Enumerative geometry

In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.

History

The problem of Apollonius is one of the earliest examples of enumerative geometry. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. In general, the problem for three given circles has eight solutions, which can be seen as 2³, each tangency condition imposing a quadratic condition on the space of circles. However, for special arrangements of the given circles, the number of solutions may also be any integer from 0 (no solutions) to six; there is no arrangement for which there are seven solutions to Apollonius' problem.

Key tools

A number of tools, ranging from the elementary to the more advanced, include:

• Dimension counting
• Bézout's theorem
• Schubert calculus, and more generally characteristic classes in cohomology
• The connection of counting intersections with cohomology is Poincaré duality
• The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology.

Enumerative geometry is very closely tied to intersection theory.

Schubert calculus

Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of Hermann Schubert. He introduced for the purpose the Schubert calculus, which has proved of fundamental geometrical and topological value in broader areas. The specific needs of enumerative geometry were not addressed, in the general assumption that algebraic geometry had been fully axiomatised, until some further attention was paid to them in the 1960s and 1970s (as pointed out for example by Steven Kleiman). Intersection numbers had been rigorously defined (by André Weil as part of his foundational programme 1942–6, and again subsequently). This did not exhaust the proper domain of enumerative questions.

Fudge factors and Hilbert's fifteenth problem

Naïve application of dimension counting and Bezout’s theorem yields incorrect results, as the following example shows. In response to these problems, algebraic geometers introduced vague “fudge factors”, which were only rigorously justified decades later.

William Fulton gives the following example: count the conic sections tangent to five given lines in the projective plane. The conics constitute a projective space of dimension 5, taking their six coefficients as homogeneous coordinates. Tangency to a given line L is one condition, so determined a quadric in P⁵. However the linear system of divisors consisting of all such quadrics is not without a base locus. In fact each such quadric contains the Veronese surface, which parametrizes the conics

(aX + bY + cZ)² = 0

called 'double lines'. The general Bézout theorem says 5 quadrics will intersect in 32 = 2⁵ points. But the relevant quadrics here are not in general position. From 32, 31 must be subtracted and attributed to the Veronese, to leave the correct answer (from the point of view of geometry), namely 1. This process of attributing intersections to 'degenerate' cases is a typical geometric introduction of a 'fudge factor'.

It was a Hilbert problem (the fifteenth, in a more stringent reading) to overcome the apparently arbitrary nature of these interventions; this aspect goes beyond the foundational question of the Schubert calculus itself.

References

• H. Schubert, Kalkul der abzählenden Geometrie (1879) reprinted 1979.
• William Fulton, Intersection Theory (1984), Chapter 10.4