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This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, then at least the answers might be of use not just for me.

To differentiate this from some questions already asked, let me clarify:

  • I am talking only about modern algebraic geometry, as in: everything that is better dealt with in terms of sheaves and schemes rather than varieties and curves. I know well enough that classical ("Italian") algebraic geometry has lots of applications; I am interested in knowing a reason to study (and a golden thread to follow in that) the kind of algebraic geometry that started with Serre, Leray, Grothendieck.

  • A "combinatorial/constructive algebraist" is a notion I cannot really formalize, but I mean an algebraist who is interested in actual computable things and their "fine structure" rather than topological abstracta and their "crude structure"; for example, actual polynomial identities rather than equality of zero-sets; actual isomorphisms instead of isomorphy; "for every point not on the zero-set of some particular ideal" rather than "for almost every point". The "combinatorial/constructive algebraist" (himself an abstraction) is fine with abstraction and formalism as long as he knows how to transform the abstract results into concrete equations and algorithms in case of need. He is not fine with nonconstructive existence results, although he is wary of declaring proofs unconstructive at first sight merely due to their formulation...

I believe I know of one example of this kind, a problem on matrix factorization solved using cohomology of sheaves somewhere on MathOverflow (any help with finding it is appreciated). There is also the interpretation of commutative Hopf algebras as coordinate Hopf algebras of affine schemes - but affine schemes are not really what I consider to be modern algebraic geometry; they correspond 1-to-1 to rings and are more frequently considered as functors than as locally ringed spaces in Hopf algebra theory. I would personally be more convinced by applications to invariant theory (viz., results from classical invariant theory proved with geometric methods) or the combinatorial kind of representation theory. I used to think that Swan's paper I linked in question 68071 is another application of scheme theory, but after understanding Seiler's proof it seems rather unnecessary to me.

Community
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darij grinberg
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    Can you give some names of people you consider "combinatorial/constructive algebraists?" – JSE Oct 01 '11 at 21:08
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    It is not always easy to find out the underlying philosophy of a mathematician just by looking at some of its works, but these are very obvious cases: Gian-Carlo Rota, Doron Zeilberger, Donald Knuth, Henri Lombardi. – darij grinberg Oct 01 '11 at 21:23
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    There were many reasons for introducing the tools of sheaves and cohomology, but an important one was to solve the classical problems in algebraic geometry that were not already solved. See Zariski's report on sheaves from the 1950s. (Zariski was a brilliant geometer from the era before sheaves, trained in the Italian school but not limited by its perspectives, and his report --- which is essentially a report on Serre's paper --- explains how the new sheaf-theoretic methods allow one to recover and generalize many results of the Italians and of Zariski himself.) – Emerton Oct 01 '11 at 23:28
  • Would a combinatorial/constructive algebraist care about equations over finite fields? – Felipe Voloch Oct 01 '11 at 23:51
  • Felipe: if there are exact results to be had, likethe number of solutions, then yes. Examples? – darij grinberg Oct 02 '11 at 00:55
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    Emerton: Do you mean this? http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183520530 The applications sound interesting, but arithmetic genera isn't anything I have a good intuition for. Will try to learn – darij grinberg Oct 02 '11 at 00:59
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    How about Weil's conjectures? At least one of them is a bit hard to prove without "modern" algebraic geometry. – algori Oct 02 '11 at 04:34
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    Dear Darij, Yes, this is what I meant. I had in mind in particular the discussion of sheaf theory and Riemann--Roch (which has the most geometric content, I think). Riemann--Roch-type formulas, and related results like the lemma of Enriques--Severi--Zariski (and related notions like that of arithmetic genus) are concerns that were central to the Italian school, and were greatly clarified by sheaf-theoretic methods. Regards, Matthew – Emerton Oct 02 '11 at 04:41
  • Motives ?.... – M.G. Oct 02 '11 at 11:04
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    How about existence of a fixed point of an explicit automorphism of an explicit variety? Although these theorems originate earlier, SGA 5 includes beautiful, general theorems which are phrased in terms of sheaf theory. – Jason Starr Oct 02 '11 at 12:09
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    Very nice question. I'm indeed facing a very similar dilemma, my answer is that i will try to learn about schemes because I'm just naturally trying to view things "in a combinatorial way", but I feel the need of getting acquainted with different points of view myself. And your old point of view can help you to have insights that people usually working in a certain topic may have missed. – Maurizio Monge Oct 02 '11 at 15:02
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    Is there some reason why this is not community wiki? – Igor Rivin Oct 02 '11 at 15:17
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    @ex falso quodlibet: Please, a few more words, not to say a readable exposition paper for non-number theorists... – darij grinberg Oct 02 '11 at 16:21
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    @Igor: I never make questions like that CW, because I believe that most answers to such a question do deserve the votes they get (the question itself, of course, doesn't; it is bound to be overrated). But if you want to make it CW, no problem. – darij grinberg Oct 02 '11 at 16:23
  • @Jason: Hmm, OK. I don't believe I am able to fish these results out of SGA 5 at the moment (French being not my only problem), but it looks like I'll have to look at that one day. – darij grinberg Oct 02 '11 at 16:25

