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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This semester I will teach a
topics course.
The topic will be etale cohomology of schemes.
I intend to work through the chapters [Arcata] and [Rapport] of SGA 4 1/2.
The class meets Tuesday and Thursday from 11-12:15 in Room 307 of the Mathematics Building.
There are now
class notes,
maintained by Thibaut Pugin et al.
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This semester the
graduate student algebraic geometry seminar is on algebraic stacks.
We meet on Fridays from 10:30-12:30 in Room 312 of the Mathematics Building.
I will give the first two lectures to introduce the material, the first one on Friday, September 11.
After the first two lectures the graduate students will lecture on examples
of algebraic stacks,
especially those which are relevant to moduli theory,
and more generally arithmetic algebraic geometry.
Here is a (preliminary) list of topics for lectures by students:
pdf |
dvi |
tex.
List of talks (see file above for more details):
- Sept 25. Quotient stacks. Alice Rizzardo.
- Oct 2. Stacky Curves. Xue Hang.
- Oct 9. Finite coverings of DM stacks. Jarod Alper.
- Oct 16. Vector bundles on stacks. Qi You.
- Oct 23. Picard Stacks. Yanhong Yang.
- Oct 30. Coarse Moduli Spaces. Yi Zhu.
- Nov 6. GIT STacks. Alon Levy.
- Nov 13. No Seminar.
- Nov 20. Cohomology of stacks. Thibaut Pugin.
- Nov 27. No Seminar.
- Dec 4. Intersection theory stacks. Zachary Maddock.
- Dec 11. Algebraic Spaces.
There is a guide to the literature, written by Jarod Alper, see
stacks_guide.pdf.
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The ALGEBRAIC STACKS PROJECT.
Please visit its
project page to learn more.
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):
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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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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:
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