Facts from algebraic geometry that are useful to non-algebraic geometers

Question

A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic information about Galois theory and algebraic geometry, which, according to him, are important even for non-algebraists.

What are some useful facts from algebraic geometry that are useful for non-algebraic geometers? Ideally, the statements at least should be accessible without knowing much algebraic geometry.

Edit: Please do not post results that are only relevant to people who already know massive amounts of algebraic geometry anyway. In particular: Be very cautious about posting statements whose only applications are in number theory.

Example: Here is a basic statement that I have seen applied outside algebraic geometry, if not necessarily outside of algebra:

Let $U \subset \mathbb{C}^n$. If there is some nonzero polynomial satisfied by every point of $\mathbb{C}^n \smallsetminus U$, then $U$ is dense in $\mathbb{C}^n$ (with the usual topology), and in fact contains a dense open subset of $\mathbb{C}^n$.

[Sketch of proof: Given any point $p \in \mathbb{C}^n$, find a complex line $L$ passing through $p$ that intersects $U$. Then $L \cap (\mathbb{C}^n \smallsetminus U)$ is algebraic, hence contains only finitely many points of $L$, and so $p$ is a limit point of $U$.]

Answer 1

I would vote for Chevalley's theorem as the most basic fact in algebraic geometry:

The image of a constructible map is constructible.

More down to earth, its most basic case (which, I think, already captures the essential content), is the following: the image of a polynomial map $\mathbb{C}^n \to \mathbb{C}^m$, $z_1, \dots, z_n \mapsto f_1(\underline{z}), \dots, f_m(\underline{z})$ can always be described by a set of polynomial equations $g_1= \dots = g_k = 0$, combined with a set of polynomial ''unequations'' (*) $h_1 \neq 0, \dots, h_l \neq 0$.

David's post is a special case (if $m > n$, then the image can't be dense, hence $k > 0$). Tarski-Seidenberg is basically a version of Chevalley's theorem in ''semialgebraic real geometry''. More generally, I would argue it is the reason why engineers buy Cox, Little, O'Shea ("Using algebraic geometry"): in the right coordinates, you can parametrize the possible configurations of a robotic arm by polynomials. Then Chevalley says the possible configuration can also be described by equations.

(*) Really it seems that "inequalities" would be the right word her...might be a little late to change terminology though...

Answer 2

If $p_1$, $p_2$, ..., $p_m$ are polynomials in $n$ variables, with $m>n$, then there is a polynomial $q$ such that $q(p_1, p_2, \ldots, p_m)$ is identically zero.

Answer 3

In the classical theory of semisimple Lie algebras over the complex numbers (and elsewhere in Lie theory), it's convenient to apply easy Zariski density arguments for some underlying affine spaces. For instance, a natural proof of Harish-Chandra's basic theorem on the structure and characters of the center of the universal enveloping algebra involves restriction of polynomial functions from the Lie algebra to a Cartan subalgebra. Here the density of "regular" elements makes it possible to focus just on their behavior. Similarly, some classical conjugacy theorems for the Lie algebras relative to the adjoint group action are most easily studied in geometric terms. The point is that polynomials play a prominent role, making even the most elementary parts of algebraic geometry helpful.

Answer 4

Any compact Riemann surface is projective and algebraic.

Riemann surfaces are studied in analysis an differential geometry, and of course is easier to work with polynomial equations. This statement is useful also for studying non compact Riemann surfaces.

Answer 5

The Tarski-Seidenberg theorem states that if you project a semialgebraic set (i.e. given by polynomial in/equalities) from $\mathbb{R}^{n+1}$ to $\mathbb{R}^{n}$ you obtain another semialgebraic set. This was used by Lars Hormander (maybe even earlier by Lars Garding) in a few spectacular applications to characterize the solvability and regularity properties of constant coefficient PDEs. I understand that the TS theorem has applications to logic, model theory, functional analysis, and other fields, but I have no direct knowledge of them so maybe some expert might be willing to comment.

Answer 6

I would mention Bézout's theorem. Forgetting the complicated general definition of the intersection index one of the consequences is: whenever two curves in $\mathbf{P}^2(K), K$ an algebraically closed field have no common component, the number of the intersection points is finite and is always at most the product of the degrees. Moreover, it is the product of the degrees provided no intersection point is a singular point and all intersections are transversal. To state this over $\mathbf{C}$ we essentially only need multivariable calculus. But a proof requires a bit of algebraic geometry.

Let me also mention two algebraic geometry books that present precisely the kind of material people in other areas are likely to use. One is "Algebraic geometry, a first course" by Joe Harris; the other is "Undergraduate algebraic geometry" by Miles Reid. If memory serves, it says somewhere in the latter book that it covers (together with Atiyah-MacDonald) all algebraic geometry questions that the author was ever asked by his colleagues who specialize in other areas.

Answer 7

Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated with some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${\mathcal C}^\infty$ is the so-called Lopatinskii Condition, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${\mathcal C}^\infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied uniformly. In several interesting cases, LC or ULC condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called dissipative symmetrizer, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULC) and the books by R. Sakamoto (LC) or by S. Benzoni-Gavage and myself (ULC).

Answer 8

Grobner basis calculation is very practical for engineering, but also theoretically, in combinatorics (e.g. to prove colorability of given graph classes) and theoretical computer science (e.g. for polynomial interpretations to prove termination of programs).

Answer 9

The basic theory of curves is essential to modern communication theory, particularly in the construction of error-correcting codes and elliptic-curve cryptosystems.

Answer 10

Here is an application of the Hironaka's resolution of singularities theorem to functional analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial $f$ on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$.

