of the next entry. JasonsTrick_final.pdf
• J. Starr, A.J. de Jong, Almost proper GIT-stacks and discriminant avoidance, Preprint with missing references and misleading title. This paper also handles the period-index problem for Brauer groups of surfaces over any algebraically closed field. 5-torsor.pdf
• A.J. de Jong, Shioda cycles in families of surfaces, Preprint on work in progress: Shioda.dvi
• A.J. de Jong and N.M. Katz Counting the number of curves over a finite field. This unpublished note was written probably around the time I was in Princeton (1998-2000) since it resulted from discussions with Nick katz. It gives the trivial upper bound for the number of isomoprhism classes of genus g curves over F_q. Here is a link: curves.dvi.

OLD Calculus I web-pages:

• The webpage (spring 2008 -- this contains links to the Fall 2007 tests).
• Spring 2008 exams: first_midterm.dvi, second_midterm.dvi, final.dvi
• The webpage (fall 2007).

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