What motivates modern algebraic geometry for a combinatorial/constructive algebraist?

Question

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.

Answer 1

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.

Answer 2

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.)

Answer 3

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.

Answer 4

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.