Algebraic geometry concerns itself with the zero sets of polynomials. For a polynomial p(z) = a₀ + a₁ z + a₂ z² + … + a_k z^k of one variable the matter is trivial: The fundamental theorem of algebra tells us that there are k roots (counting multiplicities). So the zero set is a finite set of complex numbers.

Polynomials of several variables are considerably more subtle and more interesting. If P = {p₁(z₁, …, z_n), … p_m(z₁, …, z_n)} are m polynomials in n variables then we may consider the set of common zeros of P. This will be a variety, which is a set that is a manifold (generically) of dimension n - m with a singular set of lower dimension.

A simple example of a variety is that determined by the two polynomials

z₃² + z₁³ = 0 and z₂ = z₃²

in C³. This will be a complex curve with a cusp at the origin.

Algebraic geometers are interested in classifying varieties, and in determining when two varieties--with apparently different equations--are the same.

An interesting and useful invariant that can be used to address some of these questions is the idea of a "rational curve" in the variety. Here a rational curve is a curve that can be parametrized using rational algebraic functions.

Example Consider the variety x² + y² = 1 in two-dimensional space. Many mathematicians would parametrize this curve (a circle, obviously) using the sine and cosine functions. But an algebraic geometer wishes to do things algebraically. We may do so by considering the slope parametrization of lines through the point (1,0) in the variety. See Figure 1.

Figure 1

Thus we have

y = t(x - 1) , (*)

where t is a parameter that denotes the slope of the line through (1,0). Taking into account the equation of the circle, we then write

x² + t² (x - 1)² = 1

or

(1 + t²) x² - 2 t² x = 1 - t² . (**)

The equation (**) has the obvious solution x ≡ 1 and the more interesting solution

x = [t² - 1]/[t² + 1] .

The equation for y then yields

y = -2t/[t² + 1] .

These last two equations taken together certainly give a rational parametrization of the unit circle.

A more complicated (or higher-dimensional) variety may have a rational curve as a proper subset. As an instance, consider the variety X given by

x³ + y³ + z³ = 1

in 3-dimensional space. Then the curve

x = t y = - t z = 1

lies in X. And it is certainly a proper subset of X, for the point (1,1,-1) of X does not lie on the rational curve.

As previously indicated, the presence or absence of rational curves in a variety is a useful device for classification of varieties. An important notion in these studies is that of "hyperbolicity". Hyperbolicity is a pervasive idea in geometry; in the present context it is related to whether the variety possesses finitely many rational curves.

Many varieties, like abelian varieties, have no rational curves. Abelian varieties and complex tori have no rational curves but are not hyperbolic. The presence of infinitely many rational curves is also a valuable tool for distinguishing varieties.

It is difficult to prove that any particular variety is hyperbolic. But if a variety is not hyperbolic then this is usually easy to verify. This AIM workshop focuses on varieties for which rational curves are ubiquitous.

Recall that in classical Euclidean geometry it is a useful paradigm that through any two points p, q in the plane there is a line. Just so, in algebraic geometry it is helpful to consider varieties with the property that through any two points there is a rational curve in the variety. Such a variety is called rationally connected. The consideration of rationally connected varieties leads to interesting invariants from other parts of mathematics: Betti numbers (from algebraic topology) and notions from the structure of a complex manifold.

It is conjectured that rationally connected varieties can be characterized in terms of which globally defined holomorphic differential forms live on the variety. It appears that the rationally connected ones have few such forms. Here differential forms are the technical device by which we do calculus on a surface. An example of this phenomenon is complex projective space (i.e., the space of all lines in complex Euclidean space). There are no differential forms on projective space (essentially by the classical maximum principle), and it is certainly possible in projective space to join any two points by a line. This simple example may be a model for the general phenomenon.

By contrast, if A is Cⁿ modulo a lattice, then A will have no rational curves. And such an A will have plenty of differentials.

The ideas discussed here have exciting connections with number theory. For instance, if X is a variety that is rationally connected over a finite field F_q then X(F_q) will not be empty. So there will always be solutions of these equations. This important result of Esnault is related to further ideas of Graber, Harris, and Starr. Specifically, let X be a rationally connected smooth projective variety defined over a field k. Esnault showed that if k = F_q (a finite field), then X(k) ≠ ∅ (that is, the variety has a rational point). Graber, Harris, and Starr showed that if k is the field of meromorphic functions on a Riemann surface then X(k) ≠ ∅. These two sets of results extend earlier work of Tsen, Lang, Chevalley, and Warning.

The theory of rational curves on algebraic varieties has proved to be a meeting point for techniques from many branches of mathematics. The workshop at AIM was a vehicle for bringing together workers with many different points of view, and generating new attacks on old problems.