Aise Johan de Jong -- Homepage

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

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CURRENT AND FUTURE PROJECTS

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

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ALGEBRAIC STACKS PROJECT. Please visit its project page to learn more.

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PAST PROJECTS

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This 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.

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This past semester the graduate student algebraic geometry seminar was organized by Maksym Fedorchuk and me. The topic was be Introduction to higher dimensional algebraic geometry. Here is some more information:

Please email me to get on the mailing list.
Lectures will be Fridays 10:45 - 12:30 in Math 312.
First organizational meeting: Friday January 23.
First actual lecture: Friday January 30.
List of talks: positivity.pdf
Main reference: R. Lazarsfeld, Positivity in Algebraic Geometry, Ergebnisse der Mathematik, 3 Folge, Volume 48.
Other reference: R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics 156.

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This past semester I taught algebraic geometry, and in particular schemes (again, see below). See the webpage. As material I wrote some notes on schemes. See some of the files in the stacks project directory. This is a moving target (and probably contains errors) and I hope you will help me update it as the semester goes along.

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Two semesters ago the graduate student algebraic geometry seminar is organized by Frans Oort and me. The topic was ``Abelian Varieties over Finite Fields''. Main Literature Sources, with links:

J. Tate, Classes d'isog\'enies de vari\'et\'es ab\'eliennes sur un corps fini (d'\`apres T. Honda), S\'em. Bourbaki 21 (1968/69), Exp. 352. Link.
F. Oort, Abelian varieties over finite fields, In: Higher-dimensional geometry over finite fields, Advanced Study Institute 2007 Proceedings (Editors: Y. Tschinkel and D. Kaledin). Link1 and Link2.
J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144. Link.
Very detailed exposition of Complex Multiplication written up as notes for the 2005-05 VIGRE number theory working group. Organized by Brian Conrad and Chris Skinner. Link

Other links:

Homepage of Frans Oort at Columbia University Link.
Homepage of Frans Oort at Utrecht University Link.
List of talks and overview of the seminar: pdf-file or dvi-file .
Here (pdf-file and dvi-file) is a short note describing how to find all q-Weil numbers of degree exactly 4 over Q. Here is a pari script to list all minimal polynomials of q-Weil numbers of degree 4 over Q, based on the note above. And finally here are the lists for q = 2, 3, 4.

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Last semester I taught commutative algebra (again, see below). As material I wrote some algebra notes. See the file entitled algebra.pdf or algebra.dvi which you can find in the following directory. This is a moving target (and probably contains errors) and I hope you will help me update it as the semester goes along.

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The Fall semester of 2007 we discussed motives in the graduate student algebraic geometry seminar. Here are some related documents:

and a list of literature is here.
A short discussion of intersection theory (as used in the seminar) is here.
A short discussion of Weil cohomology theories (as used in the seminar) is here.
A short discussion of algebraic de Rham cohomology as a Weil cohomology theory is here.

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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

Feel free to email me with questions/comments/etc.

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A while ago I taught algebraic geometry which was a second semester of an introductory graduate course in algebraic geometry. Having gone through a substantial amount of commutative algebra in the first semester, we used this material to build the foundations of modern algebraic geometry (schemes). Here is the old webpage. Here is a link to the old webpage for the first semester of this course.

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I am nerdier than 94% of all people. Are you nerdier? Click here to find out!

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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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