Ravi Vakil's Publications and Preprints, etc.

I use this page for distributing articles, preprints and work in progress. On this page are:

Publications and preprints

Any comments, corrections or suggestions would be greatly appreciated. I haven't posted TeX files of articles with complicated figures.

1. "On Conway's recursive sequence", with T. Kubo, Discrete Math. 152 (1-3) (1996) pp. 225-252.
article

2. "Enumerative geometry of curves via degeneration methods". Ph.D. Thesis, Harvard University, 1997, under the supervision of Joe Harris.

3. "On the Steenrod length of real projective spaces: finding longest chains in certain directed graphs", in the Gould Anniversary Volume of Discrete Math. 204 (1-3) (1999) pp. 415-425.
article

4. "The enumerative geometry of rational and elliptic curves in projective space", J. Reine Angew. Math. (Crelle's Journal) 529 (2000), 101--153.
dvi ps pdf, much older arxiv, journal

5. "Recursions for characteristic numbers of genus one plane curves", Arkiv for Matematik, 39 (2001), no. 1, 157--180.
dvi ps pdf, older arxiv

6. "Characteristic numbers of quartic plane curves", Can. J. Math. 51 (1999), no. 5, 1089--1120.
dvi ps, earlier arxiv, article
This article won the G. de B. Robinson Award for best article in the Can. J. Math. in 1998 and 1999.

7. "Genus 0 and 1 Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations", Trans. Amer. Math. Soc., 353 (2001), 4025--4038.
dvi ps pdf, older arxiv

8. "Counting curves on rational surfaces", manuscripta math. 102 (2000), 53--84.
dvi ps pdf, much older arxiv version of first half and second half)

9. "Twelve points on the projective line, branched covers, and rational elliptic fibrations", Math. Ann. 320 (2001) no. 1, 33--54.
dvi ps pdf arxiv

10. "The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals", with I. P. Goulden and D. M. Jackson, Proc. London Math. Soc. (3) 83 (2001), no. 3, 563--581.
dvi ps pdf arxiv

11. "Hodge integrals, Hurwitz numbers, and virtual localization", with T. Graber, Compositio Math. 135 (1) (January 2003), 25--36.
dvi ps pdf, older arxiv
Correction: Maryam Mirzakhani points out that the formula for Hurwitz numbers with two parts, just after the statement of the main theorem, should not have a factor of d^d.

12. "On the tautological ring of ¯Mg,n", with T. Graber, in (refereed) Proceedings of the Gokova Geometry-Topology conference 2000, Akbulut, Onder and Stern eds., International Press, 2001.
dvi ps pdf

13. "A tool for stable reduction of curves on surfaces", in "Advances in Algebraic Geometry Motivated by Mathematical Physics", E. Previato ed., Contemp. Math. 276 (2001), 145-154 (refereed).
dvi ps pdf

14. "The moduli space of curves and its tautological ring", Notices of the Amer. Math. Soc. (feature article), vol. 50, no. 6, June/July 2003, p. 647-658.
dvi ps pdf, article

15. "A geometric Littlewood-Richardson rule" (with an appendix joint with A. Knutson), Annals of Math. 164 (2006), 371-422.
pdf, earlier arxiv

16. "Relative virtual localization, and vanishing of tautological classes on moduli spaces of curves", with T. Graber, Duke Math. J., vol. 30, no. 1, 2005, 1--37.
dvi ps pdf, earlier arxiv

17. "Schubert induction", Annals of Math. 164 (2006), 489-512.
pdf, earlier arxiv

18. "Towards the geometry of double Hurwitz numbers", with I. P. Goulden and D. M. Jackson, Advances in Mathematics, (special issue in honour of Michael Artin) 198 (2005), 43-92.
dvi ps pdf, earlier arxiv

19. "The affine stratification number and the moduli space of curves", with M. Roth, (refereed) Proceedings of "Workshop on algebraic structures and moduli spaces", CRM Proceedings and Lecture Notes, Universite de Montreal, Volume 38, 2004, 213-227.
dvi ps pdf arxiv, addenda and corrigenda

20. "Murphy's Law in algebraic geometry: Badly-behaved deformation spaces", Invent. Math. 164 (2006), 569--590.
dvi ps pdf, older arxiv (includes arguments later excised)

20a. "Murphy's Law for the Hilbert scheme (and the Chow variety, and moduli spaces of surfaces of general type, and nodal and cuspidal plane curves, and ...)", Oberwolfach Reports Vol. 1 No. 3, 2004, p. 1676-1678.
electronic publication version (click on "report" and go to page 22)

21. "Intersections of Schubert varieties and other permutation array schemes", with S. Billey, in "Algorithms in Algebraic Geometry", IMA Volume 146, p. 21-54, 2008.
dvi ps pdf, original arxiv, maple code available on Sara Billey's webpage.