4 Answers4

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A combinatorial motivation is the n! conjecture, whose proof by Haiman uses Hilbert schemes. An account of this work written by Haiman for the Current Developments in Mathematics conference in 2002 is at math.berkeley.edu/~mhaiman/ftp/cdm/cdm.pdf. Haiman emphasizes at the start of the paper that the main geometric results which had to be proved were motivated by combinatorial evidence. Around the time that Haiman first announced his results on the n! conjecture (before he moved to Berkeley) I had heard from other people that this conjecture motivated Haiman to learn modern algebraic geometry. Haiman's response to receiving the Moore prize in the AMS Notices April 2004, p. 432, more or less seems to confirm this, so it's analogous to the way that the Weil conjectures were a concrete open problem which motivated Grothendieck's work.

kram1032
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KConrad
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    Very nice (I was aware of the conjecture but not of how it is proven), thanks! – darij grinberg Oct 02 '11 at 01:47
  • It is of course true that this is a nice concrete application of modern algebraic geometry. But people interested in explicit formulas have the right to expect more than these techniques give: we don't want an abstract reason for positivity as much as a manifestly positive formula. – Stephen Oct 02 '11 at 14:17
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    SPG: By "positivity", which wasn't part of my answer, I assume you're referring to the Macdonald positivity conjecture, which was one of the corollaries of Haiman's proof of the n! conjecture. The positivity comes from an identification of the numbers in Macdonald's conjecture with character multiplicities, as character multiplicities are manifestly positive (well, manifestly nonnegative). See Conjecture 2.2.2 in Haiman's paper (Journal AMS 14 (2001), p. 945). I don't understand your complaint about not wanting an abstract reason for positivity. The positivity is not for an abstract reason. – KConrad Oct 02 '11 at 16:27
  • KConrad: One could (and some combinatorialists do) complain that a proof that a certain vector space has dimension d should actually exhibit a basis with d elements. That is still open for the $n!$ problem. Also, for Macdonald positivity, one would like a combinatorial set that actually $q,t$-counts the coefficients of the Schur expansion (and ideally with it a way of creating a basis from this combinatorial set, and more ideally this basis respects the decomposition into $S_n$-irreps), and that question is still open (at least if one wants a direct definition of the set). – Alexander Woo Oct 03 '11 at 08:04
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    Alexander: If the desire for an alternate proof fitting a certain style successfully inspires someone, that's fair. Lots of second (or third) proofs arise for that reason. At the same time, even some elementary theorems only have proofs with an indirect character. For example, the only reason we know that the unit group $({\mathbf Z}/p{\mathbf Z})^\times$ is cyclic for every prime $p$ is by a non-explicit existence proof. If we refused to consider this theorem until such a time as someone found an explicit recipe for a generator, it may never be available for all $p$. :) – KConrad Oct 03 '11 at 12:43
  • Alex, thanks for clarifying my assertion somewhat. KConrad, I don't see a binary choice here, and it's not a matter of taste: Haiman's theorem is a very good theorem. An even better theorem would give an effective procedure for calculating the expansion of Macdonald polynomials in terms of Schur functions, with coefficients that have (1) a concrete geometric and/or representation theoretic interpretation that (2) visibly deforms the usual tableaux combinatorics (which should be recovered by setting q=1=t). Unfortunately, this is not the type of result that is typically provided by AG... – Stephen Oct 04 '11 at 22:08
