# Ravi Vakil's Publications and Preprints, etc.

## Research-related 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.
, much older , journal

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

6. "Characteristic numbers of quartic plane curves", Can. J. Math. 51 (1999), no. 5, 1089--1120.
, earlier , 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.
, older

8. "Counting curves on rational surfaces", manuscripta math. 102 (2000), 53--84.
, much older arxiv version of first half and second half)
Update 2011: For an interesting follow-up, which can deal with the case of the "degree 3 Fano = cubic surface" more directly, and may lead to the degree 2 case, see Shoval and Shustin's paper here.

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

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.

11. "Hodge integrals, Hurwitz numbers, and virtual localization", with T. Graber, Compositio Math. 135 (1) (January 2003), 25--36.
, older
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 ", with T. Graber, in (refereed) Proceedings of the Gokova Geometry-Topology conference 2000, Akbulut, Onder and Stern eds., International Press, 2001.

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).

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.
, article

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

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.
, earlier

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

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.
, earlier

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.

20. "Murphy's Law in algebraic geometry: Badly-behaved deformation spaces", Invent. Math. 164 (2006), 569--590.
, older (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.
, original , 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, Duke Math. J. 146 no. 2 (2009), 175-226.
The accepted version (Mar. 7 '08): . Based on older preprints and .
The code used in the paper is available here.

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 1947, Springer, Berlin, 2008.
, earlier

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, J. Reine Angew. Math. (Crelle's Journal), 637 (2009), 175-191.
Jul 15 '08 version , earlier

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, 27/2006, 1643-1645.
, 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, Annals of Combinatorics, 15 no. 3 (2011), 381-436.
, earlier

27. "Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus", with I. Coskun, in "Algebraic Geometry --- Seattle 2005" Part 1, 77--124, Proc. Sympos. Pure Math., 80, Amer. Math. Soc., Providence, RI, 2009.
, earlier .

28. "A desingularization of the main component of the moduli space of genus-one stable maps to projective space", with A. Zinger, Geom. Topol. 12 (2008), no. 1, 1-95.
, earlier

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 IMRN 2007, no. 20, Art. ID rnm080, 10 pp.
, earlier

30. "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, J. Comb. Theory Ser. A, 115 (2008), no. 7, 1296-1303.
, earlier
Erratum: in section 2.2, the quotient of 6 points in 3-space should be described as the Segre cubic, not the Igusa quartic. (We thank Daniel Huybrechts for catching this.)
This note relates to the baby toy on the right. I discovered later that this toy, made by Hands on Toys (who make other cool things), was created by a sculptor and mathematician, Dick Esterle. At some point I hope to write more about this. You can order the toy on the web: google "Nobbly Wobbly".

31. "Algebraic structures on the topology of moduli spaces of curves and maps", with Y.P. Lee, in Surveys in Differential Geometry Vol. XIV: "Geometry of Riemann surfaces and their moduli spaces" in honor of the 40th anniversary of Deligne and Mumford's paper, 197-216, Sur. Differ. Geom. 14, Int. Press, Somerville, MA, 2009.
, earlier

32. "The geometry of eight points in projective space: Representation theory, Lie theory, dualities", with B. Howard, J. Millson, and A. Snowden, Proc. Lond. Math. Soc. (3) 105 (2012), no. 6, 1215-1244.

33. "Universal covering spaces and fundamental groups in algebraic geometry as schemes", with K. Wickelgren, J. Theor. Nombres Bordeaux 23 (2011), no. 2, 489-526.
Adrian Langer kindly told us about an earlier description of nonexistence of group structure on etale covers of group schemes; Maruyama has a great example, which is Remark 4 in Miyanishi's paper "On the algebraic fundamental group of an algebraic group" (J. Math. Kyoto Universit, 12 (1972) 351-367). See "Covering spaces of algebraic groups" by Andy Magid (Am. Math. Monthly 83 (1976) 614-621) for more.
, earlier

34. "The relations among invariants of points on the projective line", with B. Howard, J. Millson, and A. Snowden, C.R. Math. Acad. Sci. Paris 347 (2009), no. 19-20, 1177-1182. This is an announcement of the results of papers #35 and #32.