(Example: on $\mathbb{R}$ the meromorphic continuation of the function $|x|^\lambda$ to $\lambda=-1$ has a pole, and to $\lambda=-2$ equals to $(ln|x|)''$, where the second derivative is understood in the sense of generalized functions.)

To the best of my knowledge, the first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities of algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf It has far reaching extensions. The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

Bernstein has constructed a module over the ring of differential operators (the module has a formal generator $|f|^\lambda$) for which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.

Answer 11

Hyperelliptic curves play a basic role in the construction of solutions to completely integrable systems (soliton equations), e.g. KdV. But it should be noted that these solutions are of non-soliton type.

The basic idea is, that such a system can be written as a Lax pair: $$ \dot{L} = [P_j, L] $$ for some $j$. Different $j$ correspond to different members of the hierachy. Now to construct such an algebro-geometric solution. Consider some $\ell > j$, and look for an $L$ such that $$ [L, P_{\ell}] = 0 $$ then some general theory implies that this solution satisfies a polynomial equation, and can thus be written in terms of data on a curve.

Answer 12

Since no one has done it so far let me also mention the Riemann-Roch theorem for complex analytic or algebraic curves. (True, if one is mainly interested in the analytic case, then the algebraic version will not be of much use since to apply it one needs to show first that any Riemann surface is algebraic, which is most easily done by taking a projective embedding, which requires in turn the analytic Riemann-Roch theorem.) One needs only 1-variable complex analysis to define smooth compact complex curves and rational functions and divisors on them. Now if the degree of a divisor $D$ is $>2g-2$ where $g$ is the genus, then $\dim\mathcal{L}(D)=d-g+1$ where $d=deg(D)$. One does not even need the canonical divisor to state this.

As a consequence one gets many results on meromorphic functions on Riemann surfaces. For instance

1. there exist nonconstant meromorphic functions.

2. the Mittag-Leffler problem (find a meromorphic differential form with given poles and given principal parts at the poles) has a solution iff the sum of the residues is 0. There is a similar statement for meromorphic functions that can be proved in a similar way.

3. any algebraic curve (or Riemann surface) is projective and moreover can be embedded in $\mathbf{P}^3$.

Answer 13

This falls outside the scope of "basic facts", but it seems interesting enough to mention here: apparently the seminar in our AG group in Hannover this week will be about applying resolution of singularities (more specifically, the concept of log canonical thresholds) to a problem of Bayesian statistics. Maybe this is a standard technique, but it certainly surprised me.

Abstract: http://www.iag.uni-hannover.de/de/oberseminar/abstracts/abstract.php?in=lin_de.html

Answer 14

This is another answer, in a different direction (about which I personally know little and can't offer any value judgments). While applied mathematicians are unlikely to be interested in the more abstract parts of modern algebraic geometry, some fairly sophisticated ideas have found their way into the literature of "systems theory". Chris Byrnes, one of our enterprising UMass Ph.D. students in the 1970s, went in that direction after learning from John Fogarty and others about moduli spaces. Chris spent time later with Roger Brockett's group at Harvard, then had an active career in university teaching and administration. One of his early papers gives an indication of how geometric ideas interacted with more applied problems:

Christopher I. Byrnes, On the control of certain deterministic, infinite-dimensional systems by algebro-geometric techniques. Amer. J. Math. 100 (1978), no. 6, 1333–1381.

Answer 15

In computer-aided geometric design, we use algebraic geometry a little. The curves and surfaces that are used in CAD (and therefore in engineering and manufacturing) are usually described by low degree polynomial or rational functions. We "implicitize", by which we mean constructing implicit equations from parametric ones. This makes certain computations easier, sometimes. The theory of resultants and elimination theory helps us with implicitization. We compute intersections, and Bezout's theorem tells us how many intersections to expect. Groebner bases are useful, occasionally.

I'm not sure we ever use any algebraic geometry that was developed in the last 100 years, but the old-fashioned stuff is useful. Our favorite text-book is George Salmon's "Lessons introductory to the modern higher algebra" from 1885.

There is a sample here. All baby stuff, by modern standards, I suspect.

Answer 16

Not a fact but a philosophy: to me the most important way of thinking in algebraic geometry that would be useful in many areas is that of a moduli space. I.e. the idea that the set of isomorphism classes of certain objects should be viewed with structure of that same type, and its properties studied as a tool for understanding the original types of objects. This I believe is basic to the work of Chris Byrnes alluded to above. This philosophy is perhaps not due to or original with algebraic geometry, but is practiced systematically there. It may derive from algebraic topology, (classification of vector bundles, E.H. Brown's representability of cohomology,....), like many other things in AG.

It might be of interest e.g. to some high school students to know that Euclid proved the set of congruence classes of circles is an open half line, and that the set of all triangles modulo similarity is parametrized by the interior of an isosceles right triangle, modulo the reflection in the altitude on the hypotenuse (via the unordered coordinates AA given by the two largest angles), hence also the interior of an isosceles right triangle, but with the interior of one edge added in. Then the set of congruence classes of triangles can be seen as the product of this triangular region with an infinite open half line, i.e. an infinite parallelpiped, (via the ASA theorem). They might then compare this with the realization of this same space by the coordinates SSS.

Answer 17

Bounds for minimal sized frames which are injective for phase retrieval (a problem in signal processing) have been obtained using the fact that the variety of $m \times n$ matrices of rank at most $k$ is irreducible of codimension $(m - k)(n - k)$ in the projective space of nonzero $m \times n$ matrices (specifically in the case $n=m$ and $k=2$). See the paper of Balan, Casazza and Edidin or Conca, Edidin, Hering, and Vinzant for details.