  • BTW, I think the "real" thm proved by Haiman in this connection is the identification of $S_n$-Hilb of $\mathbb{C}^{2n}$ with the Hilbert scheme of points in the plane. This is of course of interest for its own sake---it's just that I don't see that the combinatorial corollary is all that one might hope for. – Stephen Oct 04 '11 at 22:13
  • SPG: In Haiman's paper (Journal AMS 14 (2001), pp. 944-945), equation (7) expresses transformed (q,t)-Macdonald polynomials in terms of Schur functions with transformed (q,t)-Kostka coefficients. Equation (15) expresses the (q,t)-Kostka coefficients as sums of (q,t)-monomials whose numerical coefficients are character multiplicities (a concrete representation-theoretic interpretation). Admittedly -- and maybe this is the non-effective part which Alexander is referring to -- the $S_n$-modules $(D_\mu)_{r,s}$ in equation (14) which occur in that character multiplicity formula [contd...] – KConrad Oct 05 '11 at 00:40
  • are defined externally, so to speak, rather than internally by some explicit spanning set, but it nevertheless appears to me that substituting equation (15) into equation (7) provides a not-too-abstract representation-theoretic explanation for the non-negativity of the coefficients in Macdonald's conjecture. – KConrad Oct 05 '11 at 00:45
  • KConrad, I don't think we have a fundamental disagreement here. You are right, the Garsia-Haiman modules $D_\mu$ give a satisfying rep.thy. explanation for positivity, and it's amazing that 3 of the 4 known proofs (Haiman via geometry of Hilbert schemes, Grojnowski-Haiman, and now Gordon, via Hodge theory) use so much modern machinery. Producing an explicit bigraded basis might also be possible via AG or RT; at any rate this is an obvious challenge for people working in the area. Haiman himself regards this as important: his (former) student Sami Assaf has been working on a CO approach. – Stephen Oct 05 '11 at 13:30
  • Anyone who is puzzled by the above disagreement: Section 1.1 of Stanley's Enumerative Combinatorics textbook gives a very good explanation. – Alexander Woo Oct 06 '11 at 21:03
  • There was recently a purported 5th proof (or variant of the 1st) which seemed to have some hope of leading to an explicit basis, but I just looked on the ArXiv and there is a note that the authors have withdrawn it due to a serious error found by the referee. – Alexander Woo Oct 06 '11 at 21:07
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Positivity of Kazhdan--Lusztig polynomials (and all the other positivity results in Kazhdan--Lusztig theory in general).

Consider the Hecke algebra $H_n(q)$. It is a particular deformation of the group algebra of the symmetric group (or some other Coxeter group). As such, it has a basis $T_w$ indexed by permutations, and multiplication is given by $$T_wT_{s_i}=T_{ws_i}$$ if $\ell(ws_i)=\ell(w)+1$, and $$T_wT_{s_i}=qT_{ws_i}+(1-q)T_w$$ if $\ell(ws_i)=\ell(w)-1$.

Define an involution on $H_n(q)$ (usually called the bar involution) by $\overline{q}=q^{-1}$ and $\overline{T_w}=(T_{w^{-1}})^{-1}$. Kazhdan and Lusztig proved that there exists a unique basis $C^\prime_w$ such that

1) $\overline{C^\prime_w}=C^\prime_w$

2) If we write $C^\prime_w=\sum_x P_{x,w}(q)T_x$, then the degree of $P_{x,w}(q)$ is bounded above by $(\ell(w)-\ell(x)-1)/2$.

3) $P_{w,w}(q)=1$.

The polynomials $P_{x,w}(q)$ (and in particular their coefficient in the maximum degree they are allowed) turn out to give a very nice combinatorial way to construct representations of $S_n$ (or the Coxeter group in question), and a similar theory also constructs representations of finite groups of Lie type.