35. "The ideal of relations for the ring of invariants of n points on the line", with B. Howard, J. Millson, and A. Snowden, J. Eur. Math. Soc., 14 (2012) no. 1, 1-60. This is the main paper in the series we four have written, and resolves the structure hinted at in #22. Code is available here.
, older

36. "The ideal of relations for the ring of invariants of n points on the line: integrality results", with B. Howard, J. Millson, and A. Snowden, Comm. Algebra 40 (2012), no. 10, 3884-3902.
older arXiv version:

37. "Solving Schubert Problems with Littlewood-Richardson Homotopies", with F. Sottile and J. Verschelde, in "Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation", Stephen W. Watt ed., ACM 2010, p. 179-186.

38. "Formal pseudodifferential operators and Witten's r-spin numbers", with K. Liu and H. Xu, J. Reine Angew. Math. (Crelle's Journal), to appear.

39. "Mnev-Sturmfels universality for schemes", with S. H. Lee, in "A celebration of algebraic geometry", 457-468, Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, 2013, 457-468.

40. "Discriminants in the Grothendieck ring", with M. M. Wood, Duke Math. J. {164 (2015), no. 6, 1139-1185.

41. "The Chow ring of the moduli space of curves of genus six", with N. Penev, Algebr. Geom. 2 (2015), no. 1, 123--136.

42. "On the Chow ring of the Hurwitz space of degree three covers of the projective line", with A. Patel, submitted.

Current research interests: deformation theory, Schubert varieties, invariant theory, moduli of curves, Gromov-Witten theory, fundamental groups, universal covers, the Grothendieck ring of varieties.

## Articles unrelated to research

Here are some. I've not bothered linking to many others (in my column in the Mathematical Intelligencer with Michael Kleber, or in Mathematical Mayhem, for example).

43. "πp, the value of π in lp", with J. Keller, Amer. Math. Monthly, 116 (Dec. 2009), no. 10, 931-935. An interesting article we learned of after this article was published: p-arclength of the q-circle, by Lindqvist and Peetre, in the Mathematics Student in 2003. A reference to this and a couple of other articles will appear soon, and I'll add a link here.

44. "The Mathematics of Doodling", Amer. Math. Monthly, February 2011. This article has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. (Vi Hart has another terrific take on doodling and mathematics.)
article

45. MathOverflow, with A. Geraschenko and S. Morrison, Notices of the Amer. Math. Soc., June/July 2010, p. 701.

46. "Algebraic geometry and the ongoing unification of mathematics", Eur. Math. Soc. Newsl. No. 89 (2013), 24-30.

## Books

47a. A Mathematical Mosaic: Patterns and Problem-Solving, first ed., 1997 (sold out). Here is the American Mathematical Monthly Review.

47b. A Mathematical Mosaic: Patterns and Problem-Solving, second expanded edition, 2007. For an amazon link, click here --- searching for the title on amazon will only yield the first edition due to a flaw in amazon's software. Here is a small pic of the cover by Henry Segerman, so you can see the intricate details. Here is a bigger pic. Here is a youtube video of the Fibonacci pinecone being built. Here is a video of a related "Fibonacci landscape" --- the HD button below the video switches it to high definition, and is better for seeing what's going on.

48. 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.

49. 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.

50. (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.

51. Enumerative Invariants in Algebraic Geometry and String Theory (Lectures from the C.I.M.E. Summer School held in Cetraro, June 6-11, 2005), with D. Abramovich, M. Marino, M. Thaddeus; K. Behrend and M. Manetti ed., Lecture Notes in Math. 1947, Springer, Berlin; Fondazione C.I.M.E., Florence, 2008, p. 143-198.

52. (co-edited with Brendan Hassett, James McKernan, and Jason Starr) "A celebration of algebraic geometry", Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, 2013.

