I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of combinatorics and algebraic geometry, and gave some examples like the combinatorial Nullstellensatz, the affirmative answer to the conjecture of Read and Rota-Heron-Welsh and the graph-theoretic analogue of the Riemann-Roch theorem, but then, it seems that all these interactions are one way. I would like to know of an important algebraic geometry theorem proved using combinatorics, or at least used combinatorics in a critical part.
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4$\begingroup$ Does Schubert Calculus qualify? $\endgroup$– Alex R.Commented Mar 26, 2015 at 21:59
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5$\begingroup$ Toric varieties and polyhedra? $\endgroup$– Justin HilburnCommented Mar 26, 2015 at 22:19
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3$\begingroup$ Somewhat along the lines of Turbo's comment (in fact, all the comments before this), it's a little hard to know what combinatorics is exactly, or how one should define its scope. For example, matroids from one point of view is combinatorics. From another, it's more or less what model theorists call a "geometry" or "pre-geometry". $\endgroup$– Todd TrimbleCommented Mar 27, 2015 at 1:02
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1$\begingroup$ arxiv.org/abs/1001.2774 $\endgroup$– Sam HopkinsCommented Mar 27, 2015 at 1:17
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4$\begingroup$ So more people feel curious about @SamHopkins's link: it points to a paper called A tropical proof of the Brill-Noether Theorem by Filip Cools, Jan Draisma, Sam Payne, Elina Robeva. $\endgroup$– Omar Antolín-CamarenaCommented Mar 27, 2015 at 2:41
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8 Answers
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Jan Draisma's chapter "Noetherianity up to symmetry" in the book Combinatorial Algebraic Geometry (Springer LNM 2108) presents various finiteness theorems that are based on Kruskal's tree theorem (or actually the special case known as Higman's lemma).
The rest of the book also contains some potential examples, although there's some risk of getting tangled up in debates about where the combinatorics ends and the algebra or geometry begins.
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Haiman's study of the isospectral Hilbert scheme (the reduced fiber product of $\mathbb{C}^{2n}$ and $\mathrm{Hilb}_n(\mathbb{C}^2)$ over $\mathrm{Sym}^n \mathbb{C}^2$) features a lengthy combinatorial argument about the combinatorics of hyperplane arrangements and their coordinate rings (Section 4, on polygraphs).
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Many things in algebraic geometry can be proved using a degeneration to combinatorial objects like hyperplane arrangements, monomial ideals or toric varieties.
For instance, de Fernex-Ein-Mustata proved an inequality involving certain invariants of a singularity (e.g., the Samuel multiplicity and log canonical threshold) by degenerating to a monomial ideal. For monomial ideals the inequality is a simple consequence of the arithmetic means geometric means inequality!
Another example: A generic smooth hypersurface has no automorphisms, as can be shown by degenerating to a union of hyperplanes (which is rigid for high degree!).
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$\begingroup$ (A generic smooth hypersurface, I suppose you mean: there are certainly hypersurfaces of each degree and dimension with nontrivial automorphism groups.) $\endgroup$ Commented Mar 28, 2015 at 2:40
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$\begingroup$ There's some work by MacPherson and his collaborators that may fall into this category. For example, the equivariant cohomology of certain varieties can be computed from a purely combinatorial object called the moment graph. But I'm not sure if @criel would accept examples where something an algebraic geometer wants to compute is reduced to a combinatorial calculation (as opposed to directly quoting a standard theorem from combinatorics). If not then there may not be too many examples since combinatorics is stronger on methods and ad hoc reasoning than on ready-made theorems. $\endgroup$ Commented Dec 2, 2015 at 2:32
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The equivalence of several definitions for some Donaldson-Thomas invariants was first established combinatorially via equality of certain classes of plane partitions. Similar techniques have been used to prove further results in this direction. See for example this paper by Benjamin Young, and discussion thereof in Chapter 7 of this book.
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Does the theory of codes (e.g. "Hamming" codes) count? If so, then Beauville used a lemma about them crucially in his paper establishing the maximum number of ordinary double points possible on a quintic surface in P^3 with only such singularities.
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I won't fight hard about "important", but here's a theorem that was definitely combinatorial before it was geometric.
Consider the basis of $K(G/P)$ consisting of $K$-classes of structure sheaves of Schubert varieties, $\{[\mathcal O_{X_\lambda}]\}$. Brion proved that the coefficients $c_{\lambda\mu}^\nu$ in the product structure are appropriately positive, or more precisely, nonnegative times $(-1)^{|\lambda|+|\mu|-|\nu|}$. The proof is by a vanishing theorem in sheaf cohomology and does not compute the coefficients.
However, shortly before that, Buch gave an actual formula for these coefficients (though only in the case $G/P$ is a Grassmannian), with the minor corollary being that they had this predictable sign. The proof is pretty much completely combinatorial and doesn't explain "why" the sign should be predictable in (to me) as satisfying a way as Brion's does. But it's much more precise, and was earlier.
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A milestone result in the moduli theory of stable surfaces by Alexeev on boundedness is often cited as a result using "big" combinatorics. I'd guess the interesting part about this is that this application is less obvious than something in toric geometry or Schubert calculus.
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Disclaimer: I am quite ignorant of algebraic geometry. I am also most opinionated about subjects that I do not know. Caveat lector!
Ehrhart theory is a fundamental tool in toric geometry. In my opinion Ehrhart polynomials and Ehrhart generating functions belong to the land of combinatorics. When Eugene Ehrhart invented these beautiful things, he was doing combinatorics.