Now, the only way to prove that $P_{x,w}(q)$ have positive integer coefficients so far is to show that they are the Poincare polynomials for local intersection cohomology on Schubert varieties. Even better, one should interpret the Hecke algebra as a kind of Grothendieck group on the category of perverse sheaves on the flag variety. Springer in the early 1980s used this interpretation to show that, if one takes a product $C_vC_w$ and expands this product in the $C$ basis, the coefficients are all polynomials with positive integer coefficients. (The $C^\prime$ basis is a variant of the $C$ basis that is a little easier to write.)

(The best references I know are Humphrey's book on reflection groups and Coxeter groups and Bjorner and Brenti's book on Combinatorics of Coxeter groups, both of which have a chapter devoted to this subject.)

darij grinberg
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Alexander Woo
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  • Thanks a lot - posts like this are precisely the reason I am not making my questions community wiki. (Ironically, I had to edit it a bit - there was faulty LaTeX.) – darij grinberg Oct 02 '11 at 16:33
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If you are interested in actual computations using modern algebraic geometry, there are plenty to be had in Gromov-Witten theory and enumerative geometry. For example, Kontsevich's formula counting rational plane curves is a famous example. The proof itself does not use any scheme theory, but it was based on the structure of a very delicate object called moduli space of stable maps, which could not be constructed without using schemes. Basically, counting problems in enumerative geometry are usually transformed into intersection theory on moduli spaces of the objects being counted. We want the moduli space to be compact so we can use the "invariant of numbers" principle (example : two lines intersect at one point in projective plane, but not necessarily so in affine line). Compactifying the moduli space meaning you allow your objects to have limits, thus even if your objects of interested are smooth varieties, their limits may be deformed and have unreduced structures (read : schemes and sheaves). For example, a conic may be denegrate to a double line which only makes sense as a scheme. You can read a nice exposition of Kontsevich's formula in : http://arxiv.org/abs/alg-geom/9608011.

As another example, in http://arxiv.org/abs/alg-geom/9612004 , Getzler computed the intersection pairing matrix of $\mathcal M_{1,4}$ to obtain relation between cycles and then use it to compute, for example, the number of elliptic curves of degree $5$ passing through $20$ lines in $\mathbb P^3$ to be $2,583,319,387,968$, among other thing.

Tmonk
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The paper http://arxiv.org/abs/math/0309121 by Borisov and Sapir uses schemes to prove that mapping tori of free group endomorphisms are residually finite.

Benjamin Steinberg
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  • Just from glancing through the paper, it seems to me that it uses only affine schemes, which makes me wonder whether it couldn't as well be worded in terms of Hopf algebras. Have I missed some non-affine things? – darij grinberg Oct 02 '11 at 16:29
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    @Darij: Even if only affine schemes are used, it is worthwile to mention that there are group theoretic questions which can be attacked by thinking in terms of modern algebraic geometry. – Martin Brandenburg Oct 02 '11 at 20:43
  • Given that they are looking at quasi-fixed points of polynomial dynamical systems over finite fields, I don't think it is natural to change the language to commutative rings. I haven't read the paper in awhile, but I think schemes, maybe affine, over Z are used. This is in a sense modern algebraic geometry since classical algebraic Italian style algebraic geometry was over fields and considered only reduced affine schemes. – Benjamin Steinberg Oct 03 '11 at 01:59
  • They also mention how free groups can be replaced by f.g. linear groups by using a result of Hrushovski that does seem to use modern algebraic geometry. – Benjamin Steinberg Oct 03 '11 at 02:14
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    Our paper with Borisov is using commutative algebra and affine schemes. We tried to use algebraic geometry (and arbitrary schemes) but we did not succeed because there is no satisfactory intersection theory in positive characteristic. In the paper, we indicate precisely the intersection theory facts we needed. That and much more has been done by Hrushovski in his paper about Frobenius in characteristic $0$. The main results of Hrushovski are geometric but the methods are from logic and model theory. –  Oct 05 '11 at 01:10