53. and 54. The notes 2 and 5 below are gradually turning into approximations of books (the first joint with Max Lieblich, Martin Olsson, and Brian Osserman).

## 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. The Rising Sea (Foundations of algebraic geometry) '05-'06 notes, '07-'08 notes, '09-'10 notes. Full notes have mostly appeared here.

## 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).

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.

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 had been unaware of any such example in the literature, and no one I'd asked knew one. However, see the last update...) K. Schwede, then a graduate student at the University of Washington and now a professor at the University of Utah, 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. Update (Nov. '08): Simon Schieder kindly told me (months ago) what's behind Hochster's example in his article Prime Ideal Structures in Commutative Rings: Hochster calls a topological space X "spectral" if it is of the form Spec(A) for a ring A. He then proves that a topological space X is spectral if and only if: X is T_0 and quasi-compact, the quasi-compact open subsets are closed under finite intersections and form an open basis, and every non-empty irreducible closed subset has a generic point.

Knowing this, one can e.g. check easily that the following simple topological space is spectral, and removing the unique closed point gives a scheme without any closed points.

As a set, take X to be the set of natural numbers {1, 2, 3, ...} together with an extra point P ("infinity"). As open sets, take the empty set, X, X - {P} and all sets of the form [1,n), where n is a natural number.

• 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.

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 at some point.
Addendum Mar. '11, from Brian Conrad: "you mention that the essentially unique non-split commutative extension of an elliptic curve E by G_a over C is non-affine but Stein. This actually underlies something more interesting: Serre's example of a pair of non-isomorphism smooth algebraic surfaces over C that are analytically isomorphic. This is in a footnote in Mumford's book on abelian varieties. The assertion is that this extension H satisfies H^{an} = C* \times C* (= Stein), whence G_m x G_m is another "algebraization". (There are at least two different ways to prove that H cannot be isomorphic to G_m x G_m as a variety, so you can pick your own preferred proof of that.) To see that H^{an} = C* x C*, one first uses that H^{an} is a commutative connected 2-dimensional complex Lie group to identify H^{an} with V/L for a 2-dimensional C-vector space V and a discrete subgroup L. Since H^{an}[n] = H[n] (n-torsion) due to the extension structure on the group H), we see that L = Z^2. But since the non-split extension structure on H cannot be analytically split either (as H^1(E,O) --> H^1(E^{an},O) is injective), it's easy to check that the basis of L cannot be in a single C-line, so they're also a C-basis of V and hence we get the asserted description."
Addendum, Dec. '12: Qing Liu kindly informed me of more of the history of this problem. Nagata gave an example of a quasi-affine integral (non-normal) surface. It is described in p. 47-48 of these notes of Nagata. Other examples can be found in Mumford's lecture on Hilbert's 14th Problem in 1976.
• 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. Oct. 2, 2008: Simon Pepin Lehalleur at ENS points out another example, of a normal proper surface cannot be a closed subvariety of a smooth scheme, in Koll'ar's Lectures on resolution of singularities, Aside 3.46. He reports that it is a carefully chosen blow-down of a blow-up of P^1*C where C is a non-rational curve, and admits too few'' Cartier divisors, and that it looks straightforward.
• 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.
• 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) I've always wanted an easy but complete introduction into Witt vectors and their more important intricacies. Joe Rabinoff explained them to me cleanly, and his notes are posted here.
(b) 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. I'd initially posted a link to his notes, but they are now taken down; you can ask him for a copy. Here are his complete notes.
(c) (The 290 Theorem) Many years ago, Manjul Bhargava and Jonathan Hanke gave a remarkable proof of a conjecture of John H. Conway, which says that if a positive-definite quadratic form with integer coefficients represents a certain set of twenty-nine integers (of which 290 is the largest), then it represents all positive integers. In other words, if you can solve this diophantine equation in these 29 cases, you can solve it for all positive integers. For some reason (unfathomable to me), their proof has not (as of this writing) been published, or posted on the arXiv, so I decided to post a copy here. (It is also available on Jon Hanke's homepage, here.)