22. "The equations for the moduli space of n points on the line", with B. Howard, J. Millson, and A. Snowden, submitted.
The updated resubmitted version (Sept. 24 '07): pdf. Based on older preprints arxiv and arxiv.
The code used in the paper is available here. NEW!

23. "The moduli space of curves and Gromov-Witten theory", in "Enumerative invariants in algebraic geometry and string theory" (Behrend and Manetti eds.), Lecture Notes in Mathematics, Springer, to appear.
ps pdf, earlier arxiv

24. "A short proof of the λg-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves", with I.P. Goulden and D.M. Jackson, submitted.
dvi ps pdf earlier arxiv

25. "A natural smooth compactification of the space of elliptic curves in projective space", with A. Zinger, Electronic Research Announcements of the Amer. Math. Soc. 13 (2007) 53--59.
article

25a. "A natural smooth compactification of the space of elliptic curves in projective space via blowing up the space of stable maps", with A. Zinger, Oberwolfach Reports, to appear.
dvi ps pdf, preliminary version

26. "The moduli space of curves, double Hurwitz numbers, and Faber's intersection number conjecture", with I.P. Goulden and D.M. Jackson, submitted for publication.
Nov. 21, 2006 version dvi ps pdf arxiv

27. "Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus", with I. Coskun, submitted to the volume coming out of the Seattle monster conference.
dvi ps pdf, earlier arxiv. Revised May '08: pdf

28. "A desingularization of the main component of the moduli space of genus-one stable maps to projective space", with A. Zinger, Geometry and Topology, to appear.
pdf, earlier arxiv

29. "Absolute Galois acts faithfully on the components of the moduli space of surfaces: A Belyi-type theorem in higher dimension", with R. Easton, Int. Math. Res. Notices, to appear.
dvi ps pdf, earlier arxiv

30. "πp, the value of π in lp", with J. Keller, submitted for publication.
Feb. '08 version: pdf toy

31. "A new description of the outer automorphism of S6, and the invariants of six points in projective space", with B. Howard, J. Millson, and A. Snowden, Journal of Comb. Theory A, to appear.
dvi ps pdf, earlier arxiv
(This note relates to the baby toy on the right.)

32. "Universal covering spaces annd fundamental groups in algebraic geometry are schemes", with K. Wickelgren, in preparation.

33. "Murphy's Law in algebraic geometry, and other universality theorems", intended for the Bulletin of the Amer. Math. Soc., in preparation (it will be a while!). This will probably be written with Mike Roth.

Current research interests: deformation theory (20, 29, 31), Schubert varieties (21, 27), Geometric Invariant Theory (22), moduli of curves (23, 24, 26), Gromov-Witten theory (23, 25, 28), fundamental groups and universal covers (no papers yet).

Mosaic 2nd ed Mosaic 1st ed

Books

34a. A Mathematical Mosaic: Patterns and Problem-Solving, first ed., 1997 (sold out). Here is the American Mathematical Monthly Review.

34b. A Mathematical Mosaic: Patterns and Problem-Solving, second expanded edition, 2007. I will soon add a big pic of the cover by Henry Segerman, so you can see the intricate details.

Snowbird Putnam MirSym 35. Mirror Symmetry, with K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, and E. Zaslow, Clay Math. Inst., Amer. Math. Soc., August 2003.

36. The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary, with K. Kedlaya and B. Poonen, Mathematical Association of America, 2002. Review. Happy campers. Errata.

37. (edited) Snowbird Lectures in Algebraic Geometry (a refereed collection of the Seminaire Bourbaki-style articles based on lectures at the summer 2004 Snowbird conference), Contemp. Math. 388, Amer. Math. Soc., Providence, RI, 2005.

Course notes

1. Introduction to algebraic geometry (fall '99) notes.

2. Deformation theory in algebraic geometry (fall '00) notes.

3. Complex algebraic surfaces (fall '02) notes.

4. Intersection theory in algebraic geometry (fall '04) notes.

5. Foundations of algebraic geometry: '05-'06 notes, '07-'08 notes.

Miscellaneous

These are notes that I've passed on to people at various times. I make no claims of originality, correctness, completeness, etc., so read at your own risk. None of these are intended for publication. In fact, you probably shouldn't look at them. As always, suggestions and corrections would be appreciated.

