
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:
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 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), but this did not exhaust the proper domain of enumerative questions.
Fudge factors and Hilbert's fifteenth problem.
Naïve application of dimension counting and Bézout'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.
As an 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, and five points determine a conic, if the points are in general linear position, as passing through a given point imposes a linear condition. Similarly, tangency to a given line "L" (tangency is intersection with multiplicity two) is one quadratic condition, so determined a quadric in "P"5. 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
called 'double lines'. This is because a double line intersects every line in the plane, since lines in the projective plane intersect, with multiplicity two because it is doubled, and thus satisfies the same intersection condition (intersection of multiplicity two) as a nondegenerate conic that is "tangent" to the line.
The general Bézout theorem says 5 general quadrics in 5-space 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 ".
Hilbert's fifteenth problem was to overcome the apparently arbitrary nature of these interventions; this aspect goes beyond the foundational question of the Schubert calculus itself.
Clemens conjecture.
In 1984 H. Clemens studied the counting of the number of rational curves on a quintic threefold formula_1 and reached the following conjecture.
This conjecture has been resolved in the case formula_6, but is still open for higher formula_3.
In 1991 the paper about mirror symmetry on the quintic threefold in formula_8 from the string theoretical viewpoint gives numbers of degree d rational curves on formula_5 for all formula_10. Prior to this, algebraic geometers could calculate these numbers only for formula_11.
Examples.
Some of the historically important examples of enumerations in algebraic geometry include: 