1. A beginner's guide to jet bundles from the point of view of algebraic geometry. This note provides a functional introduction to jet bundles (and flags of jet bundles) from the point of view of enumerative algebraic geometry (with lots of exercises). It dates from early in graduate school (1994 or 1995, when I was beginning to learn algebraic geometry), and the details were worked out (with others) without referring to the literature, so the terminology may be very nonstandard, and the references are very incomplete (e.g. don't include EGA).
dvi ps pdf

2. A short proof of the irreducibility of the moduli space of curves in characteristic 0, using semistable reduction. Again, this was written down at some point in graduate school (first version 1995 or 1996). In retrospect, it is related to Fulton's beautiful argument in an appendix to a paper of Harris and Mumford, although it doesn't use admissible covers.
dvi ps pdf

3. Here are some interesting examples in algebraic geometry. The existence of some of these pathologies is ``common knowledge'', but I had never known what they were. The only claim of originality I make for these are local in nature: they are new to me, but were most likely known to someone somewhere at some time. (If you know of more precise references, please let me know.)

  • It is "well-known" that there are schemes without closed points, and I had always wanted an example. (I'm actually unaware of any such example in the literature, and no one I'd asked knew one. However, see the last update...) K. Schwede, a graduate student at the University of Washington, came up with a beautiful example; here it is. Update (Nov. '04): B. Osserman tells me that there may be another example in Hochster's thesis, where he precisely classifies possible topological spaces underlying affine schemes. That is, one of the spaces in his classification clearly has an open subspace with no closed points (in that subspace). At some point I hope to investigate this (or have someone else tell me more). Update (Nov. '06): There is a very short, beautiful example in Qing Liu's book Algebraic geometry and arithmetic curves, in Exercises 3.3.26 and 3.3.27.
  • A fun example of a variety whose ring of global sections is not finitely generated (and discussion of interesting resulting geometry), done with A. J. de Jong, motivated by a question of B. Osserman's and a suggestion of B. Conrad's. It is quasi-affine: an affine minus a Weil divisor. If a similarly simple (or simpler, or the same) example exists somewhere in the literature, please let me know. D. Allcock told me that something similar to this example may appear earlier in an article of Rees, mentioned in Nagata's counterexample to that Hilbert problem. I'll look this up at some point.
    dvi ps pdf
    Addendum, February '06: Antoine Chambert-Loir has kindly pointed out the following reference: in his 1988 Annals of Math paper, "Steins, affines, and Hilbert's fourteenth problem", Amnon Neeman proves (among other things): (a) if X is a complex algebraic variety, then X is affine if and only if X is Stein and its ring of functions is finitely generated; and (b) the total space of the (essentially unique) nontrivial G_a-bundle on an elliptic curve is Stein but not affine. Addendum, Oct. '07: Tom Nevins has kindly pointed the way in which I phrased the previous comment is misleading, and explained how. I will fix this soon!
  • There is an example of an irreducible reduced proper threefold that cannot be a closed subvariety of any proper smooth variety! (Example 4.9 of paper 19 above. The example is by M. Roth and myself, based on another example of J. Starr.) In particular, the example is not projective. There are two earlier (different) examples in the literature, due to Nori and Horrocks (as pointed out by W. Fulton); they are more complicated.
  • The previous example also gives an example of another interesting phenomenon: for each n, there is such a threefold such that the threefold requires at least n affine open sets to cover it.
  • The previous example is related to Hironaka's example of a proper nonprojective variety (see the appendix in Hartshorne). A. Knutson has told me a completely different (elementary and beautiful) example.
    dvi ps pdf
  • There is an example of a Cartier divisor that is not the difference of effective Cartier divisors. Sam Payne gave me a toric example (which is also an example of other interesting things), and I will eventually post it here.

    4. Other peoples' notes
    (a) NEW! I've always wanted an easy but complete introduction into Witt numbers and their more important intricacies. Joe Rabinoff explained them to me cleanly, and his notes are posted here.
    (b) NEW! I've also always wanted to see an understandable proof of the theorem comparing etale and analytic cohomology. Jack Hall explained this in a SAGS talk. Here are his complete